Solving Percent Problems

Percents are a ratio of a number and `100`, so they are easier to compare than fractions, as they always have the same denominator, `100`. A store may have a `10%` off sale. The amount saved is always the same portion or fraction of the price, but a higher price means more money is taken off. Interest rates on a saving account work in the same way. The more money you put in your account, the more money you get in interest. It’s helpful to understand how these percents are calculated.

Jeff has a coupon at the Guitar Store for `15%` off any purchase of `$100` or more. He wants to buy a used guitar that has a price tag of `$220` on it. Jeff wonders how much money the coupon will take off the original `$220` price.

Problems involving percents have any three quantities to work with: the percentA ratio that compares a number to `100`. “Per cent” means “per `100`,” or “how many out of `100`.”, the amountIn a percent problem, the portion of the whole corresponding to the percent., and the baseIn a percent problem, the base represents how much should be considered `100%` (the whole); in exponents, the base is the value that is raised to a power when a number is written in exponential notation. In the example of `5^3`, `5` is the base. .

The percent has the percent symbol (%) or the word “percent.” In the problem above, `15%` is the percent off the purchase price.

The base is the whole amount. In the problem above, the whole price of the guitar is `$220`, which is the base.

The amount is the number that relates to the percent. It is always part of the whole. In the problem above, the amount is unknown. Since the percent is the percent off, the amount will be the amount off of the price.

You will return to this problem a bit later. The following examples show how to identify the three parts: the percent, the base, and the amount.

Example
Problem	Identify the percent, amount, and base in this problem. `30` is `20%` of what number?
	Percent: The percent is the number with the % symbol: `20%`.
	Base: The base is the whole amount, which in this case is unknown.
	Amount: The amount based on the percent is `30.`
Answer	`"Percent" = 20%` `"Amount" = 30` `"Base" = "unknown"`

The previous problem states that `30` is a portion of another number. That means `30` is the amount. Note that this problem could be rewritten: `20%` of what number is `30`?

Percent problems can be solved by writing equations. An equation uses an equal sign (=) to show that two mathematical expressions have the same value.

Percents are fractions, and just like fractions, when finding a percent (or fraction, or portion) of another amount, you multiply.

In the examples below, the unknown is represented by the letter `n`. The unknown can be represented by any letter or a box An image of an empty box.

or even a question mark.

Example
Problem	Write an equation that represents the following problem. `30` is `20%` of what number?
	`20%` of what number is `30`?	Rewrite the problem in the form “percent of base is amount.”
	Percent is: `20%` Base is: unknown Amount is: `30`	Identify the percent, the base, and the amount.
	`text(Percent)text(Base)=text(Amount)` `20% n = 30`	Write the percent equation. using `n` for the base, which is the unknown value.
Answer	`20% * n = 30`

Once you have an equation, you can solve it and find the unknown value. To do this, think about the relationship between multiplication and division. Look at the pairs of multiplication and division facts below, and look for a pattern in each row.

Multiplication	Division
`2 * 3 = 6`	`6-:2 = 3`
`8 * 5 = 40`	`40-:8 = 5`
`7 * 4 = 28`	`28-:7 = 4`
`6 * 9 = 54`	`54-:6 = 9`

Multiplication and division are inverse operations. What one does to a number, the other “undoes.”

When you have an equation such as `20% * n = 30`, you can divide `30` by `20%` to find the unknown: `n = 30 -:20%`.

You can solve this by writing the percent as a decimal or fraction and then dividing.

Example
Problem	What percent of `72` is `9` ?
	Percent: unknown Base: `72` Amount: `9`	Identify the percent, base, and amount.
	`n * 72 = 9`	Write the percent equation: `"Percent" * "Base" = "Amount"`. Use `n` for the unknown (percent).
	`n = 9-:72`	Divide to undo the multiplication of `n` times `72`.
		Divide `9` by `72` to find the value for `n`, the unknown.
	`n = 0.125` `n = 12.5%`	Move the decimal point two places to the right to write the decimal as a percent.
Answer	`12.5%` of `72` is `9`.

You can estimate to see if the answer is reasonable. Use `10%` and `20%`, numbers close to `12.5%`, to see if they get you close to the answer.

Notice that `9` is between `7.2` and `14.4`, so `12.5%` is reasonable since it is between `10%` and `20%`.

Example
Problem	What is `110%` of `24` ?
	Percent: `110%` Base: `24` Amount: unknown	Identify the percent, the base, and the amount.
	`110% * 24 = n`	Write the percent equation. `"Percent" * "Base" = "Amount"`. The amount is unknown, so use `n`.
	`1.10 * 24 = n` `1.10 * 24 = 26.4 = n`	Write the percent as a decimal by moving the decimal point two places to the left. Multiply `24` by `1.10` or `1.1`.
Answer	`26.4` is `110%` of `24`.

This problem is a little easier to estimate. `100%` of `24` is `24`. And `110%` is a little bit more than `24`. So, `26.4` is a reasonable answer.

Percent problems can also be solved by writing a proportion. A proportion is an equation that sets two ratios or fractions equal to each other. With percent problems, one of the ratios is the percent, written as `n/100`. The other ratio is the amount to the base.

Example
Problem	Write a proportion to find the answer to the following question. `30` is `20%` of what number?
	`20/100=("amount")/("base")`	The percent in this problem is `20%`. Write this percent in fractional form, with `100` as the denominator.
	`20/100=30/n`	The percent is written as the ratio `20/100`, the amount is `30`, and the base is unknown.
	`20 * n = 30 * 100` `20 * n = 3,000` `n = 3,000-:20` `n = 150`	Cross multiply and solve for the unknown, `n`, by dividing `3,000` by `20`.
Answer	`30` is `20%` of `150`.

Example
Problem	What percent of `72` is `9` ?
	`text(Percent)=("amount")/("base")`
	`n/100=9/72`	The percent is the ratio of `n` to `100`. The amount is `9`, and the base is `72`.
	`n * 72 = 9 * 100` `n * 72 = 900` `n = 900-:72` `n = 12.5`	Cross multiply and solve for `n` by dividing `900` by `72`.
Answer	`12.5%` of `72` is `9`.	The percent is `12.5/100=12.5%`.

Example
Problem	What is `110%` of `24`?
	`text(Percent)=("amount")/("base")`
	`110/100=n/24`	The percent is the ratio `110/100`. The amount is unknown, and the base is `24`.
	`24 * 110 = 100 * n` `2,640-:100=n` `26.4 = n`	Cross multiply and solve for `n` by dividing `2,640` by `100`.
Answer	`26.4` is `110%` of `24`.

Let’s go back to the problem that was posed at the beginning. You can now solve this problem as shown in the following example.

Example
Problem	Jeff has a coupon at the Guitar Store for `15%` off any purchase of `$100` or more. He wants to buy a used guitar that has a price tag of `$220` on it. Jeff wonders how much money the coupon will take off of the `$220` original price.
	How much is `15%` of `$220`?	Simplify the problems by eliminating extra words.
	Percent: `15%` Base: `220` Amount: `n`	Identify the percent, the base, and the amount.
	`15% * 220 = n`	Write the percent equation. `"Percent" * "Base" = "Amount"`
	`0.15 * 220 = 33`	Convert `15%` to `0.15`, then multiply by `220`. `15%` of `$220` is `$33`.
Answer	The coupon will take `$33` off the original price.

You can estimate to see if the answer is reasonable. Since `15%` is half way between `10%` and `20%`, find these numbers.

Example
Problem	Evelyn bought some books at the local bookstore. Her total bill was `$31.50`, which included `5%` tax. How much did the books cost before tax?
	What number `+ 5%` of that number is `$31.50`? `105%` of what number `= 31.50`?	In this problem, you know that the tax of `5%` is added onto the cost of the books. So if the cost of the books is `100%`, the cost plus tax is `105%`.
	Percent: `105%` Base: `n` Amount: `31.50`	Identify the percent, the base, and the amount.
	`105% * n = 31.50`	Write the percent equation. `"Percent" * "Base" = "Amount"`.
	`1.05 * n = 31.50`	Convert `105%` to a decimal.
	`n = 31.50-:1.05 = 30`	Divide to undo the multiplication of `n` times `1.05`.
Answer	The books cost `$30` before tax.

Example
Problem	Susana worked `20` hours at her job last week. This week, she worked `35` hours. In terms of a percent, how much more did she work this week than last week?
	`35` is what percent of `20`?	Simplify the problem by eliminating extra words.
	Percent: `n` Base: `20` Amount: `35`	Identify the percent, the base, and the amount.
	`n * 20 = 35`	Write the percent equation. `"Percent" * "Base" = "Amount"`.
	`n = 35-:20`	Divide to undo the multiplication of `n` times `20`.
	`n = 1.75 = 175%`	Convert `1.75` to a percent.
Answer	Since `35` is `175%` of `20`, Susana worked `75%` more this week than she did last week. (You can think of this as, “Susana worked `100%` of the hours she worked last week, as well as `75%` more.”)

Percent problems have three parts: the percent, the base (or whole), and the amount. Any of those parts may be the unknown value to be found. To solve percent problems, you can use the equation, `"Percent" * "Base" = "Amount"`, and solve for the unknown numbers. Or, you can set up the proportion, `text(Percent)=("amount")/("base")`, where the percent is a ratio of a number to `100`. You can then use cross multiplication to solve the proportion.

Learning Objectives

Introduction

Parts of a Percent Problem

Solving with Equations

Using Proportions to Solve Percent Problems

Summary