A **percent **is a value divided by 100. For example, 80% and 45% are equal to 80/100 and 45/100, respectively. Just as a percent is a portion of 100, an actual quantity is part of an unknown whole.

This article focuses on using percent and proportions to solve for that unknown whole.

### Finding the Whole in Real Life: Commissions

Real estate agents, car dealers, and pharmaceutical sales representatives earn commissions. A commission is a percentage, or part, of sales. For example, a real estate agent earns a portion of the selling price of a house that she helps a client purchase or sell. A car dealer earns a portion of the selling price of an automobile that she sells.

Example: Real Estate Agent

Noë aims to earn at least $150,000 as a realtor this year. He earns a 3% commission. What's the total dollar amount of houses that he must sell to reach his goal?

What do you know?

Noë will earn 3 dollars per 100;

Noë will earn 150,000 dollars per ?

3/100 = 150,000/x**Cross multiply.** *Hint*: Write these fractions vertically to get the full understanding of cross multiplying. To cross multiply, take the first fraction's numerator and multiply it by the second fraction's denominator. Then take the second fraction's numerator and multiply it by the first fraction's denominator.

3 * *x* = 150,000 * 100

3*x* = 15,000,000

Divide both sides of the equation by 3 to solve for *x*.

3*x*/3 = 15,000,000/3*x* = $5,000,000

Verify the answer.

Does 3/100 = 150,000/5,000,000

3/100 = .03

150,000/5,000,000 = .03

### Answers and Solutions to the Exercises

1. Ericka, a real estate agent, specializes in leasing apartments. Her commission is 150% of her client's monthly rent. Last week, she earned $850 in commission for an apartment that she helped her client to lease. How much is the monthly rent? **$566.67**

What do you know?

150 dollars per 100 of the monthly rent is paid to Ericka as a commission;

850 dollars per ? of the monthly rent is paid to Ericka as a commission

150/100 = 850/*x*

Cross Multiply.

150 * *x* = 850 * 100

150*x = *85,000

Divide both sides of the equation by 150 to solve for x.

150*x/*150* = *85,000/150*x* = $566.67

Verify your answer.

Does 150/100 = 850/* *566.67

150/100 = 1.50

850/* *566.67 = 1.50

2. Ericka wants $2,500 for each leasing transaction. For each transaction, she earns 150% of her client's monthly rent. How much must her client's rent be for her to earn $2,500? **$1,666.67**

What do you know?

150 dollars per 100 of the monthly rent is paid to Ericka as a commission;

2,500 dollars per ? of the monthly rent is paid to Ericka as a commission

150/100 = 2,500/*x*

Cross Multiply.

150 * *x* = 2,500 * 100

150*x* = 250,000

Divide both sides of the equation by 150 to solve for *x*.

150*x*/150 = 250,000/150*x* = $1,666.67

Verify your answer.

Does 150/100 = 2,500/1,666.67

150/100 = 1.5

2,500/1,666.67 = 1.5

3. Pierre, an art dealer, earns 25% commission of the dollar value of the art pieces that he sells at the Bizzell Gallery. Pierre earns $10,800 this month. What is the total dollar value of the art that he sells? **$43,200**

What do you know?

25 dollars per 100 of Pierre's art sales is paid to him as commission;

10,800 dollars per ? of Pierre's art sales is paid to him as commission

25/100 = 10,800/*x*

Cross Multiply.

25 * *x* = 10,800 * 100

25*x* = 1,080,000

Divide both sides of the equation by 25 to solve for *x*.

25*x*/25 = 1,080,000/25*x* = $43,200

Verify your answer.

Does 25/100 = 10,800/43,200?

25/100 = .25

10,800/43,200 = .25

4. Alexandria, a car dealer, earns 40% commission of her luxury vehicles sales. Last year, her salary was $480,000. What was the total dollar amount of her sales last year? **$1,200,000**

What do you know?

40 dollars per 100 of the car sales is paid to Ericka as a commission;

480,000 dollars per ? of the car sales is paid to Ericka as a commission

40/100 = 480,000/*x*

Cross Multiply.

40 * *x* = 480,000 * 100

40*x* = 48,000,000

Divide both sides of the equation by 40 to solve for *x*.

40*x*/40 = 48,000,000/40*x* = $1,200,000

Verify your answer.

Does 40/100 = 480,000/1,200,000?

40/100 = .40

480,000/1,200,000 = .40

5. Henry is an agent for movie stars. He earns 10% of his clients’ salaries. If he made $72,000 last year, how much did his clients make in all? **$720,000**

What do you know?

10 dollars per 100 of the movie stars' salaries is paid to Henry as a commission;

72,000 dollars per ? of the movie stars' salaries is paid to Henry as a commission

10/100 = 72,000/*x*

Cross Multiply.

10 * *x* = 72,000 * 100

10*x* = 7,200,000

Divide both sides of the equation by 10 to solve for *x*.

10*x*/10 = 7,200,000/10

*x* = $720,000

Verify your answer.

Does 10/100 = 72,000/*x*?

10/100= .10

72,000/720,000= .10

6. Alejandro, a pharmaceutical sales representative, sells statins for a drugmaker. He earns a 12% commission of the total sales of the statins that he sells to hospitals. If he earned $60,000 in commissions, what was the total dollar value of the drugs that he sold? **$500,000**

What do you know?

12 dollars per 100 of the dollar value of the drugs is paid to Alejandro as a commission;

60,000 dollars per ? of the dollar value of the drugs is paid to Alejandro as a commission;

12/100 = 60,000/*x*

Cross Multiply.

12 * *x* = 60,000 * 100

12*x* = 6,000,000

Divide both sides of the equation by 12 to solve for *x*.

12*x*/12 = 6,000,000/12

*x* = $500,000

Verify your answer.

Does 12/100 = 60,000/500,000?

12/100= .12

60,000/500,000 = .12