Ratios & Proportions: Combining Ratios with Different Multipliers - Equate the Common Member

The solution - expand or reduce the ratios so that they are comparable. The goal is to get the single member common to both ratios (goats) to the same ratio units (and the same multiplier).

Back to our diagram:

Camels : Goats : Sheep

1^st ratio         ( 5    :   2 )

2^nd  ratio                  ( 1    :    3 )

What should you do now?

Incorrect.

This will get both Camels and Sheep to the same number of ratio units, but will not solve the main problem - the goats, which are the common member of both ratios, are still represented by two different ratios and multipliers.

To sum up:

[[summary]]

While this will indeed solve the problem, i.e. get the common member (Goats) to the same multiplier in both ratios, dividing in this way creates an uncomfortable fraction in the first ratio: 5/2 : 1. Camels like comfortable numbers, and so do you. Is there another way of achieving the same goal?

You've already seen and solved ratio questions using the ratio box. Those questions involved considering a Ratio value with its corresponding Real value. From these two values, you could find the Multiplier, which is the same multiplier for all members of the ratio. The ratio box is an invaluable tool, both as a way to organize the information and as a teaching aid to understand the concept of the multiplier - the connecting factor between ratio and real.

However, tougher GMAT ratio problems involve several ratios with different multipliers.

A petting zoo holds Camels, Goats and Sheep. The ratio of Camels to Goats is 5:2, while the ratio of Goats to sheep is 1:3. What is the ratio of camels to sheep in the petting zoo?

This ratio problem requires you to build a third ratio out of two given ratios. To see what's going on, let's organize the information. With two or more different ratios, the following organization works best:

Camels : Goats : Sheep

1^st ratio         ( 5    :   2 )

2^nd  ratio                  ( 1    :    3 )

The question asks for the ratio of Camels to Sheep. Now the average test taker, looking at the diagram above, will jump up and yell "5:3!".

And he would, of course, be WRONG.

The reason we cannot simply connect the camels and sheep to form a new ratio is that the two ratios are not COMPARABLE. A Ratio is not a Real number - it merely describes a real number. Although the two ratios above describe the same Real number of Camels, Goats and Sheep, they do so in different ways.

To see what we mean, suppose that there are 10 Goats in the petting Zoo. Using the Ratio number of Goats and the Real number, we can figure out the Multiplier and the real numbers of the other animals. To illustrate the problem, here are the ratio boxes for both ratios:

Notice how the same number of goats yields different multipliers for each case (×5 for camels, ×10 for sheep)?

Remember the secret of the Ratio Box: The multiplier must be the same for all members of the ratio.

This exemplifies why the ratios are not comparable - each "ratio unit" does not represent the same number in Real.

Correct.

Multiply both sides of the ratio by two to get 2:6.

Camels : Goats : Sheep

1^st ratio         ( 5    :   2 )

2^nd  ratio                  ( 2    :    6 )

Now that the goats have 2 ratio units in both ratios, the two ratios are comparable. The final ratio of Camels to Sheep is 5:6.