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. 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 $20\%$ tip. What amount would the tip be?

To solve this, we want to find what amount is $20\%$ of $\text{\textdollar}80$. The $\text{\textdollar}80$ is called the base. The amount of the tip would be $0.20(80)$, or $\text{\textdollar}16$. 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. What number is $35\%$ of $90$?

Translate into algebra, letting $n$ be the number: $n = 0.35 \cdot 90$. Multiply: $n = 31.5$. So $31.5$ is $35\%$ of $90$.

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

Translate into algebra, letting $a$ be the number: $1.25 \cdot 28 = a$. Multiply: $35 = a$. So $125\%$ 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. Translate and solve: $36$ is $75\%$ of what number?

Translate, letting $b$ be the number: $36 = 0.75 \cdot b$. Divide both sides by $0.75$: $\tfrac{36}{0.75} = \tfrac{0.75b}{0.75}$. Simplify: $48 = b$. So $36$ is $75\%$ of $48$.

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

Translate, letting $b$ be the number: $0.065 \cdot b = 1.17$. Divide both sides by $0.065$: $\tfrac{0.065b}{0.065} = \tfrac{1.17}{0.065}$. Simplify: $b = 18$. So $6.5\%$ of $\text{\textdollar}18$ is $\text{\textdollar}1.17$.

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

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

Translate into algebra, letting $p$ be the percent: $p \cdot 36 = 9$. Divide by $36$: $\tfrac{36p}{36} = \tfrac{9}{36}$. Simplify: $p = \tfrac{1}{4}$. Convert to decimal form: $p = 0.25$. Convert to percent: $p = 25\%$. So $25\%$ of $36$ is $9$.

Example. $144$ is what percent of $96$?

Translate, letting $p$ be the percent: $144 = p \cdot 96$. Divide by $96$: $\tfrac{144}{96} = \tfrac{96p}{96}$. Simplify: $1.5 = p$. Convert to percent: $150\% = p$. So $144$ is $150\%$ of $96$.

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 the ones we solved above. We’ll use the same problem-solving strategy we used earlier in applications, updated to include equations.

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

Let $t =$ the amount of tip. The tip is $18\%$ of the total bill, so $t = 0.18 \cdot 68.50$. Multiply: $t = 12.33$. Check: if we approximate the bill to $\text{\textdollar}70$ and the percent to $20\%$, we would expect a tip of $\text{\textdollar}14$, so a tip of $\text{\textdollar}12.33$ seems reasonable. The couple should leave a tip of $\text{\textdollar}12.33$.

Example. 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?

Let $a =$ the total recommended amount of potassium. $85$ mg is $2\%$ of the total amount, so $85 = 0.02 \cdot a$. Divide both sides by $0.02$: $\tfrac{85}{0.02} = \tfrac{0.02a}{0.02}$. Simplify: $4{,}250 = a$. Check: $2\%$ is a small percent and $85$ is a small part of $4{,}250$, so this is reasonable. The amount of potassium that is recommended is $4{,}250$ mg.

Example. 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?

Let $p =$ the percent from fat. What percent of $480$ is $240$? $p \cdot 480 = 240$. Divide both sides by $480$: $\tfrac{p \cdot 480}{480} = \tfrac{240}{480}$. Simplify: $p = 0.5$. Convert to percent form: $p = 50\%$. Check: $240$ is half of $480$, so $50\%$ makes sense. 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. In $2011$, the California governor proposed raising community college fees from $\text{\textdollar}26$ per unit to $\text{\textdollar}36$ per unit. Find the percent increase. (Round to the nearest tenth of a percent.)

Let $p =$ percent. Find the amount of increase: $36 - 26 = 10$. Find the percent increase — the increase is what percent of the original amount? $10 = p \cdot 26$. Divide both sides by $26$: $\tfrac{10}{26} = \tfrac{26p}{26}$. Round to the nearest thousandth: $0.385 = p$. Convert to percent form: $38.5\% = p$. 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 new amount. Then we find what percent the amount of decrease is of the original amount.

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

Let $p =$ percent. Find the amount of decrease: $3.71 - 3.64 = 0.07$. Find the percent of decrease — the decrease is what percent of the original amount? $0.07 = p \cdot 3.71$. Divide both sides by $3.71$: $\tfrac{0.07}{3.71} = \tfrac{3.71p}{3.71}$. Round to the nearest thousandth: $0.019 = p$. Convert to percent form: $1.9\% = p$. The price of gas decreased $1.9\%$.

Key terms

base — in a percent equation, the whole amount that a percent is taken of. percent increase — the amount of increase, expressed as a percent of the original amount. percent decrease — the amount of decrease, expressed as a percent of the original amount.

This section is adapted from Prealgebra 2e, Section 6.2: Solve General Applications of Percent by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: omitted the Be Prepared quiz, the restaurant-receipt and nutrition-label photos, Self Check checklist, media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.