Solve Percent Applications

Translate and Solve Basic Percent Equations

We will solve percent equations using the methods we used to solve equations with fractions or decimals. Without the tools of algebra, the best method available to solve percent problems was by setting them up as proportions. Now, as an algebra student, you can just translate English sentences into algebraic equations and then solve the equations.

We can use any letter you like as a variable, but it is a good idea to choose a letter that will remind us of what we are looking for. We must be sure to change the given percent to a decimal when we put it in the equation.

Example. Translate and solve: What number is $35\%$ of $90$?

$31.5$ is $35\%$ of $90$.

We must be very careful when we translate the words in the next example. The unknown quantity will not be isolated at first, like it was above. We will again use direct translation to write the equation.

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

$6.5\%$ of $18 is $1.17.

In the next example, we are looking for the percent.

Example. Translate and solve: $144$ is what percent of $96$?

$144$ is $150\%$ of $96$. Note that we are asked to find percent, so we must have our final result in percent form.

Solve Applications of Percent

Many applications of percent — such as tips, sales tax, discounts, and interest — occur in our daily lives. To solve these applications we translate to a basic percent equation, just as we solved above. Once we translate the sentence into a percent equation, we know how to solve it.

We will restate the problem-solving strategy we used earlier for easy reference.

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 will involve everyday situations, you can rely on your own experience.

Example. Dezohn and his girlfriend enjoyed a nice dinner at a restaurant and his bill was $\text{\textdollar}68.50$. He wants to leave an $18\%$ tip. If the tip will be $18\%$ of the total bill, how much tip should he leave?

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?

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?

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, the difference of 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 $\text{\textdollar}26$ a unit to $\text{\textdollar}36$ a unit. Find the percent increase. (Round to the nearest tenth of a percent.)

Notice that we rounded the division to the nearest thousandth in order to round the percent to the nearest tenth.

Finding the percent decrease is very similar to finding the percent increase, but now the amount of decrease is the difference of 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.

Solve Simple Interest Applications

Do you know that banks pay you to keep your money? The money a customer puts in the bank is called the principal, $P$, and the money the bank pays the customer is called the interest. The interest is computed as a certain percent of the principal; this is called the rate of interest, $r$. We usually express rate of interest as a percent per year, and we calculate it by using the decimal equivalent of the percent. The variable $t$ (for time) represents the number of years the money is in the account.

To find the interest we use the simple interest formula, $I = Prt$.

Interest may also be calculated another way, called compound interest. This type of interest will be covered in later math classes.

The formula we use to calculate simple interest is $I = Prt$. To use the formula, we substitute in the values the problem gives us for the variables, and then solve for the unknown variable. It may be helpful to organize the information in a chart.

Example. Nathaly deposited $\text{\textdollar}12{,}500$ in her bank account where it will earn $4\%$ interest. How much interest will Nathaly earn in $5$ years?

There may be times when we know the amount of interest earned on a given principal over a certain length of time, but we don’t know the rate. To find the rate, we use the simple interest formula, substitute in the given values for the principal and time, and then solve for the rate.

Example. Loren loaned his brother $\text{\textdollar}3{,}000$ to help him buy a car. In $4$ years his brother paid him back the $\text{\textdollar}3{,}000$ plus $\text{\textdollar}660$ in interest. What was the rate of interest?

Notice that in this example, Loren’s brother paid Loren interest, just like a bank would have paid interest if Loren invested his money there.

Example. Eduardo noticed that his new car loan papers stated that with a $7.5\%$ interest rate, he would pay $\text{\textdollar}6{,}596.25$ in interest over $5$ years. How much did he borrow to pay for his car?

Solve Applications with Discount or Mark-up

Applications of discount are very common in retail settings. When you buy an item on sale, the original price has been discounted by some dollar amount. The discount rate, usually given as a percent, is used to determine the amount of the discount. To determine the amount of discount, we multiply the discount rate by the original price.

Example. Elise bought a dress that was discounted $35\%$ off of the original price of $\text{\textdollar}140$. What was (a) the amount of discount and (b) the sale price of the dress?

(a)

(b) Read the problem again.

There may be times when we know the original price and the sale price, and we want to know the discount rate. To find the discount rate, first we will find the amount of discount and then use it to compute the rate as a percent of the original price.

Example. Jeannette bought a swimsuit at a sale price of $\text{\textdollar}13.95$. The original price of the swimsuit was $\text{\textdollar}31$. Find (a) the amount of discount and (b) the discount rate.

(a)

(b) Read the problem again.

Applications of mark-up are very common in retail settings. The price a retailer pays for an item is called the original cost. The retailer then adds a mark-up to the original cost to get the list price, the price he sells the item for. The mark-up is usually calculated as a percent of the original cost. To determine the amount of mark-up, multiply the mark-up rate by the original cost.

Example. Adam’s art gallery bought a photograph at original cost $\text{\textdollar}250$. Adam marked the price up $40\%$. Find (a) the amount of mark-up and (b) the list price of the photograph.

(a)

(b) Read the problem again.

Key terms

percent equation — an equation of the form “part $=$ percent $\times$ whole,” translated directly from an English sentence using “of” for multiplication and “is” for equals. percent increase — the amount of increase (new amount minus original amount), expressed as a percent of the original amount. percent decrease — the amount of decrease (original amount minus new amount), expressed as a percent of the original amount. principal — the amount of money deposited or borrowed. interest — money paid on a principal, computed as a percent of the principal. rate of interest — the percent used to compute interest, usually stated per year. simple interest — interest computed using the formula $I = Prt$. discount rate — the percent of the original price that is subtracted to give the sale price. mark-up — the amount added to a retailer’s original cost, usually calculated as a percent of that cost, to get the list price.

This section is adapted from Elementary Algebra 2e, Section 3.2: Solve Percent Applications 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: recreated the step-by-step solution tables as markdown tables; omitted the Be Prepared quiz, Self Check checklist, media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.