# 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 w____hat 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.

** 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. **

** 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