The Many Ways the GMAT Tests One Concept

Introduction

The quantitative section of the GMAT tests a discrete number of fairly basic math topics. The difficulty of the test hinges more so on the complex presentation of these concepts than on the concepts themselves. In fact, sometimes these concepts appear in such a way that they are completely hidden to the average test-taker. One such topic that often goes unnoticed on the GMAT is ratios.

Ratios show up on the GMAT in a number of formats. There are some questions that explicitly ask about ratios. These questions are usually pretty straightforward. Perhaps of greater interest are the questions for which ratios or a related concept can be used to solve, but in which the word ratio is never mentioned. For example, many data sufficiency questions have a statement that provides a ratio between two variables. In a large percentage of cases, this type of statement will be sufficient because a ratio innately provides more information about two variables than one would expect. As another example, many word problems on the GMAT contain the concepts of direct or inverse proportionality. With a good understanding of how ratios can be applied in these cases, these problems can be solved with much greater ease. In this month's strategy series we will take an in depth look at ratios, both in their explicit and implicit presentation.

We will begin our discussion with a look at the basics of ratios. Simply put, a ratio expresses the relationship between two or more things. The relationship that is described is a multiplicative one. This means that if you are given one of the values, the relationship will dictate a factor by which you need to multiply that value to obtain the other value(s). Let's take a look at an example.

The ratio of men to women in a room is 5 to 4 (NOTE that this ratio can also be expressed as 5: 4 and as the fraction, 5/4)

What exactly does this ratio mean? It means that for every five men in the room, there are four women. The number of men and the number of women in the room are not necessarily 5 and 4, but they must be some multiple of 5 and 4 (e.g. 10 men / 8 women, 15 men / 12 women, etc). One of the most helpful ways to notate ratios is by using something called the "unknown multiplier." If the ratio of men to women is 5 to 4, algebraically this can be denoted as 5x: 4x or 5x/4x, where x represents the "unknown multiplier." Note that in the case of indivisible units like people, x must be an integer.

How does the notation of the unknown multiplier help us? Let's extend our simple example to a question. If we are told now that the total number of people in the room is 54, how many more men then women are there?

According to our ratio, the number of men = 5x and the number of women = 4x.
The total number of people = 5x + 4x = 54, so 9x = 54, and x = 6.
The question asks us for the difference between the number of men and women, which must be 5x - 4x = x, or simply 6.


Let's look at a slightly more difficult application of the unknown multiplier.

The ratio of men to women in a room is 5 to 4. If 9 additional women enter the room, the new ratio of men to women will be 10 to 11. How many women are there in the room originally?

The original ratio can be expressed as 5x: 4x. With the addition of 9 women, the new ratio becomes 5x: 4x + 9, which we are told is equal to 10:11. We can set these two ratios equal to each other in a proportion to solve for x.

5x / (4x + 9) = 10/11
55x = 40x + 90
15x = 90
x = 6
This means that the number of women in the room originally is 4x or 4(6) = 24.

Use of the "unknown multiplier" and the ability to set up proportions are key to solving explicit ratio questions on the GMAT.

Ratios

One of the challenges that the GMAT test-writers face when writing Data Sufficiency questions is how to conceal the information - that is how to make statements look like they contain less information than they actually do. A commonly used means to achieve that end is to put the information in a statement that provides a ratio. The average test-taker is taken aback because no values are offered for the variables in the question, yet somehow, the relationship that the ratio provides between the two variables is enough to solve the question.

What exactly do we mean by a statement that contains a ratio? The obvious answer is a statement that contains the word "ratio." However, the use of ratios in Data Sufficiency statements goes beyond the obvious. In fact any statement which establishes an algebraic relationship that can be used to derive a ratio qualifies. Let's give some examples.

STATEMENTS WHICH PROVIDE A RATIO
y = 2x. This gives us a ratio of x to y (or y to x) because it can be solved for the expression x/y.

Peter's speed is twice that of John's (p = 2j, which is a ratio for the same reason as is the previous example).

The profit from this year's accounts increased by 30% over the profit from last year's accounts. (t = 1.3l)

STATEMENTS WHICH DO NOT PROVIDE A RATIO
Peter traveled for 2 hours longer than John. (p = j + 2, which cannot be solved for the expression p/j).

If Andrew had sold 10,000 more dollars this month, he would have sold twice as much as Henry did this month (a + 10,000 = 2h, which cannot be solved for the expression a/h)

Now let's take a look a two sample Data Sufficiency questions that contain ratios in the statements. The first example is an algebra Data Sufficiency question, the second a word problem. EXAMPLE: What is the value of p if if rq is not equal to 0?

(1) r = 4
(2) q = 3r

A Statement (1) ALONE is sufficient, but statement (2) is not sufficient
B Statement (2) ALONE is sufficient, but statement (1) is not sufficient
C BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient
D EACH statement ALONE is sufficient
E Statements (1) and (2) TOGETHER are NOT sufficient
Notice that statement (2) provides a ratio between the variables q and r. Let's pretend for a moment that you have been working on this question for 1.5 minutes to no ado. Time is dwindling, and you are feeling the familiar pressure of "it's time to move on." Is there a way to guess intelligently on a question like this one? The answer is yes, and it involves the use of the issue we have been discussing. BECAUSE DATA SUFFICIENCY STATEMENTS THAT ESTABLISH A RATIO BETWEEN TWO VARIABLES CONTAIN MORE INFORMATION THAN THEY APPEAR TO, THEY ARE OFTEN SUFFICIENT ON THEIR OWN. This is not a surefire strategy, but it is certainly enough to guess on.

Do not take this to mean that the inverse is true; statements which do not provide a ratio may in fact also be sufficient. Each statement must be checked for its own merit. In the question above, it turns out that the only statement which is sufficient is the second, and the correct answer is B.

Let's take a look at one more example - a word problem having to do with averages.

A group of men and women competed in a marathon. Before the competition, each competitor was weighed and the average weight of the female competitors was found to be 120 lbs, while the average weight of the men was found to be 150 lbs. What was the average weight of all of the competitors?

(1) 100 men competed in the marathon.
(2) There were twice as many men as women competing in the marathon.

The second statement provides us with a ratio. To calculate a weighted average, you need the average (or value) of each subgroup along with their ratio of number. With twice as many men as women, the men's average is given twice the weight of the women's. The number of men and women is irrelevant. Again we see that the ratio alone is enough to solve the question.

The trend of sufficiency of statements that contain a ratio is in no means a hard-fast rule. Nonetheless, when it comes to guessing and general pattern recognition on the GMAT, this can be a helpful guideline.

Proportions

A proportion is a relationship that is formed by setting two ratios equal to one another. It is used when the ratio between two values is known to be a constant.

On the GMAT, the need for proportions may arise in one of two ways. (1) The problem may explicitly use the term proportionate (or say that the ratio of x to y remains the same), or (2) proportionality may be implied. This latter type is of greater interest to us here.

What kinds of questions have implied proportionality? Geometry is one. In geometry questions having to do with similar triangles or other proportionate figures, relationships of proportionality can be used to simplify the solution. What do we mean by "proportionate figures?" Proportionate figures are two figures that have proportionately equal dimensions. Put differently, they are two figures that can be formed from one another using the enlarge/reduce button on a copy machine. Let's take a look at an example of how we can use proportionality on this type of question.

At a certain barn, the horses exercise in one of two circular arenas using what is called a lunge rope. A trainer stands in the center of the circular arena holding a rope attached to the horse, and the horse runs in a circular path along the perimeter of the arena. Assuming negligible distance between the horse's path of motion and the edge of the arena, approximately how many times as much rope is needed to lunge a horse in the larger arena if the larger arena has an area three times that of the smaller one?

(A) 1.5
(B) 1.7
(C) 2
(D) 3
(E) 6

This question basically deals with two circles and their radii (i.e. the amount of rope needed to lunge the horse). The two circles in this question can be considered "proportionate figures," because any two circles are proportionate. The proportion that we can use in this question relates the ratio of the radii of two circles to the ratio of the area of those two circles. The formula for the area of a circle is . Mathematically, this means that the area of a circle is proportionate to the square of the radius. If the radius of a circle is doubled, the area of the circle will be quadrupled. If the radius of a circle is tripled, the area of the circle will multiply by a factor of 9. We can represent this relationship in a proportion:



Let's call circle 1 the smaller circle. We know that the ratio of the areas in this question is 1:3. Since it is simply the ratio of the radii that we are concerned with in this question, we can assign a random value of 1 as the radius of the smaller circle and solve for x, the radius of the larger circle:



With a ratio of 1 to 1.7, the radius of the larger circle is 1.7 times larger than that of the smaller. Therefore, approximately 1.7 times as much rope is needed to lunge a horse in the larger arena, and the correct answer is B.

While the concept of proportionality is very often useful in geometry questions, there are many other word problems for which it can be applied.

Inverse Proportionality

Two variables are inversely proportional when an increase in the value of one variable causes a proportionate decrease in the value of the other.  Algebraically, this occurs when the product of the two variables is equal to some constant. To illustrate this concept, let’s consider the common rt = d formula for rate, time and distance.  If two cars travel the same route, we could say that their distance is fixed, i.e. a constant.  The formula rt = d becomes rt = (a constant), let’s say 240 miles.  If car A travels at 80 miles per hour, it will arrive in 3 hours.  If car B travels at 40 miles per hour, however, it will arrive in 6 hours.  Notice that car B’s speed is half that of car A, and as a result, the time it takes car B to arrive is twice the time of car A.

If a third car travels the same 240 miles at 60 miles an hour, it will make the trip in 4 hours.  Notice that car C’s speed (60) is 3/4 that of car A (80), but its travel time (4 hours) is 4/3 that of car A (3 hours).  We see that for a given ratio change to the rate, the time changes as the inverse of that ratio (4/3 vs. 3/4).  The variables of rate and time are inversely proportional for a fixed distance.

This relationship of inverse proportionality applies not only to rate problems on the GMAT, but also to any type of word problem that involves a formula that can be expressed as xy = z, where z is held constant.  Some common examples include: total cost (total cost = price x unit cost), work (rate x time = work), circular motion (speed = circumference x revolutions/sec).

Let’s take a look at an example that illustrates how this can be tested on the GMAT:

Alex and Peter both walked from the school library to the post office downtown. If Alex walked at an average pace of 5 miles per hour, what was Peter’s average pace for the same route?

(1)  It took Alex 48 minutes to walk to the post office.
(2)  It took Peter 25% longer to walk to the post than Alex. 

(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.

This question can be analyzed using the concept of inverse proportionality.  Alex and Peter are walking the same route, so their travel distances must be the same.  The variables of rate and time are inversely proportionate for a fixed distance. 

Statement 1 is insufficient.  While it is enough to tell us the distance from the library to the post office, we cannot find Peter’s rate.  If Alex walked at 5 miles per hour for 48/60 or 4/5 hour, he must have walked a total of 4 miles.  With no information about Peter’s time, we cannot find his pace.  Notice that there is no relational information in this statement (i.e. a ratio relating Peter’s time to Alex’s time), so we cannot capitalize on the relationship of inverse proportionality.

Statement 2, on the other hand, is sufficient. This statement provides us with the ratio of Peter’s time to that of Alex. If it took Peter 25% longer than Alex, then Peter’s time is 1.25 or 5/4 that of Alex. As we learned above, for a fixed distance, this means that Peter’s rate must be 4/5 that of Alex (rate and time are inversely proportionate). Since we know Alex's pace, the ratio relationship is enough to solve for Peter’s pace:

Peter’s pace = 4/5 x 5 miles per hour = 4 miles per hour.

The correct answer is B: Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.

The concept of inverse proportionality may very well be one of the most useful applications of ratios on the GMAT.  While questions that deal with inverse proportionality can typically be solved another way, utilizing inverse proportionality will allow you to solve problems in a much more insightful and timely manner. 

This concludes our close-up look at ratios. Hopefully, through this month's strategy series you have seen how the GMAT can take one topic and test it in a number of ways, ranging from the simple to the subtle and complex.

Copyright 2005, MG Prep, Inc.; May not be reprinted without permission.