Percent Equations

At the bottom of this page you will find a section for making practice problems along with the answers. Additionally worksheets and answer keys can also be generated and printed. All answers are rounded to 2 decimal places when necessary.

Here are two methods for solving percent equations:

Method #1
Put the problem into "(percent)(of base)(is amount)" format and solve it -- use inverse operations.
This is the equation pattern (formula):    percent•of = is

• When given the percent value be sure to convert it to decimal or fraction form before solving the equation.
• When solving for the percent value your answer will be in decimal form. Since the problem asked for the percent value you will have to convert your answer to percent form when you are finished so the answer is in the required form.

Example -- Solving for the 'is value' (the 'is value' is also called the amount).
 Problem What is 23% of 200? Or stated another way, 23% of 200 is What? Or stated another way (the reversed way), 23% is what amount of 200? Write the formula: percent • of = is Under percent put the percent value, under of put the of value, and under is put the is value 23% • 200 = x Convert 23% to decimal form. (Percent form can't be used in equations.) 23% = .23 Replace the percent value in the equation with its decimal value. .23 • 200 = x The equation is already solved, just do the math. 46 = x Answer 23% of 200 is 46.

Example -- Solving for the 'of value' (the 'of value' is also called the base).
 Problem 46 is 23% of what? Or stated another way, 23% of what is 46? Write the formula: percent • of = is Under percent put the percent value, under of put the of value, and under is put the is value 23% • what = 46 Convert 23% to decimal form. (Percent form can't be used in equations.) 23% = .23 Replace the percent value in the equation with its decimal value. .23 • x = 46 Solve the equation by dividing both sides by .23. 200 = x Answer 23% of 200 is 46.

Example -- Solving for the 'percent value'.
 Problem 46 is what percent of 200? Or stated another way, What percent of 200 is 46? Or stated another way (the reversed way), What percent is 46 of 200? Write the formula: percent • of = is Under percent put the percent value, under of put the of value, and under is put the is value x • 200 = 46 Solve the equation by dividing both sides by 200. x = .23 The answer is in decimal form but needs to be in percent form because the quesiton asked for the percent. So converted to percent form the answer is: 23%

See the "Using method #1" section below for some examples.

Method #2
Translate the question to a proportion and solve it -- cross multiply and use inverse operations.
If you are given the value of the percent just plug it in to the proportion. It will be a percent in fraction form automatically because of the way the proportion is set up. See the "Using method #2" section below for some examples.

Notes on percent equations
Percent equations are sometimes described using base, amount, and, percent (sometimes percent is called part). This may seem confusing but you can think abouut it this way. You have a bag of doughnuts. Your friend wants half of them because your friend paid half the cost. The bag of doughnuts is the base (the thing your friend wants part of). The amount is the number of doughnuts your friend gets. You can't determine what number the amount is until you know how big the base is (how many doughnuts in the bag). You do know what part (percent) your friend wants though, 1/2, you just don't know how many (the amount) 1/2 is because you don't know how big (how many) doughnuts are in the bag.

Lets summarize, the base is the bag of doughnuts, the thing that is going to have a piece taken away from it. The percent (part) is how big that piece is going to be (what fraction of the bag). The amount is how many doughnuts your friend gets. Even though your friend gets half the bag no matter what, the amount (the number of doughnuts) your freind gets depends on how big the bag is. If the bag holds 24 doughnuts your friend gets 12, if it holds 6 your friend gets 3. Your friend gets half the bag every time but the amount (of doughnuts in that half bag) depends on the size of the bag.
The equation looks like this  percent•base = amount.

When solving percent equations you are either solving for the base, the amount, or the percent. When you are solving for the percent be aware that the percent values calculated using method #1 are in decimal or fraction form depending on how you solve the problem. For example "what percent of 100 is 50?" translates to   x•100 = 50. The answer is 1/2 as a reduced fraction and .5 as a decimal. Either way you solve it you will need to convert the answer from method #1 to percent form because the question asks for the answer to be in percent form. For this example the answer is 50%.

The percent values calculated using method #2 are in percent form already. The way the proportion is set up requires that you find a number that you put over 100 on one side to equal the ratio (fraction) on the other side. Any fraction with 100 on the bottom is a percent by definition because percent means divide by 100. So 17/100 is a percent in fraction form. If you do the division you get .17 which is now in decimal form. If you replace the /100 with % you convert 17/100 to 17% which is in percent form. All you need to do with the answer to method #2 when you are solving for the percent is put a % sign after the answer.

Using method #1
If you want to translate the problem into an equation use this method.
One pattern for percent equations is "(percent)(of base)(is amount)". The parenthesis are used to separate the key phrases in the pattern they do not imply multiplication here. This is the most common pattern for percent equations and it can be translated directly into an equation and solved.
Another pattern is "(percent)(is amount)(of base)". This is not very common but it is used sometimes when the percent is not know. An example is "What percent is 5 of 50?". To solve this the key phrases have to be rearranged so that the problem looks like "(percent)(of base)(is amount)". The (percent) phrase and the (of base) phrase must be next to each other before translating into an equation. Swaping the locations of the 'is phrase' and the 'of phrase' results in "What percent of 50 is 5?". This is now ready to translate and solve.

Click the "show examples" button for a new set of examples.

Using method #2
A different method is to translate the problem into a proportion.
Look at the problem and pick out the 'is phrase', the 'of phrase', and the 'percent phrase'. The nice thing about method #2 is that you don't have to be concerned about the order of the phrases, just put them in the right spots in the proportion, cross multiply and solve the equation.

is

of

=
percent

100
replace 'is' with -- the word or number after 'is'
replace 'of ' with -- the word or number after 'of '
replace 'percent' with -- the word or number in front of the '%' symbol

Click the "show examples" button for a new set of examples.

Problem # 1