Recognizing Relative Numbers On The GMAT

Given the statement, the ratio of men to women in the room is 3 to 5, how many men are in the room?

You probably recognize pretty quickly that it is not possible to answer the question above.  Just given a ratio, it is not possible to identify the actual number of men in the room.  At this point we know the number of men in the room must be a multiple of 3, but the actual number could be 3 or 3,000 (although I am not sure I have been in a room that large).

Along with ratios in their traditional form (3 to 5 or 3:5), there are other types of numbers that are ratios, slightly disguised

a) Fractions: The container is 2/3 full.

This statement is expressing that there are 2 full parts for every 3 total parts of the container (a ratio of 2 to 3).

b) Percentages: 33% of company employees have Master’s degrees.

This statement is expressing for every 33 employees with Master’s degrees there are 100 total employees (a ratio of 33 to 100).

c) Percentage or fractional increase: The company’s profits increased 25% (or ¼) from 2010 to 2011.

This statement expresses the company earned $1.25 profit in 2011 for every $1 profit in 2010.

Just like a traditional ratio, these fractions and percentages give us no sense of magnitude.  I have no ability to tell you exactly how large the container is, how many employees the company has, or how much profit was earned.  These types values collectively (ratios, fractions, percentages) will be referred to as relative numbers for the remainder of this post.  They are relative because they only provide information relative to another measure and do not provide the actual magnitude.   Once you can quickly recognize relative numbers, you can use this knowledge to make some important assessments in GMAT data sufficiency problems.

1) If you only have relative information, this information is not sufficient to determine a non-relative value.  So if a question asks for an non-relative value (e.g. gallons of water, total profit, number of teachers), it will never be possible to answer that question if you are only provided with relative information (percentage full, ratio of student to teachers, etc.).

For example, consider the partial data sufficiency question below.

Q: In 2010, 60% of Company X’s employees are female.  How many male employees does company X have?

1) The number of male employees increased by 1/5 from 2009 to 2010.

This question asks me to determine a non-relative number (the number of male employees).  If I look at the information provided in the question (60% female) and statement 1 (increased by 1/5), I see I am only provided with relative information.  Because I have no non-relative information, I know I will not be able to answer this question without writing any equations or doing any significant calculations.  I have no sense of magnitude – the size of the company – so I can dismiss answers A (Statement 1 is sufficient) and D (Either statement is sufficient) both confidently and quickly – and any place you can save even a few seconds is valuable on the GMAT.

I will caution you to only make this quick elimination if you are certain that you actually have no non-relative information in the question or the statement.

For example, with the question stem below statement 1 is sufficient:

Q: The number of male employees at company X increased by 20 from 2009 to 2010.  How many male employees did company X have in 2010?

1) The number of male employees increased by 1/5 from 2009 to 2010.

In this case, although the statement provides only a relative number, a non-relative number is provided in the question (increase of 20 male employees).  Now just because you have a non-relative number does not mean that the statement is sufficient (in the example above it is), it just opens up the possibility.

Let’s consider another similar example

Q: In 2010, 60% of Company X’s employees are female.  How many male employees does company X have?

1) If 10 additional male employees had been hired and no other employees left or joined the company, the ratio of male to female employees would have been 5 to 6 in 2010.

The question stem in this case provides only a relative number.  But the statement is more complex.  While it does contain a relative number (ratio of 5 to 6), it also contains an actual number (10 employees).  In this case, I would not be able to quickly dismiss this statement and would actually have to take the time to evaluate it.  This statement is actually sufficient.

Bonus Fact (A similar principle applied to geometry)

If you only have angle measures you will never be able to determine any size-related aspects of a shape (side length, perimeter, area).

2) Be suspicious if you think you need all the actual numbers to answer a question about a relative number. 

For example, consider the following question.

Q: By what percentage did Company Y’s profits increase from February to March?

This question asks us for a relative value (percentage increase).  The traditional way to calculate a percentage increase is to apply the following formula:

(New-Old)/Old × 100

But in data sufficiency, GMAT writers love making you pay for only thinking in traditional terms.  While I know that I can calculate the percentage change in this manner, I want to be aware of other possibilities.  Specifically, in this case, because the question asks for a relative value, I may be able to calculate this answer with only relative numbers.

Suppose the statement were the following:

1) In February, Company Y’s profits were $3,600.

2) The ratio of profits in February to profits in March for Company Y was 3 to 4.

Using both these statements together (I know we are not supposed look at both statements together first, but humor me for teaching purposes), I am given the profit for February ($3,600) can calculate the profit for March ($4,800).  With these two pieces of information I can apply the percentage change formula and answer.  But before I submit an answer of C (Both statements together are sufficient), a voice inside my head should say, “The folks who write the GMAT don’t usually roll that way. Do I really need all these actual numbers to calculate an answer to this relative question?”

I can look at Statement 1 and quickly realize it is not sufficient because it provides no information on March, but Statement 2 does make a comparison between February and March.  Specifically, I could say that the profit in February = 3x and the profit in March = 4x.  If I plug that into my formula for change, the x’s cancel out allowing me to calculate the percentage increase (33.3% although there is no need to complete this calculation once I know the variables cancel out).  Although based on this statement I have no idea if the company is earning in the hundreds or hundreds of millions of dollars, I can answer the relative question provided.

Sometimes you may answer a question using the traditional way (e.g. need all the actual numbers to answer a relative question), but it always pays to view GMAT questions with a healthy dose of suspicion.