Using Ratios to Solve GRE Questions

- by Katharine Rudzitis

Instead of setting up two equations for a word problem and solving for both variables, some questions can be answered using a ratio table. This is a great strategy when ratios appear in the question text. Take a moment to read the following problem.

In a certain dog kennel, the ratio of the number of poodles to the number of boxers is 5 to 2, and the ratio of the number of boxers to the number of terriers is 3 to 4. If there are 60 poodles, how many terriers are there?

The traditional math approach would be setting up two equations, solving for one variable, plugging in that value, and solving for the other variable. Instead, we’ll create our own ratio to get the answer.

Finding Ratios in the Problem

This problem gives us two key ratios: poodles to boxers and boxers to terriers.

poodles:boxers         boxers:terriers

       5:2                     3:4

We have to solve for the number of terriers, but we can’t combine the ratios unless there is a like term. Both ratios include the number of boxers. We’ll multiply both ratios to get a common term for the number of boxers. Let’s multiply the left side by 3 and the right by 2.

poodles:boxers         boxers:terriers

       5:2                     3:4

    (3)5:(3)2                (2)3:(2)4

      15:6                     6:8

Now both ratios have the same value for boxers, and we can combine them into one ratio.

poodles:boxers:terriers

15:6:8

Solving the Problem with Ratios

We want to find the number of terriers, so the number of boxers isn’t important for our answer. Let’s remove boxers from our ratio.

poodles:terriers

      15:8

Ratios are another way of representing fractions, so we’ll rewrite the ratio as:

(poodles)/(terriers)=15/8

We’ll plug in 60 for the number of poodles and solve for T, the number of terriers.

 (60)/(T) = 15/8

We can cross-multiply:

15T = (60)(8) 

   T = [(60)(8)]/15

   T = (4)(8)

   T = 32

There are 32 terriers, and we’ve solved the problem.

Finding Values using a Ratio

We’ve found the answer, but what if we had to find the number of each type of dog? Here’s the ratio again:

poodles:boxers:terriers

15:6:8

Plug in 60 and 32 for the number of poodles and terriers. All that’s left is solving for the boxers. Our original ratio is 15:6:8, and we have to multiply it by something to get 60:B:32. We’ll multiply the original ratio by 4 to get 60:24:32, which tells us that there are 24 boxers.