Ratios - Not just GMAT but useful for your post-MBA Journey

Ratio is a math concept that is vital in your post-MBA journey. It allows you to compare variables and provide a means to divide the variable with a common factor. “The ratio of boys to girls is seven to two” can be expressed as the proportion: B/G = 7/2. Do with this what you like: 7G = 2B or B = 7G/2, whatever. Forget the “:” with ratios.

GMAT writers love to provide ratios (which are multiplicative relationships) and then add an absolute component (addition/subtraction). Note that when you have a ratio like B/G = 7/2, we don’t actually know the number of girls and boys. There can be 14 boys and 4 girls, or 70 boys and 20 girls. Questions that insert absolute numbers should be taken with caution. For example:

At a certain restaurant, the ratio of the number of cooks to the number of waiters is 3 to 13. When 12 more waiters are hired, the ratio of the number of cooks to the number of waiters changes to 3 to 16. How many cooks does the restaurant have?

A. 4
B. 6
C. 9
D. 12
E. 15

The key here is setting up the equation. Since we don’t know the initial scale of the number of cooks and waiters, we can express this scale by “x”.
C/W = 3x/13x.

Notice that whatever x is, the ratio will hold true. (x must be an integer, since you can’t have a portion of a cook, unless of course he chops his finger off by accident!)

“When 12 more waiters are hired” is the insertion of an absolute. Adding the 12 waiters, the new ratio becomes:
C/W = 3x/(13x + 12)

“The ratio of the number of cooks to the number of waiters changes to 3 to 16” defines this new ratio:
C/W = 3x/(13x + 12) = 3/16

STOP! Before we cross multiply and solve for x, we want to cancel out the 3’s in both the numerator. (More on this below.) After cross-multiplying, we get:

16x = 13x + 12
3x = 12
x = 4

Sweet. Answer A, right? Well, recall that x represents the scaling factor. The stimulus asks for the number of cooks, which we originally represented by 3x. So, 3*4 = 12 cooks. That’s 120 fingers. Choice D.