After seeing quite a few Integrated Reasoning problems floating around out there, I’ve found that one of the toughest situations to deal with is when instead of providing a single solution, the GMAT constructs a world with multiple possible solutions and then asks you to pick something that works within those parameters. Let me show you an example:

x, y and z are positive integers. The sum of x and y is 40. The positive difference between y and z is 20.

In the table below, identify values for x and z that are together consistent with the information. Make only one selection in each column.

x z
15
20
25
45
60

Found the answer yet? If not, I think I might know why: You’re trying to solve for y. The problem is, y could be almost any integer from 1 to 39, as long as you pick values for x and z that work. You could figure out x and z for every single value of y, but that’s a very time-consuming strategy! Without the answer choices, there are more than 50 different solutions to this problem. So what is a better strategy than trying to solve for y?

The way the question is phrased should give you a big hint: It asks, identify values for x and z that are together consistent with the information. This means that x and z could probably be two or more of these values, but there is only one pair that works. Because of this, I solved this problem by asking: What is the relationship between x and z?

Take another crack at the problem with this question in mind, and see if you can come up with the solution. You can solve using algebra or trial and error, as long as you understand that this is not a question about y “ it’s a question about x‘s relationship to z.

The only remaining challenge is to parse the phrase The positive difference between y and z. The difference between y and z is just y minus z, and to make it positive we just need to take the absolute value. This gives us a system of equations:

x + y = 40

|y z| = 20

And the sum of x and z is therefore either 20 or 60. None of the numbers pair to make 20. Only 15 and 45 pair to make 60. x must be 15 (if x were 45, then y would be negative), so y must be 25 and z must be 45.

You could also solve the problem by making a simple table:

x y y + 20 y - 20
15 25 45 5
20 20 40 0
25 15 35 -5
45 -5 n/a n/a
60 -20 n/a n/a

Takeaway: The only way to know if an IR problem has multiple solutions is to read the question carefully and look for the signals. Once you know what you are dealing with, use the answer choices to help you solve.

#### Ryan Jacobs

Ryan Jacobs is the son of an English teacher and a math teacher, which probably means he was fated from birth to teach the GMAT. He scored a 760 on the GMAT and a perfect 1600 on the SAT, both on his first attempt. He has a bachelor's degree in Music Composition as well as an MBA, both from the University of California, San Diego.

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