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Algebra 101: Solving Ratio Word Problems

Algebra 101: Solving Ratio Word Problems

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Solving ratio word problems can be brain-twisting, but once you get the hang of the formula, any problem is easily solvable. In this guide we’ll look at what a ratio is, and go step by step through the process of solving any ratio word problem.

What is a Ratio?

The ratio is a very useful notion in everyday mathematics. A ratio gives the relative sizes of two sets but not the actual numbers of objects in those sets. For example, the fact that the ratio of green marbles to red marbles in a box is 2 to 3 tells us that for every 2 green marbles there are 3 red marbles; however, it does not tell us the number of green or red marbles.

The order of terms in a ratio is important. Notice that in the expression “the ratio of green marbles to red marbles,” “green marbles” came first. This order is important: whichever word comes first, its number must come first as well. If the expression had been “the ratio of red marbles to green marbles,” then the numbers would have been “3 to 2.”

The Basic Steps for Solving Problems

The most common ratio problems involve a comparison between two quantities. These ratios are called two-term ratios. There are three basic steps you must do first, when working any ratio problems:

  1. Change the quantities to the same units; then reduce the ratio to its simplest form.

    For example, what is the ratio of 6 minutes to 8 hours?

    First, change the hours to minutes:

    8 hours = 8 × 60 = 480 minutes

    Write the ratio as a fraction and simplify:

    Simplify the fraction

    We found that the ratio of 6 minutes to 8 hours is 1:80.

  2. Write the items in the ratio in fraction form.
  3. Make sure that there are the same items in the numerator and denominator.

    For example, if the ratio of Olga’s classical CDs to her rock CDs is 14 to 25, the right setup is this:

    Correct

    This setup is wrong:

    Incorrect

Sample Ratio Problems and Solutions

Now that you know those three basics, let’s solve some ratio problems.

Sample Problem 1

In a bag of blue and yellow candies, the ratio of blue candies to yellow candies is 3:5. If the bag contains 60 yellow candies, how many blue candies are there?

Step 1: Let the number of blue candies be x. Next, write the items in the ratio as a fraction:

Write as a fraction

Step 2: Solve the equation by cross-multiplication:

3 × 60 = 5x

180 = 5x

Divide both sides by 5:

x = 36

Solution: There are 36 blue candies in the bag.

Sample Problem 2

A room is 16 feet, 8 inches long, and the ratio of the length to the width is 4 to 5. What is the width of the room?

Step 1: Since the length is given in both feet and inches, let’s convert it to inches using the fact that 1 foot equals 12 inches. To find how many inches are in 16 feet, we multiply 16 feet by 12 inches:

16 feet, 8 inches = (16 × 12) + 8 = 192 + 8 = 200

We found that the length is 200 inches.

Step 2: Let x represent the width. We can now set up the equation:

Set up the equation

Step 3: Solve the equation by cross-multiplication:

4 × 200 = 5x

800 = 5x

Divide both sides by 5:

x = 160 inches

We found that the width is 160 inches.

Step 4: Let’s now convert inches to feet so that the units for the width are consistent with the units for the length. Since 1 foot is 12 inches, we divide 160 inches by 12 to find out how many feet are in 160 inches:

160:12 = 13, with the remainder of 4 inches. The width is 13 feet, 4 inches.

Solution: The room width is 13 feet, 4 inches.

Sample Problem 3

A school has 300 students. If the ratio of boys to girls is 31 to 44, how many more girls are there in the school?

Step 1: Since we don’t know the exact number of either boys or girls, we can’t set up the equation right away. We assign variables first. Let x be the number of girls; then the number of boys will be 300 − x, since there are 300 students overall.

Step 2: Now we can set up the equation by writing the number of students in the ratio as a fraction:

Set up the equation

Solve the equation by cross-multiplication:

31x = 44(300 − x)

Distribute the right side:

31x = 13,200 − 44x

Isolate the variable and collect like terms:

31x + 44x = 13,200

75x = 13,200

Divide both sides by 75:

x = 176

We found that there are 176 girls.

Step 3: In order to find the number of boys, subtract the number of girls from the number of all students:

300 − 176 = 124

We found that there are 124 boys in the school.

Step 4: By subtracting the number of boys from the number of girls, we can find out how many more girls there are in the school:

176 − 124 = 52

Solution: There are 52 more girls than boys in the school.

Sometimes the ratio of the objects is not given. We need to use the actual number of objects to find the ratio. For example, find the ratio of boys to girls in a class if there are 60 boys and 75 girls.

Let’s write the ratio in fraction form and then simplify by the common factor of 15:

Simplify

So the ratio of boys to girls in class is 4:5.

Sample Problem 4

At a small college, the ratio of men to women is 9:4. If there are presently 720 women, how many additional women would it take to reduce the ratio of men to women to 2:1?

Step 1: To answer the problem’s question, we need to know the present number of men at the college. Let this number be x. We use the first ratio 9:4 and the present number of women to find the present number of men:

Set up the equation

Solve the equation by cross-multiplying:

9(720) = 4x

6,480 = 4x

Divide both sides by 4:

x = 1,620

We found that there are presently 1,620 men at the college.

Step 2: To reduce the ratio to 2:1, the college needs to take in more women. Let that number of additional women be y. We use the variable y since we already used the variable x to represent the amount of men. The present number of women is 720, so the new amount of women will be 720 + y.

Step 3: Now, we know the present amount of men (this should not be changed) and we know the number of women (720 + y) it will take to change the ratio to the desirable ratio 2:1, so we can set up our equation:

Set up the equation

Step 4: Solve the equation by cross-multiplication:

2(720 + y) = 1,620

Distribute and isolate the variable:

1,440 + 2y = 1,620

2y = 1,620 − 1,440

Collect like terms:

2y = 180

Divide both sides by 2:

y = 90

Solution: It would take 90 additional women to reduce the ratio of men to women to 2:1.

Now that you’ve worked through these problems and have learned to figure out ratios, you should be able to solve any ratio word problem! For more practice problems and in-depth information, check out The Complete Idiot’s Guide to Algebra Word Problems. Have fun!

From The Complete Idiot’s Guide to Algebra Word Problems by Izolda Fotiyeva, Ph.D.