Lesson 3.2: Solve Percent Applications

New Concept

This lesson shows how to translate real-world percent problems—like tips, discounts, and interest—into simple algebraic equations. You'll learn to solve for unknown amounts, rates, or base values, making everyday math clear and manageable.

What’s next

Next up, you'll tackle interactive examples and practice cards to solidify your skills in solving these percent applications.

Property

To solve percent problems, we translate English sentences into algebraic equations and then solve them. We must be sure to change the given percent to a decimal when we put it in the equation. The three basic types of percent equations are:

1. Finding the amount: What number is 35% of 90? translates to $n = 0.35 \cdot 90$
2. Finding the base: 6.5% of what number is 1.17 dollars? translates to $0.065 \cdot n = 1.17$
3. Finding the percent: 144 is what percent of 96? translates to $144 = p \cdot 96$

Examples

• What number is 25% of 160?

Translate the sentence into an equation: $n = 0.25 \cdot 160$. Solving this gives $n = 40$. So, 40 is 25% of 160.

• 4.5% of what number is 18 dollars?

Translate this as $0.045 \cdot n = 18$. To find n, divide both sides by 0.045: $n = \frac{18}{0.045} = 400$. So, 4.5% of 400 dollars is 18 dollars.

Property

Many applications of percent occur in our daily lives. To solve these applications, we'll translate to a basic percent equation. Use this problem-solving strategy:

1. Read the problem to understand it.
2. Identify what you are looking for.
3. Name the unknown quantity with a variable.
4. Translate the problem into an equation.
5. Solve the equation.
6. Check if the answer makes sense.
7. Answer with a complete sentence.

Examples

• A dinner bill is 72.50 dollars, and you want to leave a 20% tip. How much is the tip?

Let $t$ be the tip. The problem is: What is 20% of 72.50 dollars? This translates to $t = 0.20 \cdot 72.50 = 14.50$. The tip should be 14.50 dollars.

• A cereal serving provides 120 milligrams of sodium, which is 5% of the recommended daily amount. What is the total recommended amount?

Let $a$ be the total amount. The problem is: 120 is 5% of what amount? This translates to $120 = 0.05 \cdot a$. Solving for $a$ gives $a = 2400$. The total recommended amount is 2400 mg.

Property

If an amount of money, $P$, called the principal, is invested for a period of $t$ years at an annual interest rate $r$, the amount of interest, $I$, earned is

where

• $I$ = interest
• $P$ = principal
• $r$ = rate
• $t$ = time

Interest earned according to this formula is called simple interest.

Examples

• How much interest will be earned on a principal of 8,000 dollars invested at an interest rate of 3% for 5 years?

Using the formula $I = Prt$, we substitute the values: $I = (8000)(0.03)(5) = 1200$. The interest earned is 1,200 dollars.

• A loan of 4,000 dollars was repaid with 640 dollars in interest after 2 years. What was the interest rate?

Using $I = Prt$, we have $640 = (4000)(r)(2)$. This simplifies to $640 = 8000r$. Solving for $r$ gives $r = 0.08$, so the interest rate was 8%.

Property

The formulas for calculating a discount are:

Keep in mind that the sale price should always be less than the original price.

Examples

• A jacket with an original price of 220 dollars is on sale for 25% off. What is the amount of discount and the sale price?

The amount of discount is $0.25 \times 220 = 55$ dollars. The sale price is $220 - 55 = 165$ dollars.

• A pair of shoes originally priced at 80 dollars is on sale for 60 dollars. What is the amount of discount and the discount rate?

The amount of discount is $80 - 60 = 20$ dollars. To find the rate, solve $20 = r \cdot 80$, which gives $r = 0.25$, or a 25% discount rate.

Property

The formulas for calculating a mark-up are:

Keep in mind that the list price should always be more than the original cost.

Examples

• An art gallery buys a sculpture for 400 dollars and marks the price up by 60%. What is the amount of mark-up and the list price?

The amount of mark-up is $0.60 \times 400 = 240$ dollars. The list price is $400 + 240 = 640$ dollars.

• A music store buys a guitar for 1,500 dollars and sells it for 2,100 dollars. What is the amount of mark-up and the mark-up rate?

The amount of mark-up is $2100 - 1500 = 600$ dollars. To find the rate, solve $600 = r \cdot 1500$, which gives $r = 0.4$, or a 40% mark-up rate.