Proportion Math Problems, Solutions and Explanations

Math Problems with Proportions

Proportions state that two ratios are equal, so they involve four quantities and one or two different types of units. For instance, if you knew that a car traveled 70 miles in one hour and 140 miles in two hours, you could represent this with the following proportion:

70 miles/1 hour = 140 miles/2 hours

We can state that the two ratios in the proportion are equal because in each one, the rate of speed in miles per hour (mph) is the same: 70 miles ÷ 1 hour = 70 mph and 140 miles ÷ 2 hours = 70 mph.

Solving a Proportion

Many word problems will give you three of the four quantities that make up a proportion, and you will have to identify the fourth quantity. Here's an example:

Ellen can run 0.75 miles in 15 minutes. How long will it take her to run three miles if she maintains a constant speed?

This problem gives you three of the four quantities in the proportion. Here's how you solve for the unknown quantity:

1. Identify the two ratios and the unknown quantity. In this problem, the two ratios are 0.75 miles:15 minutes and 3 miles:x minutes.

2. Set the two ratios equal to each other. Be sure that both numerators and denominators contain like units. One possible way to write the proportion is 0.75 miles/15 minutes = 3 miles/x minutes.

3. Multiply each of the denominators by the opposite numerator, and write the results on opposite sides of the equals sign, like this: 0.75x = 45.

4. Solve for the variable. In this problem, you divide both sides of the equation by 0.75 to find that x = 60 minutes.

Practice Problems

1. Frank left a tip of $5.00 for his $25.00 meal, and Sandy left a $7.50 tip for her meal. If Frank's and Sandy's tips represent equal percentages of the price of their meals, how much did Sandy's meal cost?

2. Sweaters at Mega Mall are on sale. If Eleanor saved $20 on two sweaters, how much will she save if she buys five sweaters?

3. If Shawn paid $6.40 in sales tax on a $100 purchase, how much sales tax will he pay on a $1,000 purchase?

Solutions and Explanations

Problem One

Since we know the ratio of tip to meal cost is equal for both Frank and Sandy, we can set up and solve this proportion:

$5.00/$25.00 = $7.50/x

5x = 187.5

(5x) ÷ 5 = 187.5 ÷ 5

x = 37.5

Sandy's meal cost $37.50.

Problem Two

To find Eleanor's savings, solve this proportion:

2 sweaters/$20 = 5 sweaters/x

2x = 100

(2x) ÷ 2 = 100 ÷ 2

x = 50

Eleanor will save $50 if she buys five sweaters.

Problem Three

The sales tax on a $1,000 purchase can be found by solving this proportion:

$6.40/$100 = x/$1,000

100x = 6,400

(100x) ÷ 100 = 6,400 ÷ 100

x = 64

The sales tax on $1,000 would be $64.00.