Solve General Applications of Percent

6.2 Solve General Applications of Percent

The topics covered in this section are:

  1. Translate and solve basic percent equations
  2. Solve applications of percent
  3. Find percent increase and percent decrease

6.2.1 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 $ \$ 80$. They wanted to leave a $20 \%$ tip. What amount would the tip be?

To solve this, we want to find what amount is $20 \%$ of $ \$80$. The $ \$ 80$ is called the base. The amount of the tip would be $0.20(80)$, or $ \$ 16$. See Figure 6.6. To find the amount of the tip, we multiplied the percent by the base.

The figure shows a customer copy of a restaurant receipt with the amount of the bill, $80, and the amount of the tip, $16. There is a group of bills totaling $16.
Figure 6.6 A $20 \%$ tip for an $ \$80$ restaurant bill comes out to $ \$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 1

What number is $35 \%$ of 90?

Solution

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

Example 2

$125 \%$ of $28$ is what number?

Solution

Translate into algebra. Let $a = \mathrm{the\ number}$..
Multiply.$35=a$
$125 \%$ of $28$ is $35$.

Remember that a percent over $100$ is a number greater than $1$. We found that $125 \%$ of $28$ is $35$, which is greater than $28$.

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

Example 3

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

Solution

Translate into algebra. Let $b = \mathrm{the\ number}$..
Divide both sides by $0.75$.$\large \frac{36}{0.75} = \frac{0.75b}{0.75}$
Simplify.$48=b$
$36$ is $75 \%$ of $48$.

Example 4

$6.5 \%$ of what number is $ \$ 1.17$?

Solution

Translate into algebra. Let $b = \mathrm{the\ number}$..
Divide both sides by $0.065$.$\large \frac{0.065n}{0.065} = \frac{1.17}{0.065}$
Simplify.$n=18$
$6.5 \%$ of $ \$ 18$ is $ \$1.17$.

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

Example 5

What percent of $36$ is $9$?

Solution

Translate into algebra. Let $p = \mathrm{the\ percent}$..
Divide by $36$.$\large \frac{36p}{36} = \frac{9}{36}$
Simplify.$p= \frac{1}{4}$
Convert to decimal form.$p=0.25$
Convert to percent.$p=25 \%$
$25 \%$ of $36$ is $9$.

Example 6

$144$ is what percent of $96$?

Solution

Translate into algebra. Let $p = \mathrm{the\ percent}$..
Divide by $96$.$\large \frac{144}{96} = \frac{96p}{96}$
Simplify.$1.5=p$
Convert to percent.$150 \% = p$
$144$ is $150 \%$ of $96$.

6.2.2 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.

HOW TO: Solve an application

  1. Identify what you are asked to find and choose a variable to represent it.
  2. Write a sentence that gives the information to find it.
  3. Translate the sentence into an equation.
  4. Solve the equation using good algebra techniques.
  5. Check the answer in the problem and make sure it makes sense.
  6. Write a complete sentence that answers the question.

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 7

Dezohn and his girlfriend enjoyed a dinner at a restaurant, and the bill was $ \$ 68.50$. They want to leave an $18 \%$ tip. If the tip will be $18 \%$ 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 $t= \mathrm{the\ amount\ of\ tip}$
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.$t=12.33$
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 8

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

The figures shows the nutrition facts for cereal.
Solution

What are you asked to find?the total amount of potassium recommended
Choose a variable to represent it.Let $a= \mathrm{the\ total\ amount\ of\ potassium}$
Write a sentence that give the information to find it.$85$ mg is $2 \%$ of the total amount.
Translate the sentence into an equation..
Divide both sides by $0.02$.$\large \frac{85}{0.02} = \frac{0.02a}{0.02}$
Simplify.$4,250 = a$
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 9

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 $p= \mathrm{percent\ from\ fat}$
Write a sentence that give the information to find it.What percent of $480$ is $240$?
Translate the sentence into an equation..
Divide both sides by $480$.$\large \frac{p \cdot 480}{480} = \frac{240}{480}$
Simplify.$p=0.5$
Convert to percent form.$p=50 \%$
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.

6.2.3 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.

HOW TO: Find Percent Increase.

  1. Find the amount of increase.
    • $\mathrm{increase} = \mathrm{new\ amount} – \mathrm{original\ amount}$
  2. Find the percent increase as a percent of the original amount.

Example 10

In $2011$, the California governor proposed raising community college fees from $ \$ 26$ per unit to $ \$ 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 $p= \mathrm{percent}$
Find the amount of increase..
Find the percent increase.The increase is what percent of the original amount?
Translate to an equation..
Divide both sids by $26$.$\large \frac{10}{26} = \frac{26p}{26}$
Round to the nearest thousandth.$0.384=p$
Convert to percent form.$38.4 \% =p$
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.

HOW TO: Find percent decrease.

  1. Find the amount of decrease.
    • $\mathrm{decrease} = \mathrm{original\ amount} – \mathrm{new\ amount}$
  2. Find the percent decrease as a percent of the original amount.

Example 11

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

Solution

What are you asked to find?the percent decrease
Choose a variable to represent it.Let $p= \mathrm{percent}$
Find the amount of decrease..
Find the percent decrease.The decrease is what percent of the original amount?
Translate to an equation..
Divide both sids by $3.71$.$\large \frac{0.07}{3.71} = \frac{3.71p}{3.71}$
Round to the nearest thousandth.$0.019=p$
Convert to percent form.$1.9 \% =p$
Write a complete sentence.The price of gas decrease $1.9 \%$
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