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 you are looking for. We must be sure to change the given percent to a decimal when we put it in the equation.

Example 3.12

Translate and solve: What number is 35% of 90?

Solution

Translate into algebra. Let $n$= the number.
Remember "of" means multiply, "is" means equals.
Multiply.
	$31.5$ is $\mathrm{35\%}$ 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 in Example 3.12. We will again use direct translation to write the equation.

Example 3.13

Translate and solve: 6.5% of what number is $1.17?

Solution

Translate. Let $n=$ the number.
Multiply.
Divide both sides by 0.065 and simplify.
	$\mathrm{6.5\%}$ of$\mathrm{\$18}$ is $\mathrm{\$1.17}$

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

Example 3.14

Translate and solve: 144 is what percent of 96?

Solution

Translate into algebra. Let $p=$ the percent.
Multiply.
Divide by 96 and simplify.
Convert to percent.
	$144$ is $\mathrm{150\%}$ 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’ll translate to a basic percent equation, just like those we solved in previous examples. 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 3.15

Dezohn and his girlfriend enjoyed a nice dinner at a restaurant and his bill was $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?

Solution

Step 1. Read the problem.
Step 2. Identify what we are looking for.		the amount of tip should Dezohn leave
Step 3. Name what we are looking for.
Choose a variable to represent it.		Let t = amount of tip.
Step 4. Translate into an equation.
Write a sentence that gives the information to find it.
Translate the sentence into an equation.
Step 5. Solve the equation. Multiply.
Step 6. Check. Does this make sense?
Yes, 20% of $70 is $14.
Step 7. Answer the question with a complete sentence.		Dezohn should leave a tip of $12.33.

Notice that we used t to represent the unknown tip.

Example 3.16

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?

Solution

Step 1. Read the problem.
Step 2. Identify what we are looking for.		the total amount of potassium that is recommended
Step 3. Name what we are looking for.
Choose a variable to represent it.		Let $a=$ total amount of potassium.
Step 4. Translate. Write a sentence that gives the information to find it.
Translate into an equation.
Step 5. Solve the equation.
Step 6. Check. Does this make sense?
Yes, 2% is a small percent and 85 is a small part of 4,250.
Step 7. Answer the question with a complete sentence.		The amount of potassium that is recommended is 4,250 mg.

Example 3.17

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?

Solution

Step 1. Read the problem.
Step 2. Identify what we are looking for.		the percent of the total calories from fat
Step 3. Name what we are looking for.
Choose a variable to represent it.		Let $p=$ percent of fat.
Step 4. Translate. Write a sentence that gives the information to find it.
Translate into an equation.
Step 5. Solve the equation.
Divide by 480.
Put in a percent form.
Step 6. Check. Does this make sense?
Yes, 240 is half of 480, so 50% makes sense.
Step 7. Answer the question with a complete sentence.		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, 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 3.18

In 2011, the California governor proposed raising community college fees from $26 a unit to $36 a unit. Find the percent increase. (Round to the nearest tenth of a percent.)

Solution

Step 1. Read the problem.
Step 2. Identify what we are looking for.		the percent increase
Step 3. Name what we are looking for.
Choose a variable to represent it.		Let $p=$ the percent.
Step 4. Translate. Write a sentence that gives the information to find it.
First find the amount of increase.		new amount − original amount = increase
		$36-26=10$
Find the percent.		Increase is what percent of the original amount?
Translate into an equation.
Step 5. Solve the equation.
Divide by 26.
Change to percent form; round to the nearest tenth.
Step 6. Check. Does this make sense?
Yes, 38.4% is close to $\frac{1}{3}$, and 10 is close to $\frac{1}{3}$ of 26.
Step 7. Answer the question with a complete sentence.		The new fees represent a 38.5% increase over the old fees.

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 3.19

The average price of a gallon of gas in one city in June 2014 was $3.71. The average price in that city in July was $3.64. Find the percent decrease.

Solution

Step 1. Read the problem.
Step 2. Identify what we are looking for.		the percent decrease
Step 3. Name what we are looking for.
Choose a variable to represent that quantity.		Let $p=$ the percent decrease.
Step 4. Translate. Write a sentence that gives the information to find it.
First find the amount of decrease.		$3.71-3.64=0.07$
Find the percent.		Decrease is what percent of the original amaount?
Translate into an equation.
Step 5. Solve the equation.
Divide by 3.71.
Change to percent form; round to the nearest tenth.
Step 6. Check. Does this make sense?
Yes, if the original price was $4, a 2% decrease would be 8 cents.
Step 7. Answer the question with a complete sentence.		The price of gas decreased 1.9%.

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; 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.

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.

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.

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.

We summarize the discount model in the box below.

Example 3.23

Elise bought a dress that was discounted 35% off of the original price of $140. What was ⓐ the amount of discount and ⓑ the sale price of the dress?

Solution

ⓐ $\begin{array}{ccc}\hfill \text{Original price}& =\hfill & \text{\$}140\hfill \\ \hfill \text{Discount rate}& =\hfill & 35\text{\%}\hfill \\ \hfill \text{Discount}& =\hfill & ?\hfill \end{array}$
Step 1. Read the problem.
Step 2. Identify what we are looking for.	the amount of discount
Step 3. Name what we are looking for. Choose a variable to represent that quantity.	Let $d=$ the amount of discount.
Step 4. Translate into an equation. Write a sentence that gives the information to find it. Translate into an equation.	The discount is 35% of $140. $d=0.35(140)$
Step 5. Solve the equation.	$d=49$
Step 6. Check: Does this make sense? Is a $49 discount reasonable for a $140 dress? Yes.
Step 7. Write a complete sentence to answer the question.	The amount of discount was $49.

Table 3.5 This table provides a step-by-step walkthrough for calculating a discount, demonstrating the process from problem setup to the final solution.

ⓑ
Read the problem again.

Step 1. Identify what we are looking for.		the sale price of the dress
Step 2. Name what we are looking for.
Choose a variable to represent that quantity.		Let $s=$ the sale price.
Step 3. Translate into an equation.
Write a sentence that gives the information to find it.
Translate into an equation.
Step 4. Solve the equation.
Step 5. Check. Does this make sense?
Is the sale price less than the original price?
Yes, $91 is less than $140.
Step 6. Answer the question with a complete sentence.		The sale price of the dress was $91.

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 3.24 will show this case.

Example 3.24

Jeannette bought a swimsuit at a sale price of $13.95. The original price of the swimsuit was $31. Find the ⓐ amount of discount and ⓑ discount rate.

Solution

ⓐ $\begin{array}{ccc}\hfill \text{Original price}& =\hfill & \text{\$}31\hfill \\ \hfill \text{Discount}& =\hfill & ?\hfill \\ \hfill \text{Sale Price}& =\hfill & \text{\$}13.95\hfill \end{array}$
Step 1. Read the problem.
Step 2. Identify what we are looking for.	the amount of discount
Step 3. Name what we are looking for. Choose a variable to represent that quantity.	Let $d=$ the amount of discount.
Step 4. Translate into an equation. Write a sentence that gives the information to find it. Translate into an equation.	The discount is the difference between the original price and the sale price. $d=31-13.95$
Step 5. Solve the equation.	$d=17.05$
Step 6. Check: Does this make sense? Is 17.05 less than 31? Yes.
Step 7. Answer the question with a complete sentence.	The amount of discount was $17.05.

Table 3.6 This table illustrates a seven-step process for solving a word problem to calculate the discount amount, from identifying the unknown to providing the final answer.

ⓑ
Read the problem again.

Step 1. Identify what we are looking for.		the discount rate
Step 2. Name what we are looking for.
Choose a variable to represent it.		Let $r=$ the discount rate.
Step 3. Translate into an equation.
Write a sentence that gives the information to find it.
Translate into an equation.
Step 4. Solve the equation.
Divide both sides by 31.
Change to percent form.
Step 5. Check. Does this make sense?
Is $17.05 equal to 55% of $31?
$17.05\stackrel{?}{=}0.55\left(31\right)$
$17.05=17.05✓$
Step 6. Answer the question with a complete sentence.		The rate of discount was 55%.

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.

We summarize the mark-up model in the box below.

Example 3.25

Adam’s art gallery bought a photograph at original cost $250. Adam marked the price up 40%. Find the ⓐ amount of mark-up and ⓑ the list price of the photograph.

Solution

ⓐ

Step 1. Read the problem.
Step 2. Identify what we are looking for.		the amount of mark-up
Step 3. Name what we are looking for.
Choose a variable to represent it.		Let $m=$ the amount of markup.
Step 4. Translate into an equation.
Write a sentence that gives the information to find it.
Translate into an equation.
Step 5. Solve the equation.
Step 6. Check. Does this make sense?
Yes, 40% is less than one-half and 100 is less than half of 250.
Step 7. Answer the question with a complete sentence.		The mark-up on the phtograph was $100.

ⓑ

Step 1. Read the problem again.
Step 2. Identify what we are looking for.		the list price
Step 3. Name what we are looking for.
Choose a variable to represent it.		Let $p=$ the list price.
Step 4. Translate into an equation.
Write a sentence that gives the information to find it.
Translate into an equation.
Step 5. Solve the equation.
Step 6. Check. Does this make sense?
Is the list price more than the net price? Is $350 more than $250? Yes.
Step 7. Answer the question with a complete sentence.		The list price of the photograph was $350.