6.2 Solve General Applications of Percent

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 prealgebra 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 Figure 6.6. To find the amount of the tip, we multiplied the percent by the base.

Figure 6.6 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 6.14

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

Solution

Translate into algebra. Let $n=$the number.
Multiply.
	$31.5$ is $\mathrm{35\%}$ of $90$

Example 6.15

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

Solution

Translate into algebra. Let $a=$the number.
Multiply.
	$\mathrm{125\%}$ 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 6.16

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

Solution

Translate. Let $b=$ the number.
Divide both sides by 0.75.
Simplify.

Example 6.17

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

Solution

Translate. Let $b=$ the number.
Divide both sides by 0.065.
Simplify.

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

Example 6.18

What percent of $36$ is $9?$

Solution

Translate into algebra. Let $p=$ the percent.
Divide by 36.
Simplify.
Convert to decimal form.
Convert to percent.

Example 6.19

$144$ is what percent of $96?$

Solution

Translate into algebra. Let $p=$ the percent.
Divide by 96.
Simplify.
Convert to percent.

Solve Applications of Percent

Many applications of percent occur in our daily lives, such as tips, sales tax, discount, and interest. To solve these applications we'll translate to a basic percent equation, just like those we solved in the previous examples in this section. Once you translate the sentence into a percent equation, you know how to solve it.

We will update the strategy we used in our earlier applications to include equations now. Notice that we will translate a sentence into an equation.

Now that we have the strategy to refer to, and have practiced solving basic percent equations, we are ready to solve percent applications. Be sure to ask yourself if your final answer makes sense—since many of the applications we'll solve involve everyday situations, you can rely on your own experience.

Example 6.20

Dezohn and his girlfriend enjoyed a dinner at a restaurant, and the bill was $\text{\$68.50}.$ They want to leave an $\text{18\%}$ tip. If the tip will be $\text{18\%}$ of the total bill, how much should the tip be?

Solution

What are you asked to find?	the amount of the tip
Choose a variable to represent it.	Let $t=$ amount of tip.
Write a sentence that give the information to find it.	The tip is 18% of the total bill.
Translate the sentence into an equation.
Multiply.
Check. Is this answer reasonable?
If we approximate the bill to $70 and the percent to 20%, we would have a tip of $14. So a tip of $12.33 seems reasonable.
Write a complete sentence that answers the question.	The couple should leave a tip of $12.33.

Example 6.21

The label on Masao's breakfast cereal said that one serving of cereal provides $85$ milligrams (mg) of potassium, which is $\text{2\%}$ of the recommended daily amount. What is the total recommended daily amount of potassium?

Solution

What are you asked to find?	the total amount of potassium recommended
Choose a variable to represent it.	Let $a=$ total amount of potassium.
Write a sentence that gives the information to find it.	85 mg is 2% of the total amount.
Translate the sentence into an equation.
Divide both sides by 0.02.
Simplify.
Check: Is this answer reasonable?
Yes. 2% is a small percent and 85 is a small part of 4,250.
Write a complete sentence that answers the question.	The amount of potassium that is recommended is 4250 mg.

Example 6.22

Mitzi received some gourmet brownies as a gift. The wrapper said each brownie was $480$ calories, and had $240$ calories of fat. What percent of the total calories in each brownie comes from fat?

Solution

What are you asked to find?	the percent of the total calories from fat
Choose a variable to represent it.	Let $p=$ percent from fat.
Write a sentence that gives the information to find it.	What percent of 480 is 240?
Translate the sentence into an equation.
Divide both sides by 480.
Simplify.
Convert to percent form.
Check. Is this answer reasonable?
Yes. 240 is half of 480, so 50% makes sense.
Write a complete sentence that answers the question.	Of the total calories in each brownie, 50% is fat.

Find Percent Increase and Percent Decrease

People in the media often talk about how much an amount has increased or decreased over a certain period of time. They usually express this increase or decrease as a percent.

To find the percent increase, first we find the amount of increase, which is the difference between the new amount and the original amount. Then we find what percent the amount of increase is of the original amount.

Example 6.23

In $2011,$ the California governor proposed raising community college fees from $\text{\$26}$ per unit to $\text{\$36}$ per unit. Find the percent increase. (Round to the nearest tenth of a percent.)

Solution

What are you asked to find?	the percent increase
Choose a variable to represent it.	Let $p=$ percent.
Find the amount of increase.
Find the percent increase.	The increase is what percent of the original amount?
Translate to an equation.
Divide both sides by 26.
Round to the nearest thousandth.
Convert to percent form.
Write a complete sentence.	The new fees represent a 38.4% increase over the old fees.

Finding the percent decrease is very similar to finding the percent increase, but now the amount of decrease is the difference between the original amount and the final amount. Then we find what percent the amount of decrease is of the original amount.

Example 6.24

The average price of a gallon of gas in one city in June $2014$ was $\text{\$3.71}.$ The average price in that city in July was $\text{\$3.64}.$ Find the percent decrease.

Solution

What are you asked to find?	the percent decrease
Choose a variable to represent it.	Let $p=$ percent.
Find the amount of decrease.
Find the percent of decrease.	The decrease is what percent of the original amount?
Translate to an equation.
Divide both sides by 3.71.
Round to the nearest thousandth.
Convert to percent form.
Write a complete sentence.	The price of gas decreased 1.9%.