Fractions, Proportions and Ratios, Oh My!:

by on October 30th, 2009

GMAT questions are notorious for seeming harder than they actually are. The writers recognize time is short, and will give you ostensibly time-consuming calculations. One way to mitigate this is by retaining a rockstar aptitude in manipulating fractions, which occur in a large portion of the questions.

Some Quick Tips:

Dividing by 5 is the same as multiplying by 2/10. For example:

  • 840/5 = ?
  • 840/5 = 840*(2/10) = 84*2 = 168
  • Multiplying or dividing by 10’s and 2’s is generally easier than using 5’s.

90% of the time, fractions will be easier to perform arithmetic. Decimals are sometimes more useful when comparing numbers relative to one another, such as in a number line, but these questions are the exception. Even if given a decimal (or percent) looks easy, quickly convert to a fraction. Some common ones to memorize:

  • 1/9 = 0.111 repeating
  • 1/8 = 0.125
  • 1/7 = ~0.14
  • 1/6 = 0.166 repeating
  • 1/5 = 0.20
  • 1/4 = 0.25
  • 1/3 = 0.333 repeating
  • 1/2 = 0.5 repeating
  • Note: Multiples of these, such as 3/8 (0.375) are also important to remember, but can easily be derived by multiplying the original fraction (1/8 * 3 = 3/8 = 0.125 * 3 = 0.375)

Denominators are super important. A denominator of a reduced fraction with a multiple of 7 will not have a finite decimal, for example. Keep in mind what you can logically combine, and what you cannot.

This list is by no means extensive. There are many many more shortcuts. If you have some, leave them in the comment field, but generally practice and familiarity with the numbers helps a lot in doing quick arithmetic.

Ratios

A ratio is both a comparison and division, and can simply be treated as such. “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.

Proportions

A proportion is two ratios set equal to each other like the question above. Generally, there is a variable in one of the four slots, and we are taught to cross-multiply and solve for that variable. Before you do that, however, it’s best to reduce top-bottom AND left-right before cross multiplying. This will ensure you work with the smallest (and easiest) (and fastest) numbers possible. For example:

A football field is 9600 square yards.  If 1200 pounds of fertilizer are spread evenly across the entire field, how many pounds of fertilizer were spread over an area of the field totaling 3600 square yards?

A. 450
B. 600
C. 750
D. 2400
E. 3200

The key word here is “spread evenly”. This implies that the relationship of fertilizer per square foot is uniform, and you can set equal the relationship of the wholes to the relationship of the parts.

A/F = 9600/1200 = 3600/x

Clearly, we can eliminate the zeros on the left side:

9600/1200 = 3600/x

96/12 = 3600/x

Then we can divide 96/12:

8 = 3600/x

Here, we can still reduce left-to-right, by canceling 4 in both:

2 = 900/x

Oh wait! There’s more! Both 2 and 900 are divisible by 2!

1 = 450/x
x = 450

It DOES NOT matter whether you start top-bottom or left-right, so long as you are reducing by the same factor. Also, start with small numbers. No need to go for the biggest common factor. You’ll eventually work your way down as the numbers progressively get easier. For this question we could have started by canceling 9600 and 3600 in the numerators, which are both divisible by 400 to get:

24/1200 = 9/x. You can take it from here.

Good luck!

5 comments

  • Does it matter if to begin, I wrote the equation

    1200 / 9600 = x / 3600 ?

    I got the same answer. If it does matter, how does one know which number should be in the numerator to begin with?

  • Yes, you can write it either way. Notice that in either orientation, when you cross multiply, you will get 9600x = 1200*3600. By canceling in a "box format" (up-down left-right, but not diagonal) we preempt the canceling that would have to occur after cross-multiplying.

    In short, any orientation is correct, but make sure the units you are comparing are either next to each other, or on top, not diagonal.

  • 1/2 = 0.5 (there is a typo there, but hopefully everyone knows this one already!!!)

  • In the cooks:waiters questions, I prefer to do it this way:
    C/W = 3/13 Thus 13C = 3W (*)

    C/(W+12) = 3/16 thus 16C = 3W + 36 (**)

    Then subtract * from ** and you get 3C = 36, so C = 12.

    Are there instances where your method is better than setting up two linear equations and solving?

  • Mike,

    You're 2-linear equation method is great. What's so great about math is that there are many ways to go about each question.

    The reason I use my method is that sometimes questions can be phrased in a confusing way and I think teaching this strategy is the most direct. But, no, there is no "better" way.

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