6.2 Solve General Applications of Percent

Learning objectives

Learning objectives

• Translate and solve basic percent equations

• Solve applications of percent

• Find percent increase and percent decrease

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 $\text{\textdollar 80}.$ They wanted to leave a $\mathrm{20\%}$ tip. What amount would the tip be?

To solve this, we want to find what amount is $\mathrm{20\%}$ of $\text{\textdollar 80}.$ The $\text{\textdollar 80}$ is called the base. The amount of the tip would be $0.20(80),$ or $\text{\textdollar 16}$ See Figure 6.6. 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 6.14

Solution

1. Translate the percent equation. Let $n$ be the number that is $35\%$ of $90$. The first figure shows the matching equation $n=0.35(90)$, where $0.35$ is the decimal form of $35\%$.
2. Evaluate the product. Multiplying gives $n=0.35(90)=31.5$.
3. State the result. The number that is $35\%$ of $90$ is $31.5$, as confirmed by the second figure showing $n=31.5$.

Takeaway: Translate the phrase of into multiplication and use the decimal form of the percent before calculating.

Example 6.15

Solution

1. Write the percent equation. The wording says that $a$ is $125\%$ of $28$, so $a=1.25(28)$, as shown in the first figure. The factor exceeds $1$ because $125\%$ is more than one whole.
2. Evaluate the product. Multiplying gives $a=1.25(28)=35$.
3. Interpret the result. Since $35$ is greater than $28$, the answer has the expected size; the second figure displays $35=a$, so $125\%$ of $28$ is $35$.

Takeaway: A percent greater than 100 uses a decimal factor greater than 1 and produces an amount larger than the base.

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

Example 6.16

Solution

1. Form the percent equation. Let $b$ be the number for which $36$ is $75\%$. The first figure matches $36$ with $75\%$ of $b$, giving $36=0.75b$.
2. Divide both sides by $0.75$. The second figure shows $\frac{36}{0.75}=\frac{0.75b}{0.75}$, so $b=36\div 0.75$.
3. Compute and interpret the base. Dividing gives $b=48$, and the final figure states that $36$ is $75\%$ of $48$. Therefore, the requested number is $48$.

Takeaway: If an amount equals a percent of an unknown base, divide the amount by the decimal rate to find the base.

Example 6.17

Solution

1. Form the percent equation. Let $n$ be the unknown base. The first figure translates $6.5\%$ of $n$ as $1.17$, giving $0.065n=1.17$.
2. Divide both sides by $0.065$. As shown in the second figure, $\frac{0.065n}{0.065}=\frac{1.17}{0.065}$, so $n=1.17\div 0.065$.
3. Evaluate and check. The quotient is $n=18$, and the last figure confirms that $6.5\%$ of $18$ is $1.17$. Thus, the unknown number is $18$.

Takeaway: Dividing the known amount by the decimal rate is reliable because dividing by the same nonzero factor preserves an equation.

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

Example 6.18

Solution

1. Form the percent equation. Let $p$ be the decimal rate. The first figure shows the wording $p(36)=9$, because $9$ is the unknown percent of $36$.
2. Divide both sides by $36$. The second figure gives $\frac{36p}{36}=\frac{9}{36}$, so $p=\frac{9}{36}=\frac{1}{4}$.
3. Convert to a percent. The remaining figures show $p=\frac{1}{4}=0.25=25\%$ and confirm that $25\%$ of $36$ is $9$. Therefore, $9$ is $25\%$ of $36$.

Takeaway: To find what percent an amount is of a base, divide the amount by the base and then multiply the decimal by 100.

Example 6.19

Solution

1. Form the percent equation. Let $p$ be the decimal rate. The first figure translates $144$ as $p$ times $96$, so $144=96p$.
2. Divide both sides by $96$. The second figure shows $\frac{144}{96}=\frac{96p}{96}$, which gives $p=1.5$.
3. Convert and check. Since $1.5=150\%$, the third and fourth figures show $1.5=p$ and $150\%=p$. Thus, $144$ is $150\%$ of $96$.

Takeaway: An amount larger than the base produces a rate above 100 percent, which is a useful reasonableness check.

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

Solution

1. Write the percent equation. Let $t$ represent the tip. The first figure translates the $18\%$ tip on a $68.50$ bill as $t=0.18(68.50)$, because $0.18$ is the decimal form of $18\%$.
2. Evaluate the product. Multiplying gives $t=0.18(68.50)=12.33$.
3. State the tip. The second figure records the completed calculation, so Dezohn and his girlfriend should leave a tip of $12.33$.

Takeaway: For a percent tip, multiply the bill by the decimal rate rather than adding the rate directly to the bill.

Example 6.21

Solution

1. Form the percent equation. Let $a$ be the recommended daily amount. The first figure shows $85=0.02a$, since $85$ mg is $2\%$ of $a$.
2. Divide both sides by $0.02$. The second figure shows $\frac{85}{0.02}=\frac{0.02a}{0.02}$, so $a=85\div 0.02$.
3. Evaluate the total. Dividing gives $a=4,250$, as displayed in the last figure. The total recommended daily amount of potassium is therefore $4,250$ mg.

Takeaway: A small percent rate can correspond to a much larger base, so divide carefully rather than estimating from the decimal rate.

Example 6.22

Solution

1. Form the percent equation. Let $p$ be the decimal fraction of the calories from fat. The first figure matches $240$ with $p$ times $480$, so $240=480p$.
2. Divide both sides by $480$. The second figure gives $\frac{480p}{480}=\frac{240}{480}$, so $p=0.5$.
3. Convert to a percent. The last two figures show $p=0.5$ and $p=50\%$. Therefore, $50\%$ of the calories in each brownie come from fat.

Takeaway: The part divided by the whole gives the decimal fraction, and multiplying that decimal by 100 gives the 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.

Example 6.23

Solution

1. Find the amount of increase. The first figure subtracts the original fee from the new fee: $36-26=10$, so the increase is $10$.
2. Form the percent equation. Let $p$ be the decimal rate. The second figure matches the $10$ increase with $p$ times the original amount $26$, giving $10=26p$.
3. Divide and convert. As the third and fourth figures show, $\frac{10}{26}=p=0.384615\ldots$, which rounds to $0.385=38.5\%$. Therefore, the percent increase was $38.5\%$ to the nearest tenth of a percent.

Takeaway: A percent increase uses the original amount as the base, so the increase is divided by the original value rather than the new value.

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

Solution

1. Find the amount of decrease. The first figure shows $3.71-3.64=0.07$, so the price decreased by $0.07$.
2. Form the percent equation. Let $p$ be the decimal decrease rate. The second figure matches the $0.07$ decrease with $p$ times the original price, so $0.07=3.71p$.
3. Divide and convert. The third and fourth figures give $\frac{0.07}{3.71}=p=0.0189\ldots$, which rounds to $0.019=1.9\%$. Thus, the percent decrease was approximately $1.9\%$.

Takeaway: For every percent change, the original amount is the denominator when the rate is found.

Section 6.2 Exercises

Section 6.2 Exercises

Translate and Solve Basic Percent Equations

In the following exercises, translate and solve.

Solve Applications of Percents

In the following exercises, solve the applications of percents.

Find Percent Increase and Percent Decrease

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

Everyday Math

Writing Exercises

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

1. $\mathrm{7.5\%}$ of what number is $\text{\textdollar 1.95}?$

2. $\mathrm{8.5\%}$ of what number is $\text{\textdollar 3.06}?$

3. What percent of $76$ is $57?$

1. What percent of $120$ is $96?$

2. $110$ is what percent of $88?$

3. $126$ is what percent of $72?$

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

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

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

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

2. 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.)

3. 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?

1. 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.)

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

3. 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.)

1. $\mathrm{11.5\%}$ of what number is $\text{\textdollar 108.10}?$

2. What percent of $260$ is $78?$

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

1. $30$ is what percent of $20?$

2. $840$ is what percent of $480?$

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

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

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

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

1. 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?

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

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

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

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

3. According to Time magazine $(\mathrm{7/19/2011})$ 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.)

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

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

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

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

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

3. $\mathrm{8.2\%}$ of what number is $\text{\textdollar 2.87}?$

1. Tipping At the campus coffee cart, a medium coffee costs $\text{\textdollar 1.65}.$ MaryAnne brings $\text{\textdollar 2.00}$ with her when she buys a cup of coffee and leaves the change as a tip. What percent tip does she leave?

2. Without solving the problem $\mathrm{"}44$ is $\mathrm{80\%}$ of what number”, think about what the solution might be. Should it be a number that is greater than $44$ or less than $44?$ Explain your reasoning.

3. After returning from vacation, Alex said he should have packed $\mathrm{50\%}$ fewer shorts and $\mathrm{200\%}$ more shirts. Explain what Alex meant.