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Percentages on the SAT test

Studying percentages

On the SAT test percentages topic is part of problem solving and data analysis subscore that includes 9 advanced topics (see the full topics list on the top menu).

Percentages topic is the second topic of problem solving and data analysis subscore. Start learning problem solving and data analysis subscore with its first topic called ratios, rates and proportions.

Percentages topic is divided into sections from easy to difficult (the list of the sections appears on the left menu). Each section includes detailed explanations of the required material with examples followed by a variety of self-practice questions with solutions.

Finish studying heart of algebra subscore topics before you study this topic or any other problem solving and data analysis subscore topic. (Heart of algebra subscore includes basic algebra topics which knowledge is required for understanding problem solving and data analysis subscore topics).

Percentages- summary

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.

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.

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

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