# 1.5: Solve General Applications of Percent

- Page ID
- 152025

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- Translate and solve basic percent equations
- Solve applications of percent
- Find percent increase and percent decrease

### Translate and Solve Basic Percent Equations

In the past, you may have solved percent problems by setting them up as proportions. This method may continue to be used, or we will learn another method in this section that involves solving an equation. We will use the basic equation **amount = percent (as a decimal) x base**.

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 $80. We can solve this by plugging into the formula amount = percent (as a decimal) x base. Here, we have a base of $80, and they want to tip 20% (0.20 as a decimal). So we can plug into the formula to get amount = 0.20 x 80 = 16. So Aolani and her friends should tip $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 1.5.1

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

**Answer**-
Translate into algebra. Let $n=\phantom{\rule{0.2em}{0ex}}$the number. Multiply. $31.5$ is $\mathrm{35\%}$ of $90$

### Your Turn 1.5.1

What number is $\text{45\%}$ of $80?$

### Example 1.5.2

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

**Answer**-
Translate into algebra. Let $a\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}$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 $28.$

### Your Turn 1.5.2

$\text{150\%}$ of $78$ is what number?

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

### Example 1.5.3

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

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

### Your Turn 1.5.3

$17$ is $\text{25\%}$ of what number?

### Example 1.5.4

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

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

### Your Turn 1.5.4

$\text{7.5\%}$ of what number is $\text{\$1.95}?$

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

### Example 1.5.5

What percent of $36$ is $9?$

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

### Your Turn 1.5.5

What percent of $76$ is $57?$

### Example 1.5.6

$144$ is what percent of $96?$

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

### Your Turn 1.5.6

$110$ is what percent of $88?$

### 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 use the generic problem solving strategy for solving application problems listed below.

### How To

#### Solve an application

- Identify what you are asked to find and choose a variable to represent it.
- Write a sentence that gives the information to find it.
- Translate the sentence into an equation.
- Solve the equation using good algebra techniques.
- Check the answer in the problem and make sure it makes sense.
- 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 1.5.7

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?

**Answer**-
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 gives the information to find it. The tip is 18% of the total bill. Translate the sentence into an equation. Solve by multiplying. 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.

### Your Turn 1.5.7

Cierra and her sister enjoyed a special dinner in a restaurant, and the bill was $\text{\$81.50}.$ If she wants to leave $\text{18\%}$ of the total bill as her tip, how much should she leave?

### Example 1.5.8

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?

**Answer**-
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. Solve by dividing 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.

### Your Turn 1.5.8

One serving of wheat square cereal has $7$ grams of fiber, which is $\text{29\%}$ of the recommended daily amount. What is the total recommended daily amount of fiber?

### Example 1.5.9

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?

**Answer**-
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. Solve by dividing 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.

### Your Turn 1.5.9

Veronica is planning to make muffins from a mix. The package says each muffin will be $230$ calories and $60$ calories will be from fat. What percent of the total calories is from fat? (Round to the nearest whole percent.)

### 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

- Find the amount of increase.
- $\text{increase}=\text{new amount}-\text{original}$

- Find the percent increase as a percent of the original amount.

### Example 1.5.10

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.)

**Answer**-
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 38.5% increase over the old fees.

### Your Turn 1.5.10

In $2011,$ the IRS increased the deductible mileage cost to $55.5$ cents from $51$ cents. Find the percent increase. (Round to the nearest tenth of a percent.)

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

- Find the amount of decrease.
- $\text{decrease}=\text{original amount}-\text{new}$

- Find the percent decrease as a percent of the original amount.

### Example 1.5.11

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.

**Answer**-
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%.

### Your Turn 1.5.11

The population of one city was about $\mathrm{672,000}$ in $2010.$ The population of the city is projected to be about $\mathrm{630,000}$ in $2020.$ Find the percent decrease. (Round to the nearest tenth of a percent.)