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Solve General Applications of Percent

Translate and solve basic percent equations

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 $0.20(80),$ or $\text{\$16}$ See . To find the amount of the tip, we multiplied the percent by the base.

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

Try it.

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

Solution

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

Example

Try it.

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

Solution

Translate into algebra. Let $a\ =\$the number.
Multiply.
	$125\%$ 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

Try it.

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

Solution

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

Example

Try it.

$\text{6.5\%}$ 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

Try it.

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.

Condensed — the full section is in OpenStax Prealgebra 2e.

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

Try it.

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.

Condensed — the full section is in OpenStax Prealgebra 2e.

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

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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 38.5% 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

Try it.

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%.

Condensed — the full section is in OpenStax Prealgebra 2e.

Key Concepts

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. Write a complete sentence that answers the question.
6. Check the answer in the problem and make sure it makes sense.
Find percent increase.
1. Find the amount of increase:
  $\text{increase}=\text{new amount}-\text{original amount}$
2. Find the percent increase as a percent of the original amount.
Find percent decrease.
1. Find the amount of decrease.
  $\text{decrease}=\text{original amount}-\text{new amount}$
2. Find the percent decrease as a percent of the original amount.

Solve General Applications of Percent

Translate and Solve Basic Percent Equations

In the following exercises, translate and solve.

Try it.

What number is $\text{45\%}$ of $120?$

Solution

Try it.

What number is $\text{65\%}$ of $100?$

Try it.

What number is $\text{24\%}$ of $112?$

Solution

26.88

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What number is $\text{36\%}$ of $124?$

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$\text{250\%}$ of $65$ is what number?

Solution

162.5

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$\text{150\%}$ of $90$ is what number?

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$\text{800\%}$ of $2,250$ is what number?

Solution

18,000

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$\text{600\%}$ of $1,740$ is what number?

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$28$ is $\text{25\%}$ of what number?

Solution

112

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$36$ is $\text{25\%}$ of what number?

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$81$ is $\text{75\%}$ of what number?

Solution

108

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$93$ is $\text{75\%}$ of what number?

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$\text{8.2\%}$ of what number is $\text{\$2.87}?$

Solution

$35

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$\text{6.4\%}$ of what number is $\text{\$2.88}?$

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$\text{11.5\%}$ of what number is $\text{\$108.10}?$

Solution

$940

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$\text{12.3\%}$ of what number is $\text{\$92.25}?$

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What percent of $260$ is $78?$

Solution

30%

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What percent of $215$ is $86?$

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What percent of $1,500$ is $540?$

Solution

36%

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What percent of $1,800$ is $846?$

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$30$ is what percent of $20?$

Solution

150%

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$50$ is what percent of $40?$

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$840$ is what percent of $480?$

Solution

175%

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$790$ is what percent of $395?$

Solve Applications of Percents

In the following exercises, solve the applications of percents.

Try it.

Geneva treated her parents to dinner at their favorite restaurant. The bill was $\text{\$74.25}.$ She wants to leave $\text{16\%}$ of the total bill as a tip. How much should the tip be?

Solution

$11.88

Try it.

When Hiro and his co-workers had lunch at a restaurant the bill was $\text{\$90.50}.$ They want to leave $\text{18\%}$ of the total bill as a tip. How much should the tip be?

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Trong has $\text{12\%}$ of each paycheck automatically deposited to his savings account. His last paycheck was $\text{\$2,165}.$ How much money was deposited to Trong's savings account?

Solution

$259.80

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Cherise deposits $\text{8\%}$ of each paycheck into her retirement account. Her last paycheck was $\text{\$1,485}.$ How much did Cherise deposit into her retirement account?

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One serving of oatmeal has $8$ grams of fiber, which is $\text{33\%}$ of the recommended daily amount. What is the total recommended daily amount of fiber?

Solution

24.2 grams

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One serving of trail mix has $67$ grams of carbohydrates, which is $\text{22\%}$ of the recommended daily amount. What is the total recommended daily amount of carbohydrates?

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A bacon cheeseburger at a popular fast food restaurant contains $2,070$ milligrams (mg) of sodium, which is $\text{86\%}$ of the recommended daily amount. What is the total recommended daily amount of sodium?

Solution

2,407 mg

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A grilled chicken salad at a popular fast food restaurant contains $650$ milligrams (mg) of sodium, which is $\text{27\%}$ of the recommended daily amount. What is the total recommended daily amount of sodium?

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The nutrition fact sheet at a fast food restaurant says the fish sandwich has $380$ calories, and $171$ calories are from fat. What percent of the total calories is from fat?

Solution

45%

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The nutrition fact sheet at a fast food restaurant says a small portion of chicken nuggets has $190$ calories, and $114$ calories are from fat. What percent of the total calories is from fat?

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Emma gets paid $\text{\$3,000}$ per month. She pays $\text{\$750}$ a month for rent. What percent of her monthly pay goes to rent?

Solution

25%

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Dimple gets paid $\text{\$3,200}$ per month. She pays $\text{\$960}$ a month for rent. What percent of her monthly pay goes to rent?

Find Percent Increase and Percent Decrease

In the following exercises, find the percent increase or percent decrease.

Try it.

Tamanika got a raise in her hourly pay, from $\text{\$15.50}$ to $\text{\$17.55}.$ Find the percent increase.

Solution

13.2%

Try it.

Ayodele got a raise in her hourly pay, from $\text{\$24.50}$ to $\text{\$25.48}.$ Find the percent increase.

Try it.

Annual student fees at the University of California rose from about $\text{\$4,000}$ in $2000$ to about $\text{\$9,000}$ in $2014.$ Find the percent increase.

Solution

125%

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The price of a share of one stock rose from $\text{\$12.50}$ to $\text{\$50}.$ Find the percent increase.

Try it.

According to Time magazine $(\text{7/19/2011})$ annual global seafood consumption rose from $22$ pounds per person in $1960$ to $38$ pounds per person today. Find the percent increase. (Round to the nearest tenth of a percent.)

Solution

72.7%

Try it.

In one month, the median home price in the Northeast rose from $\text{\$225,400}$ to $\text{\$241,500}.$ Find the percent increase. (Round to the nearest tenth of a percent.)

Try it.

A grocery store reduced the price of a loaf of bread from $\text{\$2.80}$ to $\text{\$2.73}.$ Find the percent decrease.

Solution

2.5%

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The price of a share of one stock fell from $\text{\$8.75}$ to $\text{\$8.54}.$ Find the percent decrease.

Try it.

Hernando's salary was $\text{\$49,500}$ last year. This year his salary was cut to $\text{\$44,055}.$ Find the percent decrease.

Solution

11%

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From $2000$ to $2010,$ the population of Detroit fell from about $951,000$ to about $714,000.$ Find the percent decrease. (Round to the nearest tenth of a percent.)

Try it.

In one month, the median home price in the West fell from $\text{\$203,400}$ to $\text{\$192,300}.$ Find the percent decrease. (Round to the nearest tenth of a percent.)

Solution

5.5%

Try it.

Sales of video games and consoles fell from $\text{\$1,150}$ million to $\text{\$1,030}$ million in one year. Find the percent decrease. (Round to the nearest tenth of a percent.)

Condensed — the full section is in OpenStax Prealgebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Translate and solve: $\frac{3}{4}$ of $x$ is $24.$
If you missed this problem, review .

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$32$
Simplify: $(4.5)(2.38).$
If you missed this problem, review .

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$10.71$
Solve: $3.5=0.7n.$
If you missed this problem, review .

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$5$
What number is $\text{35\%}$ of $90?$

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Translate into algebra. Let $n=\$the number.
Multiply.
$31.5$ is $35\%$ of $90$
What number is $\text{45\%}$ of $80?$

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36
What number is $\text{55\%}$ of $60?$

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33
$\text{125\%}$ of $28$ is what number?

Revelar a resposta

Translate into algebra. Let $a\ =\$the number.
Multiply.
$125\%$ 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 $28.$
$\text{150\%}$ of $78$ is what number?

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117
$\text{175\%}$ of $72$ is what number?

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126
Translate and solve: $36$ is $\text{75\%}$ of what number?

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Translate. Let $b=$ the number.
Divide both sides by 0.75.
Simplify.
$17$ is $\text{25\%}$ of what number?

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68
$40$ is $\text{62.5\%}$ of what number?

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64
$\text{6.5\%}$ of what number is $\text{\$1.17}?$

Revelar a resposta

Translate. Let $b=$ the number.
Divide both sides by 0.065.
Simplify.
$\text{7.5\%}$ of what number is $\text{\$1.95}?$

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$26
$\text{8.5\%}$ of what number is $\text{\$3.06}?$

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$36
What percent of $36$ is $9?$

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Translate into algebra. Let $p=$ the percent.
Divide by 36.
Simplify.
Convert to decimal form.
Convert to percent.
What percent of $76$ is $57?$

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75%
What percent of $120$ is $96?$

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80%
$144$ is what percent of $96?$

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Translate into algebra. Let $p=$ the percent.
Divide by 96.
Simplify.
Convert to percent.
$110$ is what percent of $88?$

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125%
$126$ is what percent of $72?$

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175%

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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.

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?

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$14.67
Kimngoc had lunch at her favorite restaurant. She wants to leave $\text{15\%}$ of the total bill as her tip. If her bill was $\text{\$14.40},$ how much will she leave for the tip?

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$2.16

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?

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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.

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?

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24.1 grams
One serving of rice cereal has $190$ mg of sodium, which is $\text{8\%}$ of the recommended daily amount. What is the total recommended daily amount of sodium?

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2,375 mg

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?

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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.

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.)

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26%
The brownie mix Ricardo plans to use says that each brownie will be $190$ calories, and $70$ calories are from fat. What percent of the total calories are from fat?

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37%

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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.

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.)

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8.8%
In $1995,$ the standard bus fare in Chicago was $\text{\$1.50}.$ In $2008,$ the standard bus fare was $\text{\$2.25}.$ Find the percent increase. (Round to the nearest tenth of a percent.)

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50%

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.

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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%.

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

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6.3%
Last year Sheila's salary was $\text{\$42,000}.$ Because of furlough days, this year her salary was $\text{\$37,800}.$ Find the percent decrease. (Round to the nearest tenth of a percent.)

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10%
What number is $\text{45\%}$ of $120?$

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54
What number is $\text{65\%}$ of $100?$
What number is $\text{24\%}$ of $112?$

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26.88
What number is $\text{36\%}$ of $124?$

Symbols used here

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Solve General Applications of Percent

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

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

Tente o seu próprio

Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Mais em Arithmetic

Order of operations Fractions Percentages Powers and roots

Translate into algebra. Let \(n=\\)the number.
Multiply.
	\(31.5\) is \(35\%\) of \(90\)

Translate into algebra. Let \(a\ =\\)the number.
Multiply.
	\(125\%\) of \(28\) is \(35\).

Translate. Let \(b=\) the number.
Divide both sides by 0.75.
Simplify.

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

Translate into algebra. Let \(p=\) the percent.
Divide by 96.
Simplify.
Convert to percent.