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

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), 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:

### GMATPrep Problem Solving: Kaye’s Stamps

Some people really like ratio problems while others struggle with these. What do you think?

Let’s talk about a go-to solution method when handling a problem of this type. Try this GMATPrep problem:

* ” The number of stamps that Kaye and Alberto had were in the ratio 5 : 3, respectively. After Kaye gave Alberto 10 of her stamps, the ratio of the number Kaye had to the number Alberto had was 7 : 5. As a result of this gift, Kaye had how many more stamps than Alberto?

“(A) 20

“(B) 30

“(C) 40

“(D) 60

“(E) 90”

My very first thought as I read this problem: I have to be very careful with my work here, because it would be really easy to solve for the wrong thing (and, of course, that wrong answer will probably be among the answer choices).

As an aside, I’ve found that this attitude is one of the biggest differences between someone who has the potential to hit a top score on quant and someone who won’t make it. When you see something and you think, “I know how to do this!” the top test-taker is going to go in The Zone and pay even more attention to detail, thinking “I am going to be really careful not to make a mistake on this one!” Someone who isn’t going to hit a tip-top score will instead start to coast a little mentally, thinking, “Yeah, I’ve already got this.” Even worse, someone might think, “I can speed up on this one since I know how to do it.”

No! Don’t speed up! You don’t necessarily have to take the full 2 minutes, but don’t go any faster than you’d normally go. Don’t increase the chances that you make a careless mistake!

Okay, let’s solve this thing.

First, make very clear on your scrap paper what you want: Kaye NEW minus Alberto NEW. Not just Kaye (new or old). Not Kaye’s original number of stamps minus Alberto’s original number.

Skip a few lines and write this on the scrap paper and put a big circle around it: K_{n }– A_{n}. Do the actual work up above this text and, when you’re done, you’ll “run into” the reminder that you want Kaye NEW minus Alberto NEW.

Also, make sure you organize your work carefully as you go so that you know which portions represent the original numbers versus the new ones.

Let’s see.

### Part to Part and Part to Whole Ratios

Take a look at the following problems.

Data Sufficiency: What was Company X’s percentage profit in 2011?

1) The ratio of costs to profits for Company X was 3 to 1 in 2011.

2) Company X’s costs in 2011 were $360,000.

A recipe for punch calls for 4 parts seltzer to one part juice. If John wants to make 5 gallons of punch, how many 8 ounce cans of juice does he need (1 gallon = 128 ounces)?

A) 32

B) 20

C) 16

D) 10

E) 8

Both these problems have something in common. We are given a ratio, but not the specific ratio that would be most useful to our calculation. In the first case, the percentage profit can be calculated at profit/revenue * 100 (remember revenue = profit + costs). In the second case, the ratio that would be most helpful to solving is the ratio of juice to punch, which we could then use to calculate how many ounces of juice we need. It is common for test takers to get stuck at this point not knowing how to get to the ratio needed, but with a little understanding of ratios, it is actually a rather simple task to convert from the given ratios to the ratios that would be most helpful.