Ratios: Box ‘Em Up (Or Just Pour A Drink)

Neil Thornton —  September 9, 2013 — Leave a comment

On the GMAT, you may see a 3 to 5 ratio expressed in a variety of ways:

3:5
3 to 5
x/y = 3/5
5x = 3y (Yes, that’s the same as the other 3. Think about it.)

In the real world, we encounter ratios in drink recipes more often than anywhere else (3 parts vodka, 5 parts cranberry),don't drink and derive perhaps explaining why–after drinks that strong–we forget how to handle them.

Keep in mind: ratios express a “part to part” relationship, whereas fractions and percentages express a “part to whole” relationship. So the fraction of the above drink is 3/8 vodka (or 37.5% of the whole). Either way, hold off on mixing that drink until after this post.

I like to set up ratios using a “ratio box.” The box is a variant on the “Unknown Multiplier” technique from page 65 of our FDPs book, but it’s a nice way to visually manage ratios without resorting to algebra.

Let’s take the beginning of a typical ratio question:

“The ratio of men to women in a class is 3:2…”

Instead of doing anything fancy with variables, I just set up a tracking chart:

Men Women Total
Ratio 3 2 5

From this point alone, I have sufficient information to answer a bunch of questions.

-What fraction of the students are men? (3/5)

-What percent of the students are women? (40%)

-What is the probability of choosing a man? (3/5)

-etc.

However, I have nowhere near enough information to answer anything about the REAL numbers of students in this class. Suppose the GMAT were to add a little more information:

“The ratio of men to women in a class is 3:2. If there are 35 students in the class…”

Now we can calculate almost everything about the real numbers of people. First, make a bigger box with 3 lines. The unfilled box looks like this:

Men Women Total
Ratio 3 2 5
Multiplier
Real #s 35

Now it’s a simple question: what do you multiply 5 by to get 35? Fill that in all the way across your multiplier line. The completed box looks like this:

Men Women Total
Ratio 3 2 5
Multiplier 7 7 7
Real #s 21 14 35

And we have everything else we might need to know.  (How many men? How many women? How many more men than women?)

This technique works beautifully for many questions in the Official Guide 13th Edition (OG 13). For example, Problem Solving (PS) #179:

“A fruit salad mixture consists of apples, peaches, and grapes in the ratio 6:5:2, respectively, by weight. If 39 pounds of the mixture is prepared…”

A completed chart looks like this:

Apples Peaches Grapes Total
Ratio 6 5 2 13
Multiplier 3 3 3 3
Real #s 18 15 6 39

And the answer to the question “how many more pounds of apples than grapes” is 18 – 6 = 12

Try it with questions 98, 125, 166 (and after those, experiment with #56).

With a little tweaking, we can use the box for other, more complicated ratio problems. Get out your Official Guide and read PS#66 (page 161) before looking below. This problem gives up 3 ratios to manage, so each one should get its own line:

1st 2nd 3rd 4th
Ratio 8 5
Ratio 3 4
Ratio 3 2
REAL

The trick is to find a place to plug in a “smart” REAL number, either for 2nd or 4th grade. Lets plug in 10 for 4th graders and see what happens. This time, imagine what your multiplier would have to be to keep the ratios straight.

1st 2nd 3rd 4th
Ratio 8 5
Ratio 3 4
Ratio 3 2
REAL 12 16 15 10

And the ratio of 1st to 3rd graders is 12:15 or 4:5.

Before moving on, you may want to check out Stack Koprince’s great article.

For “shifting ratio” problems, such as PS #105, the box is a great place to start (and finish). However, we’re back to using a variable for our multiplier:

Students Teachers
Ratio 30 1
Multiplier x x
Real #s 30x 1x

After the shift, the new number of students is (30x + 50), and the new number of teachers is (x + 5). Now use our good-ole proportion to work it out:

neil thornton ratio image

Leaving us with x = 15. Plug that back into the original chart to get your original number of teachers = 15!

The ratio box is a good quick way to finish out tough weighted average and weighted percent problems, when you get to equations that look like 3b = 5g or 2x = y.

For example, at the very end PS#224, that wonderful “wheat/fescue” problem, you should end up–after all your hard work–with the simple:

2x = y

I’ve seen more students make the worst mistakes right here! Does that mean 50% or what? Well let’s plug in some numbers and set up a quick box.

If x = 1, then y = 2 and our box is

X Y Total
Ratio 1 2 3

And the fraction of the mixture is 1/3, so the percent of the mixture is 33 and 1/3%.

Now go make that drink.

 

* The text excerpted above from The Official Guide for GMAT Review 13th Edition is copyright GMAC (the Graduate Management Admissions Council). The short excerpts are quoted under fair-use statutes for scholarly or journalistic work; use of these excerpts does not imply endorsement of this article by GMAC.

 

 

 

Neil Thornton

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As the son of an engineer father and English-teaching mother, Neil Thornton has developed a unique ability to break down tough math for poets, as well as to get mathematicians excited about grammar. Neil started teaching SAT classes in 1991 while in college and learned to love beating test-writers at their own game. He’s coached hundreds (thousands?) of students through the GMAT, LSAT, MCAT-verbal, and SAT subject tests (Math level 2 is his favorite) and trained instructors all over the United States. He has scored 99th percentile on the GMAT, LSAT and GRE. Other than teaching, writing, and performing stand-up comedy, Neil spends his free time reading, running, camping, cooking, snowboarding, and juggling fire.

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