In mathematics, a percentage is a number or ratio that represents a fraction of 100. It is often denoted by the symbol "%" or simply as "percent" or "pct." For example, 35% is equivalent to the decimal 0.35, or the fraction 35 100.
Percentage Formula
Although the percentage formula can be written in different forms, it is essentially an algebraic equation involving three values.
P × V1 = V2
P is the percentage, V1 is the first value that the percentage will modify, and V2 is the result of the percentage operating on V1. The calculator provided automatically converts the input percentage into a decimal to compute the solution. However, if solving for the percentage, the value returned will be the actual percentage, not it's decimal representation.
EX: P × 30 = 1.5
P =
1.5
30
= 0.05 × 100 = 5%
If solving manually, the formula requires the percentage in decimal form, so the solution for P needs to be multiplied by 100 in order to convert it to a percent. This is essentially what the calculator above does, except that it accepts inputs in percent rather than decimal form.
Percentage Difference Formula
The percentage difference between two values is calculated by dividing the absolute value of the difference between two numbers by the average of those two numbers. Multiplying the result by 100 will yield the solution in percent, rather than decimal form. Refer to the equation below for clarification.
Percentage Difference =
|V1 - V2|
(V1 + V2)/2
× 100
EX:
|10 - 6|
(10 + 6)/2
=
4
8
= 0.5 = 50%
Percentage Change Formula
Percentage increase and decrease are calculated by computing the difference between two values and comparing that difference to the initial value. Mathematically, this involves using the absolute value of the difference between two values, and dividing the result by the initial value, essentially calculating how much the initial value has changed.
The percentage increase calculator above computes an increase or decrease of a specific percentage of the input number. It basically involves converting a percent into its decimal equivalent, and either subtracting (decrease) or adding (increase) the decimal equivalent from and to 1, respectively. Multiplying the original number by this value will result in either an increase or decrease of the number by the given percent. Refer to the example below for clarification.
EX: 500 increased by 10% (0.1)
500 × (1 + 0.1) = 550
500 decreased by 10%
500 × (1 – 0.1) = 450
Calculator Use
Find a percentage or work out the percentage given numbers and percent values. Use percent formulas to figure out percentages and unknowns in equations. Add or subtract a percentage from a number or solve the equations.
How to Calculate Percentages
There are many formulas for percentage problems. You can think of the most basic as X/Y = P x 100. The formulas below are all mathematical variations of this formula.
Let's explore the three basic percentage problems. X and Y are numbers and P is the percentage:
Find P percent of X
Find what percent of X is Y
Find X if P percent of it is Y
Read on to learn more about how to figure percentages.
1. How to calculate the percentage of a number. Use the percentage formula: P% * X = Y
Example: What is 10% of 150?
Convert the problem to an equation using the percentage formula: P% * X = Y
P is 10%, X is 150, so the equation is 10% * 150 = Y
Convert 10% to a decimal by removing the percent sign and dividing by 100: 10/100 = 0.10
Substitute 0.10 for 10% in the equation: 10% * 150 = Y becomes 0.10 * 150 = Y
Do the math: 0.10 * 150 = 15
Y = 15
So 10% of 150 is 15
Double check your answer with the original question: What is 10% of 150? Multiply 0.10 * 150 = 15
2. How to find what percent of X is Y. Use the percentage formula: Y/X = P%
Example: What percent of 60 is 12?
Convert the problem to an equation using the percentage formula: Y/X = P%
X is 60, Y is 12, so the equation is 12/60 = P%
Do the math: 12/60 = 0.20
Important! The result will always be in decimal form, not percentage form. You need to multiply the result by 100 to get the percentage.
Converting 0.20 to a percent: 0.20 * 100 = 20%
So 20% of 60 is 12.
Double-check your answer with the original question: What percent of 60 is 12? 12/60 = 0.20, and multiplying by 100 to get a percentage, 0.20 * 100 = 20%
3. How to find X if P percent of it is Y. Use the percentage formula Y/P% = X
Example: 25 is 20% of what number?
Convert the problem to an equation using the percentage formula: Y/P% = X
Y is 25, P% is 20, so the equation is 25/20% = X
Convert the percentage to a decimal by dividing by 100.
Converting 20% to a decimal: 20/100 = 0.20
Substitute 0.20 for 20% in the equation: 25/0.20 = X
Do the math: 25/0.20 = X
X = 125
So 25 is 20% of 125
Double-check your answer with the original question: 25 is 20% of what number? 25/0.20 = 125
Remember: How to convert a percentage to a decimal
Remove the percentage sign and divide by 100
15.6% = 15.6/100 = 0.156
Remember: How to convert a decimal to a percentage
Multiply by 100 and add a percentage sign
0.876 = 0.876 * 100 = 87.6%
Percentage Problems
There are nine variations on the three basic problems involving percentages. See if you can match your problem to one of the samples below. The problem formats match the input fields in the calculator above. Formulas and examples are included.
What is P percent of X?
Written as an equation: Y = P% * X
The 'what' is Y that we want to solve for
Remember to first convert a percentage to a decimal, dividing by 100
Solution: Solve for Y using the percentage formula
Y = P% * X
Example: What is 10% of 25?
Written using the percentage formula: Y = 10% * 25
First convert percentage to a decimal 10/100 = 0.1
Y = 0.1 * 25 = 2.5
So 10% of 25 is 2.5
Y is what percent of X?
Written as an equation: Y = P% ? X
The 'what' is P% that we want to solve for
Divide both sides by X to get P% on one side of the equation
Y ÷ X = (P% ? X) ÷ X becomes Y ÷ X = P%, which is the same as P% = Y ÷ X
Solution: Solve for P% using the percentage formula
P% = Y ÷ X
Example: 12 is what percent of 40?
Written using the formula: P% = 12 ÷ 40
P% = 12 ÷ 40 = 0.3
Convert the decimal to percent
P% = 0.3 × 100 = 30%
So 12 is 30% of 40
Y is P percent of what?
Written as an equation: Y = P% * X
The 'what' is X that we want to solve for
Divide both sides by P% to get X on one side of the equation
Y ÷ P% = (P% × X) ÷ P% becomes Y ÷ P% = X, which is the same as X = Y ÷ P%
Solution: Solve for X using the percentage formula
X = Y ÷ P%
Example: 9 is 60% of what?
Writen using the formula: X = 9 ÷ 60%
Convert percent to decimal
60% ÷ 100 = 0.6
X = 9 ÷ 0.6
X = 15
So 9 is 60% of 15
What percent of X is Y?
Written as an equation: P% * X = Y
The 'what' is P% that we want to solve for
Divide both sides by X to get P% on one side of the equation
(P% * X) ÷ X = Y ÷ X becomes P% = Y ÷ X
Solution: Solve for P% using the percentage formula
P% = Y ÷ X
Example: What percent of 27 is 6?
Written using the formula: P% = 6 ÷ 27
6 ÷ 27 = 0.2222
Convert decimal to percent
P% = 0.2222 × 100
P% = 22.22%
So 22.22% of 27 is 6
P percent of what is Y?
Written as an equation: P% × X = Y
The 'what' is X that we want to solve for
Divide both sides by P% to get X on one side of the equation
(P% × X) ÷ P% = Y ÷ P% becomes X = Y ÷ P%
Solution: Solve for X using the percentage formula
X = Y ÷ P%
Example: 20% of what is 7?
Written using the formula: X = 7 ÷ 20%
Convert the percent to a decimal
20% ÷ 100 = 0.2
X = 7 ÷ 0.2
X = 35
So 20% of 35 is 7.
P percent of X is what?
Written as an equation: P% * X = Y
The 'what' is Y that we want to solve for
Solution: Solve for Y using the percentage formula
Y = P% * X
Example: 5% of 29 is what?
Written using the formula: 5% * 29 = Y
Convert the percent to a decimal
5% ÷ 100 = 0.05
Y = 0.05 * 29
Y = 1.45
So 5% of 29 is 1.45
Y of what is P percent?
Written as an equation: Y / X = P%
The 'what' is X that we want to solve for
Multiply both sides by X to get X out of the denominator
(Y / X) * X = P% * X becomes Y = P% * X
Divide both sides by P% so that X is on one side of the equation
Y ÷ P% = (P% * X) ÷ P% becomes Y ÷ P% = X
Solution: Solve for X using the percentage formula
X = Y ÷ P%
Example: 4 of what is 12%?
Written using the formula: X = 4 ÷ 12%
Solve for X: X = Y ÷ P%
Convert the percent to a decimal
12% ÷ 100 = 0.12
X = 4 ÷ 0.12
X = 33.3333
4 of 33.3333 is 12%
What if X is P percent?
Written as an equation: Y / X = P%
The 'what' is Y that we want to solve for
Multiply both sides by X to get Y on one side of the equation
(Y ÷ X) * X = P% * X becomes Y = P% * X
Solution: Solve for Y using the percentage formula
Y = P% * X
Example: What if 25 is 11%?
Written using the formula: Y = 11% * 25
Convert the percent to a decimal
11% ÷ 100 = 0.11
Y = 0.11 * 25
Y = 2.75
So 2.75 of 25 is 11%
Y of X is what percent?
Written as an equation: Y / X = P%
The 'what' is P% that we want to solve for
Solution: Solve for P% using the percentage formula
P% = Y / X
Example: 9 of 13 is what percent?
Written using the formula: P% = Y / X
9 ÷ 13 = P%
9 ÷ 13 = 0.6923
Convert decimal to percent by multiplying by 100
0.6923 * 100 = 69.23%
9 ÷ 13 = 69.23%
So 9 of 13 is 69.23%
