Mathematics » Introducing Percents » Solve General Applications of Percent

How to Translate and Solve Basic Percent Equations

How to Translate and Solve Basic Percent Equations

We will solve percent equations by using the methods we used to solve equations with fractions or decimals. In the past, you may have solved percent problems by setting them up as proportions. That was the best method available when you did not have the tools of algebra. Now as a pre-algebra student, you can translate word sentences into algebraic equations, and then solve the equations.

We’ll look at a common application of percent—tips to a server at a restaurant—to see how to set up a basic percent application.

When Aolani and her friends ate dinner at a restaurant, the bill came to \(\text{\$80}.\) They wanted to leave a \(\text{20%}\) tip. What amount would the tip be?

To solve this, we want to find what amount is \(\text{20%}\) of \(\text{\$80}.\) The \(\text{\$80}\) is called the base. The amount of the tip would be \(0.20\left(80\right),\) or \(\text{\$16}\) See the figure below. To find the amount of the tip, we multiplied the percent by the base.

How to Translate and Solve Basic Percent Equations

A \(\text{20%}\) tip for an \(\text{\$80}\) restaurant bill comes out to \(\text{\$16}.\)

In the next examples, we will find the amount. We must be sure to change the given percent to a decimal when we translate the words into an equation.

Example

What number is \(\text{35%}\) of \(90?\)

Solution

Translate into algebra. Let \(n=\phantom{\rule{0.2em}{0ex}}\)the number.How to Translate and Solve Basic Percent Equations
Multiply.How to Translate and Solve Basic Percent Equations
 \(31.5\) is \(35%\) of \(90\)

Example

\(\text{125%}\) of \(28\) is what number?

Solution

Translate into algebra. Let \(a\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}\)the number.How to Translate and Solve Basic Percent Equations
Multiply.How to Translate and Solve Basic Percent Equations
 \(125%\) of \(28\) is \(35\).

Remember that a percent over \(100\) is a number greater than \(1.\) We found that \(\text{125%}\) of \(28\) is \(35,\) which is greater than \(28.\)

In the next examples, we are asked to find the base.

Example

Translate and solve: \(36\) is \(\text{75%}\) of what number?

Solution

Translate. Let \(b=\) the number.How to Translate and Solve Basic Percent Equations
Divide both sides by 0.75.How to Translate and Solve Basic Percent Equations
Simplify.How to Translate and Solve Basic Percent Equations

Example

\(\text{6.5%}\) of what number is \(\text{\$1.17}?\)

Solution

Translate. Let \(b=\) the number.How to Translate and Solve Basic Percent Equations
Divide both sides by 0.065.How to Translate and Solve Basic Percent Equations
Simplify.How to Translate and Solve Basic Percent Equations

In the next examples, we will solve for the percent.

Example

What percent of \(36\) is \(9?\)

Solution

Translate into algebra. Let \(p=\) the percent.How to Translate and Solve Basic Percent Equations
Divide by 36.How to Translate and Solve Basic Percent Equations
Simplify.How to Translate and Solve Basic Percent Equations
Convert to decimal form.How to Translate and Solve Basic Percent Equations
Convert to percent.How to Translate and Solve Basic Percent Equations

Example

\(144\) is what percent of \(96?\)

Solution

Translate into algebra. Let \(p=\) the percent.How to Translate and Solve Basic Percent Equations
Divide by 96.How to Translate and Solve Basic Percent Equations
Simplify.How to Translate and Solve Basic Percent Equations
Convert to percent.How to Translate and Solve Basic Percent Equations

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