Problem solving and data analysis

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Center, spread and shape of distributions

Center, spread and shape of distributions on the SAT test

SAT Subscore: Problem solving and data analysis

Center, spread and shape of distributions are statistical measures that describe data sets, they are called summary statistics.

A center of a data set is a way of describing a location. We can measure a center of a data in 3 different ways: the mean (average), the median and the mode.

A spread of a data set describes how similar or varied the set of the observed values. We can measure a center of a data in 2 different ways: a range and a standard deviation.

Center measures

The mean is the average value of a given data set.
Mean = average = sum of the values / number of the values

The median is the middle number in a sorted in ascending order data set (the median is the value that splits the data set into two halves). To calculate the median,  arrange the values in an ascending order, count them and calculate the median. If the number of values is odd, the median is the middle value. If the number of values is odd, the median is the average of the two middle values.

The mode of a data set is the number that occurs most frequently in the set. To determine the mode, order the numbers from least to greatest, count how many times each number occurs and determine the mode. If no value appears more than once in the data set, the data set has no mode. If a there are two values that appear in the data set an equal number of times, they both will be modes etc.

Spread measures

The range measures the spread of a data inside the limits of a data set, it is calculated as a difference between the highest and lowest values in the data set. The larger the range, the greater the spread of the data.
range= the highest value – the lowest value.

The standard deviation is the measure of the overall spread (variability) of a data set values from the mean. The spread is measured as the distances (absolute values) from the mean of each value of the data sat. The more spread out a data set is, the greater are the distances from the mean and the standard deviation.

Outliers

An outlier is a value that is very different from the other values, so that it lies an abnormal distance from other values and is far from the middle of the data set.

  • Mean: Removing a big outlier will reduce the mean value and removing a small outlier will enlarge the mean value. 
  • Median: Removing a small outlier will enlarge the median; removing a large outlier will reduce the median (the median will be the same if the values in the positions after the removal are equal to the values in the positions before the removal).
  • Range: The value that will replace the outlier will be less distant therefore the range after removing the outlier will be smaller.
  • Standard deviation: Since the outlier is a value that is far from other values and the mean, its removal will reduce the spread of the data and the standard deviation.  

Continue reading this page for detailed explanations and examples.

Measuring a center of a data set

A center of a data set is a way of describing a location. We can measure a center of a data in 3 different ways: the mean (average), the median and the mode.

We can measure a center of a data in 3 different ways: the mean (average), the median and the mode.

Mean calculation

Mean is an average value of a given data set. To calculate the mean, we need to add the total values given in a data set and then divide the sum by the total number of the values.

The mean formula is:
Mean = average = sum of the values / number of the values

Note that if a single value appears in a data set number of times, we need to include it number of times when calculating the sum of the values.

Consider the following example:

What is the mean (average) of the following numbers 10, 12, 16, 5 and 2?

mean = sum of the values / number of the values

mean=(10+12+16+5+2)/5=45/5=9.

Consider the following example:

There are 3 children in 3 families and 2 children in 2 families.

What is the mean (average) of the children in a family?

We need to translate the word problem into numerical values. Since the values 2 and 3 appears in a data set number of times, we need to include them number of times when calculating the sum of the values.

The values of the number of children in the families are 3, 3, 3, 2 and 2.

mean = sum of the values / number of the values

mean=(3+3+3+2+2)/5=13/5=2.6 children in a family.

Finding missing values given the mean

The mean formula is:
Mean = average = sum of the values / number of the values

If we are given the mean value, we can solve the equation of the mean formula for 1 value that is missing. This value can be one of the numbers in the data set.

For example:
In a data set {-21, 5, x, 10} the mean is 0.5.  The value of x is:
Mean = sum of the values / number of the values
(-21+5+x+10)/4=0.5
-21+5+x+10=4*0.5
-6+x=2
x=8

Consider the following example:

The average grade of 4 students is 75.

The teacher added another grade so that the average became 5 points higher.

What was the grade that the teacher added?

average = sum of the values / number of the values

(4*75+x)/5=75+5
300+x=80*5
300+x=400
x=400-300
x=85

Checking the answer: (4*75+100)/5=(300+100)/5=400/5=80.

Median calculation

The median is the middle number in a sorted in ascending or descending order data set. In other words, the median is the value that splits the data set into two halves.

Medial calculation steps:

Step 1: Arrange the values in an ascending (or a descending) order.

Step 2: Count how many values are in the data set.

Step 3: Calculate the median: If the number of values is odd, the median is the middle value. If the number of values is odd, the median is the average of the two middle values.

Note that if a single value appears in a data set number of times, we need to include it number of times when arranging the values in an ascending (or a descending) order.

Consider the following example:

What is the median of the following numbers 10, 12, 16, 5 and 2?

Step 1: Arranging the values in an ascending order: 2, 5, 10, 12 and 16.

Step 2: Counting how many values are in the data set: there are 5 values in the data set.

Step 3: Calculating the median: the number of values is odd, therefore the median is the middle value which is 10.

2, 5, 10, 12, 16

Consider the following example:

There are 3 children in 3 families, 1 child in 3 families and 2 children in 2 families.

What is the median of the children in a family?

We need to translate the word problem into numerical values. Since the values 1, 2 and 3 appears in a data set number of times, we need to include them number of times when arranging the values in an ascending order.

The values of the number of children in the families are 3, 3, 3, 1, 1, 1, 2 and 2.

Step 1: Arranging the values in an ascending order: 1, 1, 1, 2, 2, 3, 3 and 3.

Step 2: Counting how many values are in the data set: there are 8 values in the data set.

Step 3: Calculating the median: the number of values is even, therefore the median is the average of the two middle values in the locations 4 and 5 which is (2+2)/2=2.

1, 1, 1, 2, 2, 3, 3, 3

Calculating a median of a frequency graph

Arrange the data given in the graph in a table and follow the previous steps.

Consider the following example:

The table below shows exam grades of a group of students.

What is the median grade?

a median value of a frequency graph

Arranging the data from the graph in a table:
65-70      3
70-75      1
75-80      2
80-85      0
85-90      3
90-95      1

Arranging the values in an ascending order: 65-70, 65-70, 65-70, 70-75, 75-80, 75-80, 85-90, 85-90, 85-90, 90-95.

The number of the values is 3+1+2+3+1=10. The number of the values is even, therefore the median is the average of the groups in the positions 5 and 6. This is the group 75-80.

65-70, 65-70, 65-70, 70-75, 75-80, 75-80, 85-90, 85-90, 85-90, 90-95

Mode calculation

The mode of a data set is the number that occurs most frequently in the set.

Calculating the mode steps:

Step 1: Order the numbers from least to greatest.

Step 2: Count how many times each number occurs.

Step 3: Determine the mode- a data set can have more than one mode or no mode:

If no value appears more than once in the data set, the data set has no mode.

If a there are two values that appear in the data set an equal number of times, they both will be modes etc.

Consider the following example:

What is the mode of the following numbers 10, 12, 16, 5 and 2?

Step 1: Ordering the numbers from least to greatest- 2, 5, 10, 12 and 16.

Step 2: Counting how many times each number occurs- each number occurs one time.

Step 3: Determining the mode- there is no mode in the data set.

Consider the following example:

There are 3 children in 3 families, 1 child in 3 families and 2 children in 2 families. What is the mode of the children in a family?

This word problem already describes the values in groups, therefore we don’t need to arrange and count the values. Since 3 is the largest number of families, there are two modes which are 1 and 3 children.

1, 1, 1, 2, 2, 3, 3, 3  

Measuring a spread of a data set

A spread of a data set describes how similar or varied the set of the observed values.

We can measure a center of a data in 2 different ways: a range and a standard deviation.

Range calculation

A range measures the spread of a data inside the limits of a data set, it is calculated as a difference between the highest and lowest values in the data set. The larger the range, the greater the spread of the data.

A range formula is: range= the highest value – the lowest value.

Consider the following example:

What is the range of the following numbers 10, 12, 16, 5 and 2?

range= the highest value – the lowest value

range=16-2=14.

Consider the following example:

There are 3 children in 3 families, 1 child in 3 families and 2 children in 2 families. What is the range of the number of children in a family?

range= the highest value – the lowest value

range=3-1=2.

Standard deviation calculation

Standard deviation is the measure of the overall spread (variability) of a data set values from the mean. The spread is measured as the distances (absolute values) from the mean of each value of the data sat. The more spread out a data set is, the greater are the distances from the mean and the standard deviation.

Standard deviation calculation is not covered in the SAT, but you need to know to determine which data group has a greatest standard deviation.

Comparing standard deviations of number of data sets steps:

Step 1: Calculate the mean of each data set.

Step 2: Calculate the distance of each value from the mean. Note that the distance should be calculated as an absolute value.

Step 3: Summarize the distances (absolute values from step 2) of each dataset. The data set that has the smaller sum has the smallest standard deviation.

Consider the following example:

Which of the two following groups of numbers has a higher standard deviation?

Group A: 12, 16 and 20.

Group A: 7, 10 and 16.

Step 1: Calculating the mean of each group:
Group A: mean = average = sum of the values / number of the values=(12+16+20)/3=48/3=16.
Group B: mean = average = sum of the values / number of the values =(7+10+16)/3=33/3=11.

Step 2: Calculating the distance of each value from the mean:
Group A: 12-16=-4, 16-16=0, 20-16=4.
Group B: 7-11=-4, 10-11=-1, 16-11=5.

Step 3: Summarizing the distances (absolute values from step 2) of each group:
Group A: 4+0+4=8.
Group B: 4+1+5=10.

Group B has a bigger standard deviation than group A.

Outliers and their effect on summary statistics measures

An outlier is a value that is very different from the other values, so that it lies an abnormal distance from other values and is far from the middle of the data set.

For example:
In a data set of 5, 8, 10 and 30 the outlier is 30.
In a data set -21, 5, 8, 10 the outlier is -21.

The effect of an outlier on a mean

The mean formula is: mean = average = sum of the values / number of the values

Removing a value from a data set reduces the number of values (denominator) by 1 and changes the sum of the values (numerator). If we remove a value that is smaller than the mean, the new mean will be bigger; If we remove a value that is bigger than the mean, the new mean will be smaller; if we remove a value that is equal to the mean, the new mean will be without change. We know that the outlier is significantly smaller or bigger than the mean, therefore removing a big outlier will reduce the mean value and removing a small outlier will enlarge the mean value.

Consider the following example:

Calculate the mean of the following data sets with and without the outlier:

{5, 7, 8, 40}

{-21, 5, 8, 10}

In a data set of {5, 7, 8, 40} the mean is (5+7+8+40)/4=60/4=15.
The data set without the outlier of 40 is {5, 7, 8}, its mean is (5+7+8)/3=20/3=6.333.
The mean reduced from 15 to 6.33, since the outlier 40 is larger than the mean 15.

In a data set {-21, 5, 8, 10} the mean is (-21+5+8+10)/4=2/4=0.5.
The data set without the outlier of -21 is {5, 8, 10}, the mean is (5+8+10)/4=23/4=7.67.
The mean increased from 0.5 to 7.67, since the outlier -21 is smaller than the mean 0.5.

The effect of an outlier on a median

The median is the middle number in a sorted in ascending order data set.

The median is calculated from the middle value/ values of the data set, therefore removing the outlier will change the positions of the values that are taken to calculate the median.

If we remove a small outlier (the first number), the positions of all values get smaller by 1 (value number 2 becomes number 1, value number 3 becomes number 2…). After removing a small outlier, we take values in bigger positions to calculate the median. Since the values are in ascending order, the numbers in bigger positions have bigger values, therefore the median will be bigger (the median will be same if the values in the positions after the removal are equal to the values in the positions before the removal).   

If we remove a large outlier (the last number), the positions of all values don’t change but the positions of the values that are taken to calculate the median get smaller by 1. Since the values are in ascending order, the numbers in smaller positions have smaller values, therefore the median will be smaller (the median will be same if the values in the positions after the removal are equal to the values in the positions before the removal).

Removing a small outlier will enlarge the median; removing a large outlier will reduce the median (the median will be the same if the values in the positions after the removal are equal to the values in the positions before the removal).

Consider the following example:

Calculate the median of the following data sets with and without the outlier:

{5, 7, 8, 40}

{-21, 5, 8, 10}

In a data set of {5, 7, 8, 40} the median is (7+8)/2=7.5.
The data set without the outlier of 40 is {5, 7, 8}, its median is 7.
The median decreased from 7.5 to 7 after removing a big outlier of 40.

In a data set {-21, 5, 8, 10} the median is (5+8)/2=13/2=6.5.
The data set without the outlier of -21 is {5, 8, 10}, its median is 8.
The median increased from 6.5 to 8 after removing a small outlier of -21.

Note that if the numbers in the new locations (after removing the median) are equal to the numbers in the previous locations (before removing the median), the median will be the same.

The effect of an outlier on the range

The range formula is: range= the highest value – the lowest value.

Since an outlier is a very small or a very big value, it is included in the range calculation, therefore removing the outlier will affect the value of the range. The value that will replace the outlier will be less distant therefore the range after removing the outlier will be smaller.

Consider the following example:

Calculate the range of the following data sets with and without the outlier:

{5, 7, 8, 40}

{-21, 5, 8, 10}

In a data set of {5, 7, 8, 40} the range is 40-5=35.
The data set without the outlier of 40 is {5, 7, 8}, its range is 8-5=3.
The range decreased from 35 to 3 after removing the outlier.

In a data set {-21, 5, 8, 10} the range is 10–21=10+21=31.
The data set without the outlier of -21 is {5, 8, 10}, its range is 10-5=5.
The range decreased from 31 to 5 after removing the outlier.

The effect of an outlier on a standard deviation

The standard deviation is the measure of the overall spread of a data set values from the mean. Since the outlier is a value that is far from other values and the mean, its removal will reduce the spread of the data and the standard deviation.  

Consider the following example:

Calculate the standard deviation of the following data sets with and without the outlier:

{5, 7, 8, 40}

{-21, 5, 8, 10}

In a data set of {5, 7, 8, 40} the mean is 15. The sum of the distances from the mean is |40-15|+|8-15|+|7-15|+|5-15|=25+7+8+10=50.

The data set without the outlier of 40 is {5, 7, 8}, the mean is 6.67. The sum of the distances from the mean is |8-6.67|+|7-6.67|+|5-6.67|=1.33+0.33+1.67=3.33.

The sum of the distances from the mean reduced from 50 to 3.33, therefore the standard deviation of the data set significantly reduced.

In a data set {-21, 5, 8, 10} the mean is 0.5. The sum of the distances from the mean is |-21-0.5|+|5-0.5|+|8-0.5|+|10-0.5|=21.5+4.5+7.5+9.5=43.

The data set without the outlier of -21 is {5, 8, 10}, the mean is 7.67. The sum of the distances from the mean is |5-7.67|+|8-7.67|+|10-7.67|=2.67+0.33+2.33=5.33.

The sum of the distances from the mean reduced from 43 to 5.33, therefore the standard deviation of the data set significantly reduced.

Data collection and data inference

Data collection and data inference on the SAT test

SAT Subscore: Problem solving and data analysis

Data collection

Data collection is a process of collecting and measuring information on variables of interest, that enables the researcher to test hypotheses and evaluate outcomes.

Data can be collected with a sample or with a controlled experiment:

A sample is a small group that is selected from a large population by using a pre- defined sampling method. The sample must be representative and random.

A controlled experiment is an experiment made on an experimental group, while one factor that is being tested is changed by the researchers and all other factors are held constant.
Each controlled experiment must have a control group. In the control group we don’t change the factor that is being tested in the experimental group. The participants of the control group must be randomly selected and must closely resemble the participants in the experimental group.

Data inference

Data inference is a generalization about a population that is based on statistics calculated from a small group (a sample) that is drawn from that population.

An estimate is a process of finding a value of a population that is close enough to the right value, by performing a sample on a part of that population.
A sample proportion is a variable that is calculated from the sample, that we assume reflects the whole population.
The estimate formula: estimate= sample proportion * population

A margin of error is the degree of error in results received from random sampling surveys, it exists since the sample does not exactly match the population.
The range formula: range= estimate ± margin of error

Continue reading this page for detailed explanations and examples.

Data collection

Data collection is a process of collecting and measuring information on variables of interest, that enables the researcher to test hypotheses and evaluate outcomes. Data can be collected with a sample or with a controlled experiment.

Data collection with a sample

A sample is a small group that is selected from a large population by using a pre- defined sampling method. Samples are used when the population size is too large for the test to include the whole population. The three most common types of sample surveys are e-mail surveys, telephone surveys, and interview surveys.

Sample characteristics:
The sample must be representative and random.
A representative sample is a sample that accurately reflects the examined characteristic of the whole population. A sample that includes members that don’t belong to the population is not representative.
A random sample is a sample that was chosen randomly (purely by chance) so that every member of the population has an equal chance to be selected. A sample that over presents or under presents the subgroup is not random.

Consider the following example:

The school principal wants to estimate how many parents in a group of 80 pupils want their children to participate in an activity. She was suggested five sampling methods, are the methods appropriate?

Method 1: Surveying the parents of randomly chosen 20 girls.
Method 2: Surveying 20 randomly chosen pupils.
Method 3: Surveying 20 randomly chosen parents of the students from student council.
Method 4: Sending the survey with SMS messages to randomly chosen 20 parents of students in the school.
Method 5: Surveying 3 randomly chosen parents of the students.

Method 1: Surveying the parents of a randomly chosen 20 girls. This surveying method is bad because it not chosen randomly. It over presents the sub- group of the girls and under presents the subgroup of the boys.

Method 2: Surveying 20 randomly chosen pupils. This surveying method is bad because it is not representative. It includes pupils that are not a part of the population, since the population is the parents.

Method 3: Surveying 20 randomly chosen parents of the students from student council. This surveying method is bad because it not chosen randomly. It over presents the sub- group of parents of the students from student council.

Method 4: Sending the survey with SMS messages to randomly chosen 20 parents of students in the school. This surveying method is bad because it not chosen randomly. It over presents the sub- group of parents of students that feel positive about the activity and therefore more likely to answer the SMS.

Method 5: Surveying 3 randomly chosen parents of the students. This surveying method is bad because it is not representative. It includes only 3 parents from a group of parents of 80 pupils, therefore it is too small.

 Data collection with a controlled experiment

We can conduct a controlled experiment and conclude about the population from the experiment outcomes.

A controlled experiment is an experiment made on an experimental group, while one factor that is being tested is changed by the researchers and all other factors are held constant (like they were before the experiment).

An independent variable is a variable that is being changed by the researchers in the experimental group.

A correlation means that there is a relationship between two variables (a positive correlation means the variables change at a same direction).

A causation means that a change in one variable is the cause of a change in another variable. In other words: one event is the result of the occurrence of the other event.

Note that a correlation is not a causation. If two variables correlate it does not mean that one causes the other, since the correlation may be caused by a third variable that affects both variables.

For example:
If we want to test the effect of a new medication, the experimental group should include 1,000 participants that will receive the medication for 1 month. The independent variable is taking the new medication for 1 month. All the other factors, like the use of other medications, must remail constant, because changing other factors may influence the outcome of the experiment and we might wrongly assume that the influence came from the factor being tested. (If a participant stops taking the medication for lowering the blood pressure his blood pressure will rise and the researchers can wrongly assume that the rise in the blood pressure was a result of a new medication).

Consider the following example:

Which experiment is appropriate to test the effect of a new quit smoking treatment on the population of smokers?

A. An experiment conducted on 500 smokers aged 20-40 that received the new treatment.

B. An experiment conducted on 500 smokers that received the new treatment and were put on a diet.

C. An experiment conducted on 500 smokers that received the new treatment.

D. An experiment conducted on 500 smokers that had a low blood pressure and received the new treatment.

The answer A in not correct, since the sample is not random, the sample over presents the subgroup of men aged 20-40.

The answer B in not correct, since there are 2 independent variables instead of one. In addition to receiving the new treatment, the smokers were also put on a diet.

The answer D in not correct, since the sample is not random, the sample over presents the subgroup of men that had a low blood pressure.

The answer C in correct, the sample is random and has one independent variable.

The control group in a controlled experiment

In a control group, we don’t change the factor that is being tested in the experimental group (it stays like it was before the experiment). Meaning that all the factors are identical between the two groups except for the factor being tested.

A control group properties:

  • The participants must be randomly selected to be in the control group.
  •  The participants must closely resemble the participants who are in the experimental group.

The purpose of the control group is to rule out alternative explanations of the experimental results.

For example:
What is the control group in the previous example?
Composition: The control group is composed of participants who do not receive the experimental medication. In addition, all the conditions must be unchanged (stay like they were before the experiment). The participants of the control group must resemble the participants of the experimental group in their age, medical conditions and other parameters.
Purpose: An unknown disease may occur in the experimental group and the control group. The existence of the control group will reveal the unknown disease as an unexpected factor, otherwise the researchers will wrongly assume that the unknown disease was caused by the new medicine.

Consider the following example:

Which control group is appropriate to test the effect of a new quit smoking treatment on the population of smokers?

A. A group of 500 smokers that received the new treatment.

B. A group of 500 male smokers.

C. A group of 500 smokers that were put on a diet.

D. A group of 500 smokers.

The answer A in not correct, since in the control group we don’t change the factor that is being tested, so that all the conditions must be unchanged (the smokers in the control group shouldn’t receive the treatment).

The answer B in not correct, since the participants of the control group don’t resemble the participants of the experimental group (there are only male participants in the control group).

The answer C in not correct, since in the control group we don’t change any factor, so that all the conditions must be unchanged (the smokers in the control group shouldn’t be put on a diet).

The answer D in correct, since all the factors are unchanged and the participants of the control group resemble the participants of the experimental group.

Data inference

Data inference is a generalization about a population that is based on statistics calculated from a small group (representative sample) that is drawn from that population. In other words: Instead of checking the whole population we check only a part of the population (representative sample) and assume that the conclusion that was derived from the representative sample is relevant to the whole population.

An estimate calculation

An estimate is a process of finding a value of a population that is close enough to the right value, by performing a sample on a part of that population.

A sample proportion is a variable that is calculated from the sample, that we assume reflects the whole population. A sample proportion can be written as fraction or as a percentage. For example: 10 percent of the sample have a positive opinion about the surveyed subject.

The estimate formula: If we found that a certain percentage from the sample (a sample proportion) represents the percentage in the whole population, we can calculate an estimate by multiplying that percentage by the total amount of items in the population.

estimate= sample proportion * population

Consider the following example:

The school principal wants to estimate the number of pupils that will participate in an activity. She makes a representative sample of 30 pupils and 10 of them answer that they will participate in the activity.

If there are 8 classes in the school with 20 pupils in each class, how many pupils are expected to participate in the activity?

The sample proportion is 10/30=33.33%=0.3333.

The population is 8*20=160 pupils.

estimate= sample proportion * population
estimate= 0.3333 * 160=53 pupils.

The answer is that approximately 53 pupils are expected to participate in the activity.

A range calculation

A margin of error is the degree of error in results received from random sampling surveys, it exists since the sample does not exactly match the population. The margin of error is commonly given as a percentage and is added to the estimate to increase the confidence in the estimate.

Note that:

  • Even after including a margin of error, there is no certainty that the estimation is correct.
  • A high margin of error indicates small confidence that the results represent the population.
  • The larger the sample, the smaller the margin of error (bigger sample size increases the certainty in the prediction).

The range formula: We add (or subtract) the margin of error to the estimate to display the size of the error getting an outcome of a range instead of a single estimate.

range= estimate ± margin of error

Consider the following example:

15 percent of the sample participants own a dog and the margin of error for the sample is 2 percent. If there are 3,000 residents in the town, how many of them are expected to own a dog?

sample proportion= 15%

population= 3,000

estimate= sample proportion * population
estimate= 15% * 3,000= 0.15*3,000=450

the margin of error= 2% from the estimate= 0.02*450=9

The number of dog owners is 450±9, between 441 and 459.

Table data

Table data on the SAT test

SAT Subscore: Problem solving and data analysis

A frequency table is a table that shows the number of times the items occur.

A two- way frequency table displays frequencies for two variables so that one variable is represented by rows and the other variable is represented by columns.

To calculate a ratio from a two- way frequency table we need to find the 2 relevant values in the table and divide them. If possible, simplify the result.

To calculate a percentage from a two- way frequency table we need to find the 2 relevant values in the table and plug them into the percentage formula.

To calculate a probability from a two- way frequency table we need to find the 2 relevant values in the table and divide them.

We can find missing values in tables:
Finding the total of a table.
Finding missing values inside of a table using data from other fields of the table
Finding missing values inside the table using a given ratio or probability value.

Continue reading this page for detailed explanations and examples.

Reading two- way frequency tables

Identify the name of the variable in the rows and the columns and the total values that are summed in the rows and the columns (if exist).

For example:
The table below shows the number of students that passed and failed the exam in two classes.

                   Class 1       Class 2      Total
Passed           20               15             35
Failed              10               15             25
Total                30               30             60

After reading the table we can see that:
There are 30 students in each class and 60 students in both classes.
The number of students that passed the exam in both classes is 35.
From the students that passed the exam there are 15 students from class 2.
There are 10 students that failed the exam and study in class 1.

Consider the following example:

The table below shows the number of students that passed and failed science and math exams in the class.

                              Passed science     Failed science      Total

Passed math                 10                           8                      18

Failed math                    7                            5                      12

Total                               17                          13                      30

How many students failed both exams?

How many students passed the math exam?

How many students passed only one exam (not both)?

How many students are in the class?

Calculating the number of students that failed both exams:
The number of students that failed both exams in written in the table and it is 10.

Calculating the number of students that passed the math exam:
The number of students that passed the math exam in written in the table and it is 18 (the total column). It is divided into 10 students that passed the science exam and 8 students that failed the science exam.

Calculating the number of students that passed only one exam (not both):
The number of students that passed only one exam is divided into students that passed the math exam and failed the science exam (8 students) and students that passed the science exam and failed the math exam (7 students). The number of students that passed only one exam is 8+7=15 students.

Note that we can calculate the number of students that passed only one exam by subtracting the number of students that failed both exams (5 students) and the number of students that passed both exams (10 students) from the total number of students (30 students) getting 30-10-5=15 students.

Calculating ratios, percentages and probabilities from two- way frequency tables

In these questions we need to find a data inside a table and then use it to calculate ratios, percentages and probabilities.

Before learning how to use tables you need to master the subjects of ratios, percentages and probabilities:

Ratios subject was explained on ratios, rates and proportions page.

Percentages subjects was explained on percentages page.

 

Calculating ratios from two- way frequency tables

A ratio is a comparison of two numbers, represented by a division of their amounts. The ratio between a and b can be represented as a fraction a/b. For example: The number of girls in the class is 22 and the number of boys in the class is 11. The ratio of boys to the total in the class is 11/33=1/3.

To calculate a ratio from a two- way frequency table we need to find the 2 relevant values in the table and divide them. If possible, simplify the result.

For example:

The table below shows the number of students that passed and failed the exam in two classes.

                  Class 1       Class 2      Total

Passed           20               15             35

Failed              10               15             25

Total                30               30             60

After reading the table we can see that:

The fraction of the students that failed the exam study in class 1 is:
We need to calculate the ratio of the students in class 1 that failed the exam from the total students that failed the exam.
The denominator is the number of students that failed the exam, we see in the table that 25 students failed the exam.
The numerator is the number of students that study in class 1 and failed the exam, we see in the table that 10 students study in class 1 and failed the exam.
The fraction is 10/25=2/5.

The ratio of students that study in class 1 from the total number of students is:
The denominator is the number of students in 2 classes, we see in the table that it is 60 students.
The numerator is the number of students that study in class 1, we see in the table that it is 30 students.
The ratio of the students that study in class 1 from the total numbers of students is 30/60=1/2.

The ratio of students that study in class 1 and passed the exam from the number of students that study in class 2 and passed the exam is:
The denominator is the number of students that study in class 2 and passed the exam, we see in the table that it is 15 students.
The numerator is the number of students that study in class 1 and passed the exam, we see in the table that it is 20 students.
The ratio is 20/15=4/3.

Consider the following example:

The table below shows the number of students that passed and failed the science and math exams in the class.

                              Passed science     Failed science      Total

Passed math                 10                           8                      18

Failed math                    7                            5                       12

Total                               17                         13                       30

What fraction of the students that failed science also failed math?

What fraction of the students that passed math passed science?

What is the ratio of students that passed both math and science from the total number of students?

Calculating what fraction of the students that failed science also failed math:
We need to calculate the ratio of the students that failed both science and math from the number of students that failed science.
The denominator is the number of students that failed science, we see in the table that it is 13 students.
The numerator is the number of students that failed both science and math, we see in the table that it is 5 students.
The fraction is 5/13.

 Calculating what fraction of the students that passed math passed science:
We need to calculate the ratio of the number of students that passed both math and science from the number of students that passed math.
The denominator is the number of students that passed math, we see in the table that it is 18 students.
The numerator is the number of students that passed both math and science, we see in the table that it is 10 students.
The fraction is 10/18=5/9.

Calculating the ratio of students that passed both math and science from the total number of students:
The numerator is the number of students that passed both math and science, we see in the table that it is 10 students.
The denominator is the total number of students, we see in the table that it is 30.
The ratio of students that passed both math and science from the total number of students is 10/30=1/3.

Calculating percentages from two- way frequency tables

A percentage is a number or a ratio expressed as a fraction of 100 and represents a part to whole relationship.

A percentage formula is:

percentage= 100 *  the value to be expressed
                                   _________________________
                                                   total

To calculate a percentage from a two- way frequency table we need to find the 2 relevant values in the table and plug them into the percentage formula.

For example:

The table below shows the number of students that passed and failed the exam in two classes.

                   Class 1       Class 2      Total
Passed           20               15             35
Failed             10               15             25
Total               30               30             60

After reading the table we can calculate that:

The percentage of students from class 2 that passed the exam is:
The total is the number of students that passed the exam, we see in the table that it is 35 students. The value to be expressed is the number of students that study in class 2 and passed the exam, we can see in the table that it is 15 students.

percentage = 100 *  the value to be expressed           =             15            =   43%
                                     _________________________                  ____________
                                                  total                                                  35

The percentage of students that failed the exam is:
The total is the number of students in both classes, we see in the table that it is 60 students. The value to be expressed is the number of students that failed the exam, we see in the table that it is 25 students.

percentage = 100 *  the value to be expressed           =             25             =   42%
                                   __________________________                  ____________
                                                       total                                             60

Consider the following example:

The table below shows the number of students that passed and failed the science and math exams in the class.

                              Passed science     Failed science      Total

Passed math                 10                           8                      18

Failed math                    7                            5                       12

Total                               17                          13                      30

What percent of students passed both exams?

What percent of students passed at least one exam?

Calculating what percent of students passed both exams:

The total is the number of students in the class, we see in the table that it is 30 students. The value to be expressed is the number of students that passed both exams, we see in the table that it is 10 students.

percentage = 100 *  the value to be expressed    =   10    =  33%                                                                                                               __________________________     ______
                                                   total                                30 

Calculating what percent of students passed at least one exam:

The total is the number of students in the class, we see in the table that it is 30 students. The value to be expressed is the number of students that passed at least one exam. The number of students that passed at least one exam is the students that passed only math (8 students) or the students that passed only science (7 students) or the students that passed both (10 students). The number of students that passed at least one exam is therefore 8+7+10=25 students.

percentage = 100 *  the value to be expressed   =   25  =   83%
                                    __________________________    ______
                                                    total                              30

Calculating probabilities from two- way frequency tables

The probability of an event is equal to the outcome of the event divided by the total outcomes.

To calculate a probability from a two- way frequency table we need to find the 2 relevant values in the table and divide them.

For example:

The table below shows the number of students that passed and failed the exam in two classes.

                  Class 1       Class 2      Total

Passed           20               15             35

Failed             10                15             25

Total               30                30             60

After reading the table we can calculate that if students are selected randomly:

The probability that a student failed the exam is:
The numerator is the outcome of the event, this is the number of students that failed the exam. We see in the table that 25 students failed the exam. The denominator is the total outcome, this is the total number of students. We see in the table that there are 60 students in two classes. The probability is equal to 25/60=5/12.

The probability that a student studies in class 2 and failed the exam is:
The numerator is the outcome of the event, this is the number of students that study in class 2 and failed the exam. We see in the table that 15 students study in class 2 and failed the exam. The denominator is the total outcome, this is the total number of students. We see in the table that there are 60 students in two classes. The probability is equal to 15/60=1/4.

Consider the following example:

The table below shows the number of students that passed and failed the science and math exams in the class.

                              Passed science     Failed science      Total

Passed math                 10                           8                      18

Failed math                    7                            5                      12

Total                               17                         13                      30

If students are selected randomly, what is the probability that a student passed both exams?

If students are selected randomly, what is the probability that a student failed only the science exam?

Calculating the probability that a student passed both exams:
The numerator is the outcome of the event, this is the number of students that passed both exams. We see in the table that 10 students passed both exams. The denominator is the total outcome, this is the total number of students. We see in the table that there are 30 students in the class. The probability is equal to 10/30=1/3.

Calculating the probability that a student failed only the science exam:
The numerator is the outcome of the event, this is the number of students that failed only the science exam. We see in the table that 10 students passed the math exam and failed the science exam. The denominator is the total outcome, this is the total number of students. We see in the table that there are 30 students in the class. The probability is equal to 10/30=1/3.

Finding missing values in a two- way frequency table

In these questions we are required to perform 3 tasks:
Finding the total of a table.
Finding missing values inside of a table using data from other fields of the table
Find missing values inside the table using a given ratio or probability value.

Calculating totals of a table

We can calculate the total of a row or a column of the table if we are given the values inside the row or inside the column.

The table below shows the number of students that passed and failed the science and math exams in the class.

                              Passed science     Failed science      

Passed math                 10                           8                      

Failed math                    7                             5                      

We can calculate the total of a row and a column of the table if we have the values inside the row or inside the column, getting the following table:

                               Passed science     Failed science      Total

Passed math                 10                           8                        18

Failed math                    7                            5                         12

Total                               17                         13                        30

Calculating values inside of a table using data from other fields of the table

Each row and column have inner numbers and their sum, therefore we can find a missing value if we are given all the other values in a row or a column.  

The table below shows the number of students that passed and failed the science and math exams in the class.

                          Passed science     Failed science      Total

Passed math                                                                         18

Failed math                    7                                                      12

Total                               17                            13                     30

After reading the table we can calculate that:
The number of students that failed both exams is 12-7=5.
The number of students that passed both exams is 17-7=10.
Now we can calculate the number of students that passed math and failed science: 18-10=8 or 13-5=8.

The full table is represented below:

                              Passed science     Failed science      Total

Passed math                  10                            8                     18

Failed math                      7                             5                     12

Total                                17                           13                    30

Calculating values inside of a table using data outside the table

We can calculates values inside the table if we are given additional data about their percentages or fractions.

Consider the following example:

The table below shows the number of students that passed and failed the exam in two classes.

                    Class 1       Class 2      Total

Passed                               15             

Failed                                 15             

Total                                                      60

What is the number of students that failed the exam and study in class 1 if 33.33 percent of the students passed the exam and study in class 1?

Calculating the number of students that passed the exam and study in class 1: We see in the table that there are 60 students in two classes. Since we know the total and we know the percentage, we can calculate the value to be expressed (the number of students that passed the exam and study in class 1) with the percentage formula.

percentage % = 100 *  the value to be expressed
                                         _________________________
                                                             total

33.33= 100 *  the number of students that passed the exam
                         _____________________________________________
                                                           60

The number of students that passed the exam and study in class 1= 33.33*60/100=20.

The table below shows the data including the number of students that passed the exam and study in class 1.

Class 1       Class 2      Total

Passed           20               15

Failed                                15

Total                                                    60

 

We can now calculate the total of students that passed the exam, this is 20+15=35 students.

The table below shows the data including the total of students that passed the exam.

                  Class 1       Class 2      Total

Passed           20               15             35

Failed                                 15

Total                                                     60

We can now calculate the number of students that failed the exam, it is 60-35=25 students.
We can calculate the number of students from class 1 that failed the exam, it is 25-15=10 students.

We can continue and calculate the other fields in the table:
We can calculate the number of students that study in class 1, it is 20+10=30 students.
We can calculate the number of students that study in class 2, it is 15+15=30 students.

The table below shows the data including all the calculations above:

                   Class 1       Class 2      Total

Passed           20               15             35

Failed             10                15             25

Total                30               30              60

Consider the following example:

The table below shows the number of students that passed and failed the exam in two classes.

                    Class 1       Class 2      Total

Passed                              15             

Failed          10                    

Total                                                    60

What is the number of students in class 2 that failed the exam if 1/2 of the students study in class 2?

We are given that the number of students that study in class 2 divided by the number of students in both classes is equal to 1/2. We see in the table that the total number of students in both classes is 60.

Therefore we can calculate the number of students that study in class 2:

the number of students that study in class 2             1
____________________________________________  =    _____
                                    60                                                    2

The number of students that study in class 2=60/2=30.

We can continue and calculate the other fields in the table:      
The number of students that study in class 2 and failed the exam is 30-15=15 students.
The number of the students in class 1 is 30-30=30 students.
The number of the students that passed the exam and study in class 1 is 30-10=20 students.
The number of students that passed the exam is 20+15=35 students.
The number of students that failed the exam is 10+15=25 students.

The table below shows the data including all the calculations above.

                 Class 1       Class 2       Total

Passed        20                 15              35

Failed           10                 15              25

Total             30                 30              60

Linear and exponential growth

Linear and exponential growth on the SAT test

SAT Subscore: Problem solving and data analysis

Before learning linear growth topic you should master the linear functions topic; before learning exponential growth topic you should master the graphing exponential functions topic.

Linear growth occurs then a variable is growing by the same amount in each unit of time. The linear growth formula is y=mx+b, where b is the initial value and m is the constant rate of change.

Exponential growth occurs when the variable is growing by the same relative​ amount in each unit of time. The exponential growth formula is y=a(b)x, where a is the initial value, b is the number that is multiplied and x is the number of the multiplications.

The difference between linear and exponential growth: Linear growth is always at the same rate, while the rate of the exponential growth increases over time.

Model a table with a linear equation if the value in the table changes by a constant amount.

Model a table with an exponential equation if the value in the table changes by a constant multiplication factor (the common factor).

Modeling a scenario with linear and exponential functions questions require determining the type of the connection (linear or exponential) and the direction of the connection (increasing or decreasing).

  • If the change is at a constant rate (by a constant), the relationship is linear.
  • If the change is by a percent from the initial value, the relationship is linear.
  • If the change is by a factor, the relationship is exponential.
  • If the change is by a percent (from the previous value), the relationship is exponential.

Continue reading this page for detailed explanations and examples.

Linear and exponential growth definitions

Linear growth (decrease) occurs then a variable is growing (decreasing) by the same amount in each unit of time, meaning that the variable gets larger (smaller) by a constant amount in each time unit. For example: the money in the saving account increases (decreases) by 1,000 dollars every month.

The linear growth formula is y=mx+b, where b is the initial value (the intercept) and m is the constant rate of change (the slope).

linear growth function: If the variable is growing, the function is increasing, meaning that the slope of the line (m) is positive and the line moves up from left to right. If the variable is declining, the function is decreasing, meaning that the slope of the line (m) is negative and the line moves down from left to right. If m=0 the variable stays constant and the line is flat.     

Exponential growth (decay) occurs when the variable is growing (decreasing) by the same relative​ amount in each unit of time, meaning that the growth (decay) rate gets faster in each time unit. For example: The money in the deposit accrues an interest of 5 percent per year and is deposited for 5 years.

The exponential growth formula is y=a(b)x, where a is the initial value (the intercept), b is the number that is multiplied (the common factor or the common ratio) and x is the number of the multiplications (in time questions x is the number of periods).

Exponential growth function: In exponential growth, the function goes up from left to right; in exponential decay, the function goes down from left to right. The slope of the function gets bigger (in absolute value) as we move from left to right (the graph gets steeper).

The difference between linear and exponential growth: Linear growth is always at the same rate, while the rate of the exponential growth increases over time. Meaning that in exponential growth as the variable gets larger, the rate of its growth also gets larger.

Modeling tables with linear and exponential functions

In these questions, we need to find the function that represents a given table.

Modeling a table with a linear function

Model a table with a linear equation if the value in the table changes by a constant amount.

The linear function formula is y=mx+b, where b is the initial value (the intercept) and m is the constant rate of change (the slope).

Modeling a table with a linear function steps:
Step 1: Determine if the function that represents the table is linear or exponential:
To find the correct function type check if the change in the y variable is linear or exponential. If the change in the y variable is linear then we need to find a linear function equation that represents it. Note that the change in the x variable must be linear.

Step 2: Find the function equation using the formula y=mx+b for the linear function:
To find the intercept b, look what is the value of y when x=0 in the table. If the y value for x=0 is not given, calculate it using the change in the y variable that you found in step 1.
To find the slope m, divide the change in variable y by the change in variable x.

Step 3: Check the equation from step 2 by plugging into it the x values from the table. You must get y values like in the table.

Step 4: If needed, draw the graph the function according to its equation.

Consider the following example:

Write an equation based of the following table and find its graph.

x    0    2    4     6     8

y    3    7   11   15   19

Step 1: Determining if the function that represents the table is linear or exponential:

We need to make sure that the change in the x variable is linear; the x variable values are 0, 2, 4, 8 and 8. Their changes are constant and are equal to 2 (2-0=2, 4-2=2…).

We need to check the change in the y variable; the y variable values are 3, 7, 11, 15 and 19. Their changes are constant and are equal to 4 (7-3=4, 11-7=4, 15-11=4, 19-15=4).

Step 2: Finding the function equation using the formula y=mx+b:

For the intercept b, we look what is the value of y when x=0 in the table. If x=0 then y=3, therefore b=3.

Note that if the y value for x=0 was not given, we can calculate it using the change in the y variable that we found in step 1. The change in the y variable was equal to 4 and we know from the table that for x=2 y=7. Therefore, to find what is the y for x=0 we need to subtract 4 from 7 getting 3. If the first y value is 3 then the second y value is 3+4=7.

For the slope m, we divide the change in variable y by the change in variable x. We know from step 1 that the change in the x variable is 2 and the change in the y variable is 4. Therefore, the slope of the function is m=4/2=2.

The equation of the linear function that represents the given table is y=mx+b, y=2x+3.

Step 3: Checking the equation:
y=2x+3
y(x=0)=2*0+3=3
y(x=2)=2*2+3=7

Step 4: Drawing the graph the function according to its equation.

presenting a table with a linear function

Modeling a table with an exponential function

Model a table with an exponential equation if the value in the table changes by a constant multiplication factor (the common factor).

The exponential function formula is y=a(b)x, where a is the initial value (the intercept), b is the number that is multiplied (the common factor) and x is the number of the multiplications.

Modeling a table with an exponential function steps:

Step 1: Determine if the function that represents the table is linear or exponential- To find the correct function type check if the change in the y variable is linear or exponential. If the change in the y variable is exponential, then we need to find an exponential function equation that represents it. Note that the change in the x variable must be linear.

Step 2: Find the function equation using the formula y=a(b)x for the exponential function.

To find the initial value a, look what is the value of y when x=0 in the table. If the y value for x=0 is not given, calculate it using the change in the y variable that you found in step 1.

The common factor b is equal to the change in the variable y.

Step 3: Check the equation from step 2 by plugging into it the x values from the table. You must get y values like in the table.

Step 4: If needed, draw the graph the function according to its equation.

Consider the following example:

Write an equation based of the following table and find its graph.

x   0    1     2     3

y   2    6   18   54

Step 1: Determining if the function that represents the table is linear or exponential:

We need to make sure that the change in the x variable is linear; the x variable values are 0, 1, 2 and 3. Their changes are constant and are equal to 1 (1-0=1, 2-1=1…).

We need to check the change in the y variable; the y variable values are 2, 6, 18 and 54. Their changes are exponential by a factor of 3 (6:2=3, 18:6=3, 54-18=3).

Step 2: Finding the function equation using the formula y=a(b)x:

For the initial value a, we look what is the value of y when x=0 in the table. If x=0 then y=2, therefore a=2.

Note that if the y value for x=0 was not given, we can calculate it using the change in the y variable that we found in step 1. The change in the y variable was by a factor of 3 and we know from the table that for x=1 y=6. Therefore to find what is the y for x=0 we need to divide 6 by 3 getting 2. If the first y value is 2 then the second y value is 2*3=6.

The common factor b is the number that is multiplied, it was calculated in step 1 as the change in the y variable and is equal to 3.

The equation of the exponential function that represents the given table is y=a(b)x, y=2(3)x.

Step 3: Checking the equation:
y=2(3)x
y(x=0)=2(3)0=2*1=2
y(x=1)=2(3)x=2(3)1=2*3=6
y(x=2)=2(3)x=2(3)2=2*3*3=18
y(x=3)=2(3)x=2(3)3=2*3*3*3=54

Step 4: Drawing the graph the function according to its equation.

presenting a table with an exponential function

Modeling a scenario (description) with linear and exponential functions

In these questions, we receive a verbal scenario and need to determine the type of the connection (linear or exponential) and the direction of the connection (increasing or decreasing) from a multiple choice equations.

Determining if the function that represents the scenario is linear or exponential

If the change is at a constant rate (by a constant), the relationship is linear.

For example: The value of the account each month is 100 dollars bigger than the value in the previous month. If the initial value is 1,000 dollars, after 1 month it will be 1,000+100=1,100 dollars, after 2 months it will be 1,100+100=1,200 dollars etc. The function equation is y=1,000+100x.

If the change is by a percent from the initial value, the relationship is linear.

For example: The value of the account increases by 5 percent from the initial value each month. If the initial value is 1,000 dollars, the value of the account each month will increase by 50 dollars. After 1 month it will be 1,000+50=1,050 dollars, after 2 months it will be 1,050+50=1,100 dollars etc. The function equation is y=1,000+50x.

If the change is by a factor, the relationship is exponential.

For example: The value of the account each month is 2 times bigger than the value in the previous month. If the initial value is 1,000 dollars, after 1 month it will be 1,000*2=2,000 dollars, after 2 months it will be 1,000*2*2=4,000 dollars etc. The function equation is y=1,000*2x.

If the change is by a percent (from the previous value), the relationship is exponential.

For example: The value of the account increases by 5 percent (from the value in the previous month). If the initial value is 1,000 dollars, after 1 month it will be 1,000*1.05=1,050 dollars, after 2 months it will be 1,000*1.05*1.05=1,102.5 dollars etc. The function equation is y=1,000*1.05x.

Determining if the function that represents the scenario is increasing or decreasing

All the examples above were for an increasing connection. For decreasing connections, the functions will become:

If the value of the account each month is 100 dollars smaller than the value in the previous month, the function y=1,000+100x will become y=1,000-100x.

If the value of the account decreases by 5 percent from the initial value each month, the function y=1,000+50x will become y=1,000-50x.

If the value of the account each month is 2 times smaller than the value in the previous month, the function y=1,000*2x will become y=1,000*(1/2)x.

If the value of the account decreases by 5 percent (from the value in the previous month), the function y=1,000*1.05x will become y=1,000*(1-0.05)x.

Key features of graphs

Key features of graphs on the SAT test

SAT Subscore: Problem solving and data analysis

A graph is defined as a pictorial representation of data or numeric values in an organized manner. Using graph enables us to represent large amounts of data in visual form for easy understanding.

The 2 types of questions about the key features of graphs are interpreting given graphs or selecting a graph based on a verbal description.

The common graph types on SAT are bar graphs, dot plots, histograms, line graphs and scatterplots. Scatterplot subject is covered on scatterplots page.

A Bar Graph is a graphical display of data using rectangular bars (columns) of different heights, so that the height of each bar determines its value. The structure of the bar graph: the y axis contains values, and the x axis contains categories or time periods.  

A dot plot is a simple type of graph that shows the frequency with which items appears in a data set. It displays data items as dots above values (or categories) on the x axis (each data item is represented with a dot above its value or category).

A histogram is a frequency bar graph where the data is grouped into ranges. The x axis presents the data ranges and the y axis presents the frequency (the number of values that fall into the specific range.

A line graph includes a line that connects individual data points together. Line graphs are used to show changes over periods of time, so that the x axis represents time values (like years).

Continue reading this page for detailed explanations and examples.

Bar graphs

A Bar Graph is a graphical display of data using rectangular bars (columns) of different heights, so that the height of each bar determines its value (the larger the value the higher the bar).

A bar graph purpose is to compare values between different groups or to compare values over time.

A bar graph structure: the y axis contains values, and the x axis contains variables of 2 types: group types (categories) or time periods.  

Bar graphs that compare values over time

The following bar graph shows a quarterly revenue trend of company A by comparing values (revenues) over time ( 4 quarters) (in dollars). The data that was used for the bar graph is presented in the table below the graph.

The bar graph above shows that:

  • The revenue is rising every quarter (the bar of each quarter is higher than the bar of the previous quarter).
  • The lowest quarterly revenue is 70,000 dollars and the highest quarterly revenue is 140,000 dollars.
  • The revenue increase in the third quarter (110,000-80,000 dollars) equals to the revenue increase in the fourth quarter (140,000-10,000 dollars).
  • The revenue increase in the third quarter (110,000-80,000 dollars) is 3 times bigger than the revenue increase in the second quarter (80,000-70,000 dollars).

Bar graphs that compare values between different groups

The following bar graph shows the revenue of 3 companies in the first quarter. The graph compares values (revenues) between different groups (companies) (in dollars). The data that was used for the bar graph is presented in the table below the graph.

Comparing values between different groups with a bar graph

The bar graph above shows that:

  • The company B has the highest revenue in the first quarter (140,000 dollars).
  • The company C has the lowest revenue in the first quarter (10,000 dollars).
  • The revenue of company B (140,000 dollars) is twice bigger than the revenue of company A (70,000 dollars).
  • The difference between the revenue of company A (70,000 dollars) and the revenue of company C (10,000 dollars) is 60,000 dollars.

Bar graphs that compare values between different groups over time

We can combine the 2 graphs from above into one graph by showing different groups over different time periods. To do so we need to show the data of the 3 groups (companies) in every quarter. Each company data bar is represented in different color to separate the data of that company from the other companies.

The following bar graph shows the revenue of 3 companies in 4 quarters. The graph compares values (revenues) between different groups (companies) over time (quarters) (in dollars). The data that was used for the bar graph is presented in the table below the graph.

Comparint values between different groups over time with bars graphs

The bar graph above shows that:

  • The revenue of company A is rising every quarter (the bar of each quarter is higher than the bar of the previous quarter).
  • The revenue of company B is declining every quarter (the bar of each quarter is lower than the bar of the previous quarter).
  • The revenues of company A and company B are always higher than the revenues of company C.
  • The revenue of company B in the first quarter is equal to the revenue of company A in the fourth quarter (140,000 dollars).

Stacked columns bar graphs

The following stacked bar graph shows the revenue of 3 companies in 4 quarters. In this bar graph the data series are stacked one on top of the other in vertical columns. This presentation method allows us to see the total revenue value in each quarter and perform part to whole comparisons.

Note that finding revenue values of companies B and C that are stacked above company A requires calculating a difference between two values. See example below.

The bar graph above shows that:

  • The total revenue of the 3 companies was the highest in the second quarter (225,000 dollars).
  • The revenue of company B in the third quarter is 90,000 dollars (200,000-110,000 dollars).  
  • The revenue of company C is approximately 5 percent of the total revenue in the fourth quarter (10,000/90,000 dollars= 1/19*100%=5%).

Dot plots

A dot plot is a simple type of graph that shows the frequency with which items appears in a data set. It displays data items as dots above values (or categories) on the x axis (each data item is represented with a dot above its value or category).

How to draw a dot plot:
Step 1- Arrange the data values (categories) in increasing order.
Step 2- Count the number of values (categories) for each grade.
Step 3- Write the values (categories names) on the x axis.
Step 4- Draw dots above each value (category).

Consider the following example:

The exam grades of 10 students are: 80, 90, 70, 65, 90, 75, 65, 80, 90 and 95.

Which pot plot describes these grades?

Step 1- Arranging the data:

Before drawing the dot plot we need to arrange the grades in increasing order: 65, 65, 70, 75, 80, 80, 90, 90, 90 and 95.

Step 2- Counting the number of students for each grade:

65-2, 70-1, 75-1, 80-2, 90-3, 95.

Steps 3 and 4- drawing the graph:

The x axis should present values (the grades) and each grade should be represented by a dot above the grade value. For example: since 3 students received a grade of 90 there should be 3 dots above the grade 90.

 The following dot plot represents the given data.

dot plots

The dot plot above shows that:

  • No student received a grade of 85.
  • The number of students that received a grade of 70 is equal to the number of students that received a grade of 75 (1 student).
  • The number of students that received the grade 65 is twice bigger than the number of students that received the grade 70.

Histograms

A histogram is a frequency bar graph where the data is grouped into ranges. The x axis presents the data intervals (ranges) and the y axis presents the frequency (the number of values that fall into the specific interval/ range). Taller bar shows that more data falls into the range of the bar.

Note that:

  • All bars have the same width.
  • All bars touch each other, because the scale is continuous (bars in the regular bar graph don’t touch each other).
  • The histogram does not give any information about the specific sizes of the values, it only shows how many values are in any data range/ interval.

Drawing a histogram steps:
Step 1: Arrange the data values in increasing order.
Step 2: Determine the size of the range.
Step 3: Place the ranges on the x axis.
Step 4: Place the frequencies on the y axis.
Step 5: Draw a bar in each interval. The height of each bar is equal to its frequency.

Consider the previous example:

The exam grades of 10 students are: 80, 90, 70, 65, 90, 75, 65, 80, 90 and 95.

Which histogram describes these grades?

Step 1: Arranging the data values in increasing order.
Before drawing the histogram, we need to arrange the grades in increasing order: 65, 65, 70, 75, 80, 80, 90, 90, 90 and 95.

Step 2: Determining the size of the range.
The size of the range that we choose is 5 points, and the ranges are 65-70, 70-75, 75-80, 80-85, 85-90, 90-95.

The following histogram represents the given data.

A histogram graph

Note that the histogram does not give any information about the specific values of the grades, it only shows how many grades (students) are in any grade range/ interval. We know that 1 student received a grade between 70-75, but we do not know from the histogram what was the grade value.

Note that the value 70 that appears in the first bar and the second bar is counted in the first bar (the first bar from the two bars), the value 90 that appears in the fifth and sixth bars is counted in the fifth bar (the first bar from the two bars) etc.

The histogram above shows that:

  • No students received grades between 80-85.
  • The number of students that received grades between 70-75 is equal to the number of students that received grades between 90-95 (1 student).
  • The number of students that received grades between 65-70 is 3 times bigger than the number of students that received grades between 70-75.

Line graphs

A line graph includes a line that connects individual data points together. Line graphs are used to show changes over periods of time, so that the x axis represents time values (like years).

Line graphs question types:

  • Identifying values on the graph.
  • Analyzing the direction of the change in the different areas of the graph (increasing or decreasing).
  • Finding the rate of the change between different points of the graph (done by analyzing at the steepness of the graph).

Drawing a line graph steps:
Step 1: Write the names of the axis (the x axis should represent time values).
Step 2: Plot the data points on the graph.
Step 3: Draw a line from each point to the point near it.

The following line graph shows the number of students that graduated from high school A from the year 2010 to 2020, the table that includes the data for the line graph is located below the histogram.

The line graph above shows that:

  • The number of students that graduated from high school A increased from the year 2012 to 2020.
  • The lowest number of students that graduated from high school A was in the year 2010.
  • The highest change in the number of students that graduated from high school A was between the years 2012 and 2013.

Finding a graph that represents a word problem

In these questions, we need to transfer a word problem to graphical terms. Below is an explanation for finding a line graph, graphs of other types can be built in a similar way.

Finding a line graph that represents a word problem steps:
Step 1: Find the x and y axes names- Find the axis names in the word problem question. Analyze the question to determine which variable is dependent (should appear on the y axis) and which is independent (should appear on the x axis).
Note that time is an independent variable and must be on the x axis.
Step 2: Divide the word question into parts (in each part the y value changes therefore the shape of the graph changes).
Step 3: Calculate the x and y values for each part of the word question.
Step 4: Draw the graph for each part of the word question identified in step 2 according to the x and y values calculated in step 3- The shape of the graph should be according to the information given in the word problem.

The properties of the shape of the graph:
The trend (direction) of the graph– the graph can go up (upward trend), go down (downward trend) or remain the same (flat trend).
The slope of the graph- the graph can change fast (steep slope) or slow (shallow slope).

Consider the following example:

The factory worker operates a machine that produces chocolate bars. He counts the number of the chocolate bars every hour and packs them in a box at the end of the day. In the morning, the worker saw that yesterday he forgot to put 5 bars in the box and turned the machine on. He decided to add the 5 bars to the bars produced today. After the first hour the worker counted 50 bars. After the second hour the worker calculated 20 percent less bars than after the first hour. Then he took a break for an hour. After he returned the worker operated the machine for the last hour (the fourth hour) and counted a total of 140 bars.

Draw a graph that represents the number of the chocolate bars produced as time passed.  

Step 1: Finding the x and y axes names:
The question asks about the number of the chocolate bars produced as time passes, therefore the axis names are the number of the chocolate bars produced and the time (in hours). Since the number of the chocolate bars produced changes as time passes, the independent variable is the time (should appear on the x axis) and the dependent variable is the number of the chocolate bars produced (should appear on the y axis).

Step 2: Dividing the word question into parts:
Each hour the number of chocolate bar changes, therefore we need to divide the word question into 4 parts.

Step 3: Calculating the x and y values for each part of the word question:
Starting point x=0: In the morning there were 5 chocolate bars, therefore the point is (0,5).
After 1 hour x=1: In the first hour the machine produced 50 bars, therefore the point is (1,55).
After 2 hours x=2: In the second hour the machine produced 20 percent less bars than in the first hour, therefore the machine produced 50*(100%-20%)=50*80%=50*80/100=40 bars, therefore the point is (2,95).
After 3 hours x=3: In the third hour the machine didn’t work, therefore the point is (3,95).
After 4 hours x=4: After four hours the worker counted a total of 140 bars, therefore the point is (4,140).

The following line graph shows the number of the chocolate bars produced as time passed.

Scatterplots

Scatterplots on the SAT test

SAT Subscore: Problem solving and data analysis

Before learning this subject, make sure you master the subjects of linear functions and graphing quadratic functions.

A scatterplot is a graphic representation of a data set of observations (each observation includes x and y variables and is represented by a dot in the xy plane). The purpose of the scatterplot is to visualize the relationship between the variables x and y to determine if there are patterns or correlations between the two variables.

Scatterplots can visualize the following features of the relationships:
The type of the correlation between the variables– positive or negative correlation or no correlation.
The type of the data pattern- linear (straight) or nonlinear (curved).
The strength of the relationship between the variables– strong or weak correlation.
Unusual features in the data.

The line of best fit can be drawn through the area of the dots, the line of best fit represents the trend of the relationship between the two variables.
A line of best fit may be a straight (linear) line or a curved (parabolic) line.
If the line of best fit is straight (linear), it has a slope that represents the rate of the change and an intercept that represents the initial value.
The line of best fit can be used to predict y values that are not included in the data set. The prediction can be done in 2 ways: looking at the graph of the line of best fit or calculating the values from the equation of the line of best fit.
Calculating differencies between the given value and the predicted value: For any given x value, we can calculate the difference between its y value given in the data set (represented by the dot on the scatterplot) and the y value predicted by the line of best fit.
To calculate estimated change in y values, multiply the slope value of the line of best fit equation by the given amount of the change.

Continue reading this page for detailed explanations and examples.

Scatterplot and line of best fit definitions

A scatterplot (also called a scatter diagram) is a graphic representation of a data set of observations. Each observation in the data set includes 2 variables (x,y) and represented as a dot on the scatterplot on x and y axes (the independent variable is plotted along the x axis and the dependent variable along the y axis).

The line of best fit can be drawn through the area of the dots, this is the line that best represents the dots of the scatterplot. Note that the line of best fit can be drawn only if there is a connection between x and y variables.

The purpose of the scatterplot is to visualize the relationship between the variables x and y to determine if there are patterns or correlations between the two variables. If there is a connection between x and y variables, draw a line of best fit through the area of the points to see the type of the connection.

The difference between the line of best fit versus a line (linear) graph: the line of best fit represents the trend of the relationship between the two variables and not joining the dots together like on a linear graph. In other words: all the dots of the linear graph appear on the graph in comparison to the dots of the line of best fit that appear near the graph (some of them may appear on the graph).

The relationship types represented by scatterplots

  • To analyze the relationship, it is recommended to sketch a line between the dots and look at the line instead of the dots.
  • When you interpret a scatterplot, look at the data as you go from left to right.

Scatterplots can visualize the following features of the relationships:

1. The type of the correlation between the variables- positive or negative correlation or no correlation:
In positive correlation, as one variable increases so does the other. In a positive correlation the slope of the line is positive.
In negative correlation, as one variable increases the other decreases. In a negative correlation the slope of the line is negative.

2. The type of the data pattern- linear (straight) or nonlinear (curved):
A linear correlation can be graphed as a straight line in the xy-plane. The slope of a linear line is constant all the way along the line.
A curved correlation can be graphed as a smooth line that changes its direction at least once. The slope of a curved line is constantly changing.

3. The strength of the relationship between the variables:
The more concentrated the dots are along the line or the curve, the stronger the relationship. In other words: if the points are close to the line or the curve, the relationship is considered as strong.

4. Unusual features in the data, such as gaps in the data set.

The graph below represents a linear scatterplot with a negative strong connection between the variables x and y.

Scatterplot- linear, negative, strong connection

In the above scatterplot:

  • The connection can be graphed as a straight line of best fit, therefore the connection is linear.
  • The dots are located near the line, therefore the connection between the variables x and y is strong.
  • The slope of the line is negative, therefore the connection between the variables x and y is negative (as x increases y decreases).

The graph below represents a curved scatterplot with a weak connection between the variables x and y. The slope changes its direction after the vertex point from negative to positive (look at the data as you go from left to right).

a curved scatterplot with a weak connection between the variables x and y.

In the above scatterplot:

  • The connection can be graphed as a parabolic line of best fit, therefore the connection is curved.
  • The dots are located far from the line, therefore the connection between the variables x and y is weak.
  • The slope changes its direction after the vertex point from negative to positive (look at the data as you go from left to right).

The line of best fit- properties and purpose

The line of best fit (also called the trend line) is drawn through the area of the dots, this is the line that best represents the positions of the dots of the scatterplot in the xy plane.

The properties of the line of best fit

A line of best fit can be drawn through the area of the dots only if there is a connection between x and y variables.

A line of best fit may be a straight (linear) line or a curved (parabolic) line, depending on how the dots are arranged on the x y plane.

The points on the line of best fit represent a trend of the connection and not specific observations (unless the observation represented by a dot is locates on the line).

If the line of best fit is straight (linear), it has a slope that represents the rate of the change and an intercept that represents the initial value.     

The purpose of the line of best fit

  1. The line of best fit helps us to identify the type of the connection between x and y variables. (It is possible to identify the connection by analyzing the dots without the line but analyzing the line will be easier and clearer).
  2. The line of best fit estimates the value of y for any specified value of x. This is very useful for predicting y values of x values that are not given in the data set.
  3. We can calculate estimated changes in y values using the slope value from the line  of best fit equation.

Estimating the function of the line of best fit

Some questions may ask you to choose the function of the line of best fit from 4 given functions.
Before learning this subject, make sure you master the subjects of linear functions and graphing quadratic functions.

Estimating a linear function in a form of f(x)=mx+b:
1. Sketch a straight line that fits the data (that line should continue so it intercepts the y axis).
2. Estimate the y intercept of the line- this is the value of the parameter b in the function.
3. Estimate the slope of the line- this is the value of the parameter m in the function.
The slope sign can be seen from the direction of the line (increasing line has a positive slope and decreasing line has a negative slope).
The slope value can be estimated from 2 dots on the graph. The slope formula is the difference in y values divided by the difference in x values of any 2 points on the line.

Estimating a quadratic function in a form of f(x)=ax2+bx+c:
1. Sketch a parabola that fits the data.
2. Estimate the y intercept of the parabola- this is the value of the parameter c in the function.
3. Identify the vertex sign (positive for minimum or negative for maximum)- this is the sign of the parameter a.

Consider the following example:

According to the graph shown below, which of the following best models the line of best fit?

A f(x)=-0.5x2+5x+10
B f(x)= 0.5x2+2x+10
C f(x)=-2x2+5x+10
D f(x)= 0.5x2+2x+15

finding a quadratic function of the line of best fit

We can see from the given graph of the parabola that the y intercept is y=10, therefore the answer D in not correct.
We can see from the given graph of the parabola that the parabola has a minimum point, therefore the sign of x2 parameter a is positive. The answers A and C are not correct.
The correct answer is B: f(x)= 0.5x2+2x+10. In this answer the intercept c=10 and the x2 parameter a=0.5 best fit the graph of the parabola.

Predicting values using the line of best fit

The prediction can be done in 2 ways:

1. Look at the x value on the x axis and find the corresponding y value on the line of best fit graph. If the predicted x value lies beyond the shown line, we can extend the best fit line to see the predicted value.

2. Calculate the y value from the equation of the line of best fit.

Consider the following example:

The scatterplot below shows a data from a sample. The equation of the line of best fit is y=-0.5x+5 and it is marked in red on the xy plane below.

What is the predicted value of y for x=0 and x=3?

Finding the corresponding y values on the line of best fit graph:
We need to extend the line of best fit, see the orange line on the xy plane above.
For x=0 we see that the y value is y=5.
For x=3 we see that the y value is y=3.5.

Calculating the y value from the equation of the line of best fit:
y(x=0)=-0.5*0+5=5
y(x=3)=-0.5*3+5=-1.5+5=3.5

Calculating the difference between the actual data and the line of best fit

For any given x value, we can calculate the difference between its y value given in the data set (represented by the dot on the scatterplot) and the y value predicted by the line of best fit.

The given y value can be seen from the dot of the scatterplot or from the data table (if given).

The predicted y value can be seen from the line of best fit graph or calculated from the equation of the line of best fit.

To represent the difference as a distance, calculate a positive value (write the bigger value first and then subtract the smaller value).

Consider the following example:

The scatterplot below shows a data from a sample. The equation of the line of best fit is y=-0.5x+5.

What is the difference in y values between the data points and the line of best fit for x=1 and x=3?

The difference between the actual data and the line of best fit

The equation of the line of best fit is y=-0.5x+5.

The predicted y values calculated from the equation of the line of best fit are:
y(x=1)=-0.5*1+5=-0.5+5=4.5
y(x=3)=-0.5*3+5=-1.5+5=3.5
Note that we can also look at the line of best fit instead of making calculations.

The actual y values from looking at the dots of scatterplot are:
For x=1 we see that y=4
For x=3 we see that y=4
The positive difference for x=1 is 4.5-4=0.5
The positive difference for x=3 is 4-3.5=0.5

Calculating estimated changes using the best fit line equation

To calculate estimated change in y values, multiply the slope value of the line of best fit equation by the given amount of the change.

Note that we don’t need to include the constant in the calculation of the change, since it doesn’t affect the change value (the constant value is included in the y values before and after the change and therefore is cancelled in the change calculation).

Consider the following example:

The line of best fit is represented by the equation y=0.1x+14.

 If x decreases by 500, what is the estimated change in y according to the line of best fit?

0.1*-500=-50 is the decrease in y if x decreases by 500.

Note that we don’t need to include the constant 14 in the calculation of the change, since it doesn’t affect the change value. The constant value (14) is included in the y values before and after the change and therefore is cancelled.

For example:
For x=1,000 we get y=0.1*1000+14=100+14=114
For x=500 we get y=0.1*500+14=50+14=64
The change from x=1,000 to x=500 is 50+14-(100+14)= 50+14-100-14= -50 the number 14 is cancelled.

Units and unit conversion

Units and unit conversion on the SAT test

SAT Subscore: Problem solving and data analysis

A unit is used to measure a quantity, while different kinds of quantities measured in different units.

Unit conversion changes a given quantity to a different unit of measurement, so that the relative amount does not change. Unit conversion is calculated with unit equivalencies.

Unit equivalencies show how the units of the same kind of quantity relate to each other.
Time Unit equivalencies measure time intervals that express duration (for example: 1 minute= 60 seconds).
Distance Unit equivalencies measure length (for example: 1 kilometer=1,000 meters ).
Mass Unit equivalencies measure weight of objects (for example: 1 kilogram=1,000 grams).
Volume Unit equivalencies measure capacity of an object or space in three dimensions (for example: 1 liter=1,000 milliliters).

Unit conversion is calculated by multiplication of the original quantity by a conversion factor of 1 and simplifying so that we are left with the desired units without the original units.

To Convert between 2 units a and c using a third unit b, we need to multiply by 2 conversion factors: a conversion factor between unit a and unit b and a conversion factor between unit b and unit c.

To Convert units that appear within rates, we need to construct one or two conversion factors to cancel the units that are given and not desired and create the units that are not given and desired.

Continue reading this page for detailed explanations and examples.

Unit conversion and unit equivalency definitions

A unit is used to measure a quantity, while different kinds of quantities measured in different units. For example: distance is measured in miles, weight is measured in kilograms and time is measured in hours. There can be many unites for measuring the same kind of quantity. For example: time can be measured in hours, minutes, or seconds.  

Unit conversion changes a given quantity to a different unit of measurement, so that the relative amount does not change. For example: 0.5 hours can be converted to 30 minutes (the relative amount stayed the same). Unit conversion is calculated by multiplication of the original quantity by a conversion factor of 1 and simplifying so that we are left with the desired units without the original units.

Unit equivalencies show how the units of the same kind of quantity relate to each other. For example: 1 hour is equal to 60 minutes. To convert a given quantity to a desired unit, you first need to know the unit equivalency between the given unit and the desired unit.   

Unit equivalencies lists

Below are lists of unit equivalences for time, distance, mass and volume.

Time Unit equivalencies:
Units of time measure time intervals that express duration.

1 minute= 60 seconds
1 hour= 60 minutes
1 day= 24 hours
1 week= 7 days
1 year= 12 months
1 year= 52 weeks
1 year= 365 days

Distance Unit equivalencies:
Units of distance measure length.

1 centimeter (cm)=10 millimeters (mm)
1 inch (in)= 2.54 centimeters (cm)
1 foot (ft)= 12 inches (in)
1 yard (yd)= 3 feet (ft)
1 meter (m)= 100 centimeters (cm)
1 kilometer (km)= 1,000 meters (m)
1 mile (mi)= 1.61 kilometers (km)

Mass Unit equivalencies:
Units of mass measure weight of objects.

1 gram (g)= 1,000 milligrams (mg)
1 pound = 453.59 grams (g)
1 kilogram (kg)= 2.2 pounds
1 kilogram (kg)= 1,000 grams (g)
1 ton (t)= 1,000 kilograms (kg)

Volume Unit equivalencies:
Units of volume measure capacity of an object or space in three dimensions, the objects can be fluids (like water) or bulk goods (like flour). Since volume is measured in three dimensions, the units of measure for volume are cubic units.

1 liter (l)= 1,000 milliliters
1 kiloliter= 1,000 liters

Calculating unit conversion

The unit conversion changes a given quantity to a different unit of measurement, so that the relative amount does not change.

The goal of the unit conversion is to transform the given quantity to desired units of measure, therefore we need to cancel the given units and create the desired units:
The given quantity has the given units in the nominator. To cancel the given units in the numerator we need to divide the given quantity by the given units.
The new quantity should have the desired units in the nominator. To create the desired units in the numerator we need to multiply the given quantity by the desired units.
Since the relative amount must stay the same, the only operation we can do is to multiply by 1 using unit equivalencies.

Therefore, unit conversion is calculated by multiplication of the original quantity by a conversion factor of 1 and simplifying so that we are left with the desired units without the original units.

Unit conversion steps

Remember to write the name of the unit near each value!

Step 1: Write the unit equivalence between the given units and the desired units.

Step 2: Write the conversion factor of 1 so that the desired units are in the numerator and the given unit are in the denominator.

Step 3: Write the conversion expression that is equal to the given quantity multiplied by the conversion factor of 1.

Step 4: Simplify the expression from step 3 so that you cancel out the given units and are left with the desired units.

Consider the following example:

Convert 2.7 meters (m) to centimeters (cm).

Step 1: Writing the unit equivalence between meters and centimeters:
We need to convert 2.7 meters, therefore we should write 1 meter in centimeters: 1 meter= x centimeters.
We know that 1 meter= 100 centimeters

Step 2: Writing the conversion factor of 1 so that the centimeters are in the numerator meters are in the denominator:
100 centimeters / 1 meter= 1

Step 3: Writing the conversion expression that is equal to the given quantity multiplied by the conversion factor of 1:
2.7 meters = 2.7 meters * 100 centimeters / 1 meter

Step 4: Simplifying the expression from step 3 canceling meters and leaving centimeters:
2.7 meters * 100 centimeters / 1 meter= 2.7*100 centimeters = 270 centimeters.

Converting between 2 units using a third unit

In these questions we are asked to convert unit a to unit c, while we are given 2 unit equivalencies: the unit equivalency between unit a and unit b and the unit equivalency between unit b and unit c. Therefore, to convert from unit a to unit c we must include their unit equivalency with unit b.

To Convert between 2 units using a third unit, we need to solve using the steps introduced above. In these questions we need to multiply by 2 conversion factors: a conversion factor between unit a and unit b and a conversion factor between unit b and unit c.

We are given unit a in the numerator and we need to be left with unit c in the numerator. To do so we must divide by unit a and multiply by unit c, therefore we need unit c in the numerator and unit a in the denominator. Since we are not given the direct unit equivalency between unit a and unit c, we must include their unit equivalencies with unit b. We know that unit b must cancel out, therefore we write unit b in the numerator and the denominator:

unit c                 unit b
_______    *   ________
Unit b                unit a

Consider the following example:

1 kilometer (km)= 1,000 meters (m)
1 mile (mi)= 1.61 kilometers (km)

How many meters are in 1 mile?

Step 1: Writing the unit equivalence between miles and meters:
We need to convert miles to meters, but we are given the conversion factor between miles and kilometers and the conversion factor between kilometers and meters.
1 kilometer (km)= 1,000 meters (m)
1 mile (mi)= 1.61 kilometers (km)

Step 2: Writing the 2 conversion factors of 1:
We know that we must divide by miles and multiply be meters (so that miles unit will cancel out and meters unit will be created).

                  meters
_______  * _______
miles

We don’t need kilometers in the answer and we don’t have them in the given quantity, but we must use them in the conversion factors. Therefore, we must create kilometers and then cancel them out, to do so we need to multiply and divide by kilometers.

kilometers           meters
___________    *     _________
miles                     kilometers

Now we can put values from the given unit equivalencies into the conversion factors:
1 kilometer (km)= 1,000 meters (m)
1 mile (mi)= 1.61 kilometers (km)

1.61 kilometers          1,000 meters
________________   *     _____________
1 miles                         1 kilometers

Step 3: Writing the conversion expression that is equal to the given quantity multiplied by 2 conversion factors of 1:

1 mile= 1 mile     1.61 kilometers          1,000 meters     = 1.61 * 1,000 = 1,610 meters
                           *  _______________    *    ______________
                                     1 mile                      1 kilometers

Converting units that appear within rates

Each rate is composed from 2 units, one in the numerator and the other in the denominator. Remember to write the name of the units near the values in the numerator and the denominator of the rate.

To Convert units that appear within rates, we need to solve using the steps introduced above. In these questions we need to construct one or two conversion factors to cancel the units that are given and not desired and create the units that are not given and desired.

Consider the following example:

Convert 1 dollars per pound to cents per kilograms.

Step 1: Writing the unit equivalence between the units:
1 kilogram (kg)= 2.2 pounds
1 dollar= 100 cents

Step 2: Writing the conversion factors of 1:

In this question we need to convert between 2 rates and each rate is composed from different units. Therefore, we need to cancel 2 units that appear in numerator and the denominator of the given rate (dollars per pounds) and create 2 units that appear in the numerator and the denominator of the desired rate (cents per kilograms).

pounds             cents
________   *   ___________
dollars            kilograms

We know the unit equivalence between dollars and cents and between kilograms and pounds:

2.2 pounds          100   cents
____________   *   ____________
1 kilogram               1 dollar

Step 3: Writing the conversion expression that is equal to the given quantity multiplied by the conversion factors of 1:

1 dollar           1 dollar           2.2 pound       100 cents
________  =   __________  *    ___________  *  __________  
1 pound         1 pound           1 kilogram        1 dollar

Step 4: Simplifying the expression from step 3 canceling dollars and pounds and leaving cents and kilograms:

1 dollar           2.2 pounds          100 cents         2.2*100 cents          220 cents
_________  *   _____________  *   ___________  =  _______________  =   ____________
1 pound           1 kilogram            1 dollar              1 kilogram              1 kilogram

Percentages

Percentages on the SAT test

SAT Subscore: Problem solving and data analysis

A percentage  is a number or a ratio expressed as a fraction of 100 and represents a part to whole relationship. The different forms of writing percentages are a decimal number, a fraction, and a ratio. Note that a percent term is used after we write a value (for example: 10 percent).

Complementary Percentages add up to 100%, where 100% refers to the total.
A percentage= 100%- complementary percentages

A percentage formula is
percentage % = 100 *  the value to be expressed
                                        __________________________
                                                       total

(The value to be expressed can be smaller than the total or bigger that the total).

To convert a percentage to a decimal fraction, remove the percent sign % and divide the percentage by 100:
Percentage as a decimal fraction=percentage without the % sign/100

The percentage of the change is equal to the difference divided by the initial value multiplied by 100:

The percentage of the change= (The final value – the initial value )       *100
                                                         _________________________________
                                                                     The initial value

Percentage data can be given in a table:

  • All percentages must sum to 100%.
  • Each percentage in the row can represent a value to be expressed. Calculate the value to be expressed of each row by multiplying its percentage as a decimal by the total of the table.

Continue reading this page for detailed explanations and examples.

Percentage definition and forms

A percentage (from Latin per centum “by a hundred”) is a number or a ratio expressed as a fraction of 100 and represents a part to whole relationship. A percentage is often denoted using the percent sign %. The term percentage does not refer to specific numbers and refers to a general relationship. For example: a large percentage of the participants voted.

The different forms of writing percentages are a decimal number, a fraction, and a ratio. The form that we choose depends on the question asked. For example: 40% can be written as a decimal number 0.4, a fraction 40/100 or a ratio 40:100.

Complementary Percentages add up to 100%, where 100% refers to the total. For example: The percentage of the girls in the class is 66.67% and the percentage of the boys in the class is 33.33%. The percentages of the boys and the girls are complementary percentages, since they add up to 100%.

A percent term is used after we write a value. For example: 10 percent of the participants voted. We can also write 10%.

When answering student produced response questions, write the percentage as an integer without the % sign. For example: if the answer is 10% write in the answer field 10 (don’t write 0.1 or 10%).

Calculations with the percentage formula

Percentage is calculated with the percentage formula: multiply the fraction by 100 and add a percent sign.

Percentage formula is:  percentage %= 100 *  the value to be expressed   
                                                                                  __________________________
                                                                                                 total

How is the percentage formula created?

The percentage formula is based on the proportion equation:
A proportion is an equality between 2 or more equivalent ratios and it can be represented as a fraction a/b=c/d:

The first ratio includes a ratio between 2 numbers where one of the numbers is the value to be expressed and the other number is the total

The second ratio includes a percentage divided by 100.

The proportion equation is the value to be expressed/ total=percentage/ 100.

We want to know what the percentage is equal to, therefore we multiply by 100 and get the formula written above:
percentage %= 100 * the value to be expressed/ total    

The steps for calculating percentages using the percentage formula

Step 1: Determine the total (whole) amount.

Step 2: Write the fraction: divide the value to be expressed as a percent by the total. Note that the number to be expressed can be smaller than the total or bigger that the total.

Step 3: Multiply the resulting value by 100.

 Consider the following example:

There are 22 girls and 11 boys in the class. What are the percentages of the boys and the girls?

Step 1: Determining the total (whole) amount: There are 11+22=33 pupils in the class.

Step 2: Writing the fraction: the fraction for the boys is 11/33 and the fraction for the girls is 22/33.

Step 3: Multiplying the resulting value by 100:

The percentage of the boys in the class is 11/33*100=33.33%
The percentage of the girls in the class is 22/33*100=66.67%

Calculating different parts of the percentage formula

To calculate any of the 3 parts of the formula, we need to know the values of the 2 other parts. Therefore, in addition to calculating the percentage as explained before, we can calculate the 2 other values of the formula (the total value or the value to be expressed): mark as x the unknown value and solve the percentage formula.

We can also calculate the total value or the value to be expressed directly:

To calculate the total, divide the value to be expressed by its percentage and multiply by 100.

This formula is created from the percentage formula: multiplying both sides of the percentage formula equation by the total and dividing them by the percentage will give us that the total is equal to the value to be expressed divided by its percentage multiplied by 100.

Consider the following example:

The number of tourists in the bus is 30, if the percentage of the tourists in the bus is 20% from the total tourists arriving to the hotel, what is the number of tourists that are not in the bus?

The total= 100*the value to be expressed/percentage

The total=100*(30/20)=100*1.5=150 the number of the tourists that are arriving is 150.

The number of tourists that are not in the bus is 150-30=120.

To calculate the value to be expressed, multiply the total by the percentage value and divide by 100.

This formula is also created from the percentage formula: Multiplying both sides of the percentage equation by the total and dividing by 100 will give us that the value to be expressed is equal to the total multiplied by the percentage and divided by 100.

Note that the number to be expressed can be smaller than the total or bigger that the total, as shown in the below examples. 

Consider the following example, in this example the number to be expressed is smaller than the total:

There were 150 tourists in the hotel, 20 percent of them were in the restaurant. How many tourists were in the restaurant?

The value to be expressed= the total* the percentage value/ 100

The number of tourists in the restaurant=150*20/100=30.

Checking the answer by using the percentage formula:
The percentage is equal to the value to be expressed divided by the total and multiplied by 100
20=30/150*100
20=3000/150
20=300/15
20=20

Consider the following example, in this example the number to be expressed is bigger than the total:

The show has a capacity of 1,000 participants. If yesterday the capacity was 120%, how many participants attended the show yesterday?

In this example the value to be expressed is bigger than the total (we see that the number of the participants yesterday is bigger than 1,000), this is because the percentage is larger than 100%.

The value to be expressed=the total* the percentage value/ 100.

The number of the participants yesterday=(1,000*120) /100=1,200.

Checking the answer by using the percentage formula: percentage is equal to the value to be expressed divided by the total and multiplied by 100
120=(1,200/1,000)*100
120=120,000/1,000
120=120

Calculating complementary percentages

Remember that complementary percentages add up to 100%, where 100% refers to the total.

We can calculate a missing percentage if we are given all its complementary percentages:
A percentage= 100%- complementary percentages

Consider the following example:

There are 50 balls in the box. If 10% of the balls are blue and 20% of the balls are red, how many balls are green?

A percentage= 100%- complementary percentages
The green balls %=100%-10%-20%=70%

The green balls amount is the value to be expressed.
The value to be expressed=the total*the percentage value/100
The green balls amount=50*70/100=35

Checking the answer by using the percentage formula:
The percentage is equal to the value to be expressed divided by the total and multiplied by 100
70=(35/50)*100
70=3,500/50
70=350/50
70=70

Representing percentages as decimal fractions

Before learning this topic, learn the decimal fractions topic.

To convert a percentage to a fraction, remove the percent sign % and divide the percentage by 100:
Percentage as a decimal fraction=percentage without the % sign/100
To divide by 100, add a decimal point at the end of the number (instead of the removed percent sign) and then shift the decimal point 2 places to the left.
For example:
5%=5/100=0.05
45%=45/100=0.45
105%=105/100=1.05
Note that if the percentage is greater than 100% its decimal fraction will be larger that 1.

A percentage as a decimal fraction = the value to be expressed/ total
This formula is also created from the percentage formula:
The percentage formula is:  percentage= 100 * the value to be expressed/ total
We can divide both sides of the formula by 100 getting: percentage/100= the value to be expressed/ total.
We know that percentage/100 is equal to the percentage as a decimal fraction, therefore a percentage as a decimal fraction = the value to be expressed/ total.

Note that all previous examples can be solved using the decimal fractions without dividing the percentage by 100. Solving questions using decimal fractions takes less time, therefore it is recommended.

Consider the previous example:

There are 22 girls and 11 boys in the class. What are the percentages of the boys and the girls?

The total number of pupils is 11+22=33

The decimal percentage of the boys is 11/33=0.3333 that is equal to 0.3333*100=33.33%

The decimal percentage of the girls is 22/33=0.6667 that is equal to 0.6667*100=66.67%

Consider the previous example:

The number of tourists in the bus is 30, if the percentage of the tourists in the bus is 20% from the total tourists arriving to the hotel, what is the number of tourists that are not in the bus?

We saw earlier that the total= 100 * the value to be expressed/percentage

We can simply divide the value to be expressed by the decimal fraction of the percentage:
x=30/0.2=150 the number of tourists arriving is 150

The number of tourists that are not in the bus is 150-30=120

We can also calculate the percentage of the tourists that are not in the bus using complementary percentages:

The percentage of tourists that are not in the bus is 100%-20%=80%, 80% expressed as decimal fraction are 0.8 (we removed the % sign and divided 80 by 100).

To calculate the number of tourists that are not in the bus we multiply their percentage as decimal fraction by the total amount getting 0.8*150=120.

Consider the previous example:

There are 50 balls in the box. If 10% of the balls are blue and 20% of the balls are red, how many balls are green?

A percentage= 100%- complementary percentages
The green balls as a decimal fraction=1-0.1-0.2=0.7

The green balls amount is the value to be expressed.
The value to be expressed=the total*the percentage as a decimal fraction
The green balls amount=50*0.7=35

Calculating the percentage change between two values

A percentage change can be calculated given 2 values: an initial value and a final value (the value after the change).

The percentage of the change is equal to the difference divided by the initial value multiplied by 100:

The percentage of the change= (The final value – the initial value )       *100
                                                         _________________________________
                                                                       The initial value

This formula can be created by calculating the percentage of the change as a percentage after the change minus the percentage before the change:

% of the change = % after the change – % before the change

% of the change = 100* the final value           100* the initial value
                                ____________________           _______________________
                                      the initial value                      the initial value

A common denominator of the initial value and factoring out 100 as a common factor gets us to the formula:

The percentage of the change= (The final value – the initial value )       *100
                                                           ________________________________
                                                                   The initial value

Note that:

We divide by the initial value and not the final value.

We can calculate the decimal value instead of the percentage without multiplying by 100, so that
the percentage of the change as a decimal is equal to the difference divided by the initial value.

In the equation above (the equation before applying a common denominator), the initial value divided by itself is equal to 1, therefore we can say that the percentage of the change as a decimal is equal to the final value divided by the initial value minus 1.

The sign of the change can be positive or negative:
If the final value is bigger than the initial value, than there is an increase, and the difference is positive.
If the final value is smaller than the initial value, than there is a decrease, and the difference is negative.

The steps for calculating a percentage change between two values:

Step 1: Calculate the difference between the two values:
If the final value is bigger than the initial value, than there is an increase, and the difference is positive.
If the final value is smaller than the initial value, than there is a decrease, and the difference is negative.

Step 2: Calculate the decimal change between the two values: Divide the difference by the initial value

Step 3: Calculate the percentage change between the two values:  multiply the decimal change by 100 and add a % sign.

Step 4: Check the answer: Multiply the initial value by the complementary percentage to get the final value.

Consider the following example:

The price of a product reduced from 200$ to 160$. What is the percent discount in price?

Step 1: Calculating the difference between the two values: 160-200=-40

Step 2: Calculating the decimal change between the two values: -40/200=-0.2

Step 3: Calculating the percentage change between the two values:  -0.2*100%=-20% the price decreased by 20 percent.

Step 4: Checking the answer:
The complementary percentage is 100%-20%=80%.
The final value should be 80%*200=160 is equal to the value in the question.

Calculating the final and the initial values given the difference value and the percentage of the change

The initial value is equal to the difference divided by the percentage of the change multiplied by 100 or the difference divided by the percentage of the change as a decimal.

The final value is equal to the difference plus the initial value.

The first statement formula can be created by rewriting the percentage of the change formula:

The percentage of the change is equal to the difference divided by the initial value multiplied by 100:

The percentage of the change= (The final value – the initial value)       *100
                                                         ________________________________
                                                                  The initial value

We can multiply both sides of the formula by the initial value and divide them by the percentage getting:

The initial value= (The final value – the initial value )       *100
                                ________________________________
                                          The percentage of the change

Consider the following example:

The amount of discount on the item was 10 dollars. If the discount percentage was 20 percent, what was the price after the discount?

The initial value is equal to the difference divided by the percentage of the change multiplied by 100:
The initial value=(10/20)*100=1,000/20=50
The final value=50-10=40

Checking the answer: the difference=50-40=10 or 50*20%=10

The initial value is also equal to the difference divided by the percentage of the change as a decimal:
Translating the percentage of the change to a decimal gets 20%=0.2
The initial value=10/0.2=50

Calculating the initial value given the final value and the percentage of the change

We can calculate the initial price by dividing the final price by the percentage after the change as a decimal (the percentage after the change is the complementary percentage of the percentage of the change).

We can also mark the initial value as x and solve the percentage of the change formula:

The percentage of the change= (The final value – x)   *100
                                                         ___________________
                                                                          x

We can also write this formula with the percentage of the change as a decimal:

The percentage of the change as a decimal= (The final value – x)
                                                                                 __________________
                                                                                                x

Note that you must put a minus sign before a discount percentage.

Consider the following example:

The price of a product after a 10% discount is 90$. What was the price before discount?

Solving with the percentage after the discount:
The percentage after the discount is 100%-10%=90%=0.9

The initial price= the final price/ the percentage after the discount as a decimal= 90/0.9=100 the price before discount was 100$.

Solving with the percentage of the change formula:
Note that you must put a minus sign before the discount percentage
-10=(90-x)*100/x
-10x=(90-x)*100
-x=(90-x)*10
-x=900-10x
9x=900
x=900/9=100 the price before discount was 100$.

Checking the answer: the discount=100-90=10 or 100*10%=10

Consider the following example:

The price of a product after a 10% discount is 90$. What was the price before discount?

Solving with the percentage after the increase:
The percentage after the increase is 100%+10%=110%=1.1

The initial price= the final price/ the percentage after the increase as a decimal= 1,100/1.1=1,000 the price before increase was 1,000$.

Solving with the percentage of the change formula:
10=(1,100-x)*100/x
10x=(1,100-x)*100
x=(1,100-x)*10
x=11,000-10x
11x=11,000
x=11,000/11=1,000 the price before the increase was 1,000$

Checking the answer: the increase=1,100-1,000=100 or 1,000*10%=100

Collecting percentage data from tables

Percentage data can be given in a table. Note that since the table represents the whole population, all the percentages must sum to 100%. Therefore, we can find one complementary percentage for all the percentages to sum to 100%.

Each percentage in the row can represent a value to be expressed, so that we calculate the value to be expressed of each row by multiplying its percentage as a decimal by the total of the table.

Consider the following example:

The table below shows quarterly sales of 5,452 products of a store.

Q1  27%

Q2  20%

Q3 25%

Q4  x

Calculate the fourth quarter sales.

How much the sales in the first quarter are larger than the sales in the fourth quarter? (round the answer to the nearest whole number).

Calculating x value using complementary percentage:
x=100%-27%-20%-25%=28%=28/100=0.28 the fourth quarter sales are 28% from the total annual sales.

The fourth quarter sales are equal to their decimal value of the percentage multiplied by the total sales:
Q4 sales=0.28*5,452=1,527 products.

The first quarter sales are equal to their decimal value of the percentage multiplied by the total sales:
Q1 sales=0.27*5,452=1,472 products.

The difference between the quarters is 1,527-1,472=55 products.

Checking the answer by calculating using the percentage difference from the total:
(0.28-0.27)*5,452=55 products.

Calculating a percentage of a percentage

To calculate a percentage of a percentage, convert both percentages to decimals and multiply them. To get the percentage value of the result, multiply the result by 100.

Consider the following example:

What is 20% of 10% ?

Converting the percentages to decimal fractions: 20%=0.2, 10%=0.1

Multiply the decimals: 0.2*0.1=0.02

Converting the result to percentages:  0.02*100=2%

Consider the following example:

The original price of the product was 50 dollars. During the holidays, the product was sold at a 10 percent discount. The last items left received an additional 10 percent discount.

What is the price of the product after the 2 discounts if the second discount was calculated from the original price and if the second discount was calculated from the net price after the first discount?

The price of the product if the second discount was calculated from the original price:

The discounts percentage is 10%+10%=20%

The percentage of the price after the discounts is the complementary percentage of the discount percentages: 100%-20%=80%=0.8

The price after the discount is the multiplication of the original price by of the decimal percentage of the price after the discounts: 50*0.8=40 dollars.

Another way of solution is to calculate each discount and subtract the discount from the original value:
A 10% discount=50*0.1=5 dollars
The price after the discount=50-5-5=40 dollars

 

The price of the product if the second discount was calculated from the price after the first discount:

We can calculate the final price as a multiplication of the price percentages after each discount:

The final price is 50*90%*90%=50*0.9*0.9=40.5 dollars.

Another way of solution is to calculate the percentage of the second discount as the percentage price of the product after the first discount multiplied by the percentage of the second discount:
The percentage of the second discount=90%*10%=0.9*0.1=0.09=9%
The discounts percentages=10%+9%=19%=0.19
The discounts value=0.19*50=9.5 dollars
The price after the discounts=50-9.5=40.5 dollars

Another way of solution is to calculate the price after the first discount and then calculate the second discount from the new discounted price.
A first 10% discount=50*0.1=5 dollars
The new price after the first discount is 50-5=45 dollars
The second discount is 10%*45=4.5 dollars
The price after the second discount is 50-5-4.5=50-9.5=40.5 dollars

Ratios, rates and proportions

Ratios, rates and proportions on the SAT test

SAT Subscore: Problem solving and data analysis

This subject includes 3 connected topics that deal with representing relationships using division: ratios, rates and proportions.

Since ratios, rates and proportions are written as fractions, before learning this topic you should learn solving fractions topic.

A ratio is a comparison of two numbers, represented by a division of their amounts. The ratio between a and b can be represented using a colon as a:b or as a fraction a/b. For example: The number of girls in the class is 22 and the number of boys in the class is 11. The ratio of boys to girls in the class is 11:22, reducing 11:22 gives 1:2. The ratio of girls to boys is 22:11, reducing 22:11 gives 2:1=2.

Equivalent ratios are ratios that express the same relationship between numbers. Two ratios are equivalent if we can reduce or expand one ratio and get the other ratio.

Complementary ratios are ratios that add up to a whole that is 1.

There are 2 types of ratios:  a part to part ratio and a part to whole ratio.

A rate is a quantity compared to another related quantity, where the quantities have different units. For example: The speed per hour is the rate measured by the number of miles per a unit of time of 1 hour.

A unit rate is a rate compared to a single unit quantity (the denominator is 1).

To calculate the unit rate, divide the total of one quantity (the numerator) by the number of units of the other quantity (the denominator).

 A proportion is an equality between 2 or more equivalent ratios. The proportion between a, b, c and d can be represented using a colon as a:b=c:d or as a fraction a/b=c/d. For example: A carrot cake recipe is composed of 3 cups of flour and 1 cup of sugar, if we put 3 cups of sugar in the bowl, we need to add 9 cups of flour: 3/1=9/3 or 3:1=9:3

We can write proportions in 2 ways: There are same units above the divisor line and below the division line or there are same units at the nominator and the denominator of each ratio.

Solving proportions is done using the cross product method which involves multiplying the numerator of one fraction by the denominator of another fraction and then equaling the multiplications.

Continue reading this page for detailed explanations and examples.

Ratios

A ratio is a comparison of two numbers, represented by a division of their amounts. The ratio between a and b can be represented using a colon as a:b or as a fraction a/b.

Reducing and expanding ratios: Since ratios are represented as fractions, ratios can be reduced or expanded. To expend a ratio, we multiply the numbers in the ratio by a same factor. To reduce a ratio, we divide the numbers in the ratio by a same factor.
For example: The number of girls in the class is 22 and the number of boys in the class is 11. The ratio of boys to girls in the class is 11:22, reducing 11:22 gives 1:2 (we divided by 11). The ratio of girls to boys is 22:11, reducing 22:11 gives 2:1=2 (we divided by 11).

Equivalent ratios and complementary ratios

Equivalent ratios are ratios that express the same relationship between numbers. Two ratios are equivalent if we can reduce or expand one ratio and get the other ratio.
For example: the ratios 11:22 and 1:2 are equivalent ratios, since we can multiply the ratio 1:2 by 11 (expending) and get the ratio 11:22 or we can divide the ratio 11:22 by 11 (reducing) and get the ratio 1:2.

Complementary ratios are ratios that add up to a whole that is 1. Note that, to get this some of 1, we need to include all the ratios in the group.
In the previous example, if we add the number of boys to the number of pupils that is 1/3 and the number of girls to the number of pupils that is 2/3 we get 1 (the class includes only boys and girls, therefore the ratios must add up to 1).

Types of ratios: a part to part ratio and a part to whole ratio

A part to part ratio compares the amount of one quantity against the amount of other quantity (each one of the amounts is a part from a total that is not included in the ratio calculation).
In the previous example: the number of boys to the number of girls 11:22=1:2 is a part to part ratio. The number of girls to the number of boys 22:11=2:1=2 is also a part to part ratio.

A part to whole ratio compares the amount of one quantity against the total amount.
In the previous example: the total number of pupils in the class is 11+22=33, therefore a part to whole ratio will be the number of boys to the number of pupils 11:33=1:3 or the number of girls to the number of pupils 22:33=2:3.

Calculating amounts of quantities

We may need to calculate the amount of quantity before we can calculate its ratio. This can be done only if we know the total amount and the amounts of all other complementary ratios.

Consider the following example:

A carrot cake recipe is composed of 3 cups of flour, 1 cup of sugar and oil.

If the total amount of the cake mixture is 5 cups, calculate the ratio of the oil to the flour and the ratio of the oil to the cake.

First, we need to find the amount of oil in one cake. We know that the mixture amount of one cake is 5 cups and that besides oil it includes 3 cups of flour and 1 cup of sugar, therefore the amount of oil in 1 cake is 5-3-1=1 cup.

Calculating the ratio of the oil to the flour: one cake includes 1 cup of oil and 3 cups of flour, therefore the ratio of the oil to the flour is 1:3.

Calculating the ratio of the oil to the cake: one cake includes 1 cup of oil and it contains 5 cups of mixture, therefore the ratio of the oil to the cake is 1:5.

Consider the following example:

One glass of milkshake includes 2 scoops of ice cream (one scoop is equal to 1/2 cup), 3/4 cups of milk and a syrup.

If the capacity of one glass is 2 cups, what is the ratio of syrup to milkshake?   

Translating 2 scoops of ice cream to glasses: 1 scoop=1/2 cup therefore 2 scoops of ice cream=2*1/2=1 cups of ice cream. Finding how much syrup we need to put in 1 glass of milkshake: 1 glass of milkshake=2 cups= 1 cups of ice cream + 3/4 cups of milk + x syrup. 2=1+3/4+x 2=13/4+x x=2-13/4=1/4 the amount of syrup in a glass of milkshake is 1/4 cup. The ratio of syrup to milkshake is 1/4 cup to 2 cups, that is (1/4)/2=1/8.

Finding a part to part ratio given complementary ratios

Since all complementary ratios add up to 1, we can calculate any part to part ratio given the other part to part ratios.

Consider the following example:

The ratio of staff to the total number of the people on the cruise ship is 1:5. What is the ratio of the tourists to the total number of people on a cruise ship?

Since the only people on the cruise ship are staff and tourists, we can calculate the ratio of the tourists to the total number of people on a cruise ship. We know that the unknown tourists to total ratio x and staff to total ratio 1:5 are complementary ratios, therefore they add up to 1:
1/5+x=1
x=1-1/5=4/5
The ratio of tourists to the total the people on a cruise ship is 4/5 (or 4:5).

Consider the previous example

One glass of milkshake includes 2 scoops of ice cream (one scoop is equal to 1/2 cup), 3/4 cups of milk and a syrup.

If the capacity of one glass is 2 cups, what is the ratio of syrup to milkshake?

Translating 2 scoops of ice cream to glasses:
1 scoop=1/2 cup therefore 2 scoops of ice cream=2*1/2=1 cups of ice cream.

Finding the ratio of syrup to milkshake using complementary ratios:
The ratio of milk to milkshake is 3/4 glass to 2 glasses, that is (3/4):2=3/8.
The ratio of ice cream to milkshake is 1 glass to 2 glasses, that is 1/2.
The ratio of syrup to milkshake is 1-3/8-1/2=8/8-3/8-4/8=1/8.

Calculating part to part ratios using part to whole ratios

We can calculate a part to part ratio by division of the corresponding part to whole ratios.

Consider the previous example:

One glass of milkshake includes 2 scoops of ice cream (one scoop is equal to 1/2 cup), 3/4 cups of milk and a syrup.

If the capacity of one glass is 2 cups, what is the ratio of syrup to milk?

We found earlier that:
1 scoop=1/2 cup therefore 2 scoops of ice cream=2*1/2=1 cups of ice cream.
The ratio of milk to milkshake is 3/4 glass to 2 glasses, that is (3/4):2=3/8.
The ratio of syrup to milkshake is 1-3/8-1/2=8/8-3/8-4/8=1/8.

The ratio of syrup to milk using the ratios to the whole is:
(1/8)/(3/8)= (1/8)*(8/3)=(1*8)/(8*3)=8/24=1/3

Checking the answer by calculating with the amounts from the total:
(1/4 cup)/(3/4 cup)= (1/4)*(4/3)=(1*4)/(4*3)=4/12=1/3

Calculating amounts of quantities using part to whole ratio

We can calculate amount of quantity by multiplying the total amount by the part to whole ratio.

Consider the following example:

The ratio of staff to the total number of people on the cruise ship is 1:5.

What is the number of the tourists if the total number of people on the cruise ship is 5,000?

The number of staff is the total amount 5,000 multiplied by the part to total ratio of staff 1:5, this gives us 5000*1/5=1000.
The number of tourists is the total amount 5,000 minus the number off staff 1,000, this gives us 5,000-1,000=4,000.

Another way is to calculate the part to total ratio of the tourists to the staff first, this is done by using the complementary ratios:
The ratio of staff to the total number of people on the cruise ship 1:5 and the ratio of tourists to the total number of people on the cruise ship x are equal to 1 (these ratios are complementary ratios, since there are no other people on the cruise ship).
1/5+x=1
x=1-1/5=4/5 the ratio of tourists to the total number of people is 4/5 or 4:5

The number of tourists is the total amount 5,000 multiplied by the part to total ratio of tourists 4:5, this gives us 5,000*4/5=4,000.

Calculating the total amount using ratios

We can calculate the total amount by dividing an amount of quantity by its part to total ratio.

Consider the following example:

The ratio of staff to the total number of people on the cruise ship 1:6.

If there are 500 of staff on the ship, what is the total number of people on the ship?  

500/(1/6)=500*6/1=3,000 the total number of people on the ship is 3,000.

This example can also be solved using proportions, as explained below.

Rates

A rate is a quantity compared to another related quantity, where the quantities have different units.

A unit rate is a rate compared to a single unit quantity (the denominator is 1). For example:
The speed per hour is the rate measured by the number of miles per a unit of time of 1 hour.
The price is the rate measured by the amount of dollars per 1 unit of product.
The heart rate is measured by the number of contractions of the heart per a unit of time of 1 minute.

To calculate the unit rate, divide the total of one quantity (the numerator) by the number of units of the other quantity (the denominator). Consider the following examples:

If the group walked 4 miles in 2 hours, what is the speed of the group (miles fer hour)?

 

The speed rate is the total of one quantity (the number of miles which is 4) divided by the number of units of the other quantity (the number of hours which is 2). The speed is therefore 4/2=2/1=2 miles per 1 hour or 2 miles per hour.

Note that we can also calculate the unit rate using proportion, this calculation takes longer therefore it is less recommended:
4 miles / 2 hours = X miles / 1 hour
4/2=x
x=2

The price of 4 books is 88 dollars, the price of 3 notebooks is 9 dollars. What is the price of 2 books and 2 notebooks?

First, we need to calculate the rates, which are prices in this example:
The price of 1 book= the price of the books 88 (the total of one quantity)/the number of books 4 (the number of units of the other quantity)= 88/4= 22 dollars.
The price of 1 notebook= the price of the notebooks 9 (the total of one quantity)/the number of notebooks 3 (the number of units of the other quantity)= 9/3= 3 dollars.

The price of 2 books and 2 notebooks is 2*22+2*3=44+6=50 dollars.

Proportions

A proportion is an equality between 2 or more equivalent ratios. The proportion between a, b, c and d can be represented using a colon as a:b=c:d or as a fraction a/b=c/d.

The ways of representing proportions

We can write proportions in 2 ways, note that the way of writing doesn’t change the equality of the proportion.

There are same units above the divisor line and below the division line.

For example: A carrot cake recipe is composed of 3 cups of flour and 1 cup of sugar, if we put 3 cups of sugar in the bowl, what is the amount of flour we need to add?

Writing a flour above the divisor line and a sugar below the divisor line:

3 cups of flour / 1 cups of sugar = x cups of flour / 3 cups of sugar

3/1=x/3

x=9 if there are 3 cups of sugar in the bowl, we need to add 9 cups of flour

Writing a sugar above the divisor line and a flour below the divisor line:

1 cups of sugar / 3 cups of flour = 3 cups of sugar / x cups of flour

1/3=3/x

x=9 if there are 3 cups of sugar in the bowl, we need to add 9 cups of flour

There are same units at the nominator and the denominator of each ratio.

For example: A carrot cake recipe is composed of 3 cups of flour and 1 cup of sugar, if we put 3 cups of sugar in the bowl, what is the amount of flour we need to add?

One ratio includes flour and the other sugar:

3 cups of flour / x cups of flour = 1 cups of sugar / 3 cups of sugar

3/x=1/3

x=9 if there are 3 cups of sugar in the bowl, we need to add 9 cups of flour

We can also replace the numerators and the denominators:

x cups of flour/3 cups of flour = 3 cups of sugar/1 cups of sugar

x/3=3/1

x=9 if there are 3 cups of sugar in the bowl, we need to add 9 cups of flour