Variables in the Answer Choices – Part I – Two Approaches

In this article, we’ll examine two different approaches you can take when solving questions with variables in the answer choices (also known as VIACs). To begin, please try the following question:


Townville has X residents, and Y of them are females. If half of Townville’s male residents are smokers, and 1/Z of the female residents are smokers, which of the following represents the total number of Townville’s residents who are NOT smokers?


A) X – 2Y – Z

B) X - Y/Z - Y

C) 2X – Y - ZX

D) X/2 + Y/2 – Y/Z

E) X + 2Y – 4Z


Algebraic Approach


Since the question tells us about the residents who are smokers, let’s determine the total number of smokers and then subtract that amount from the total number of residents in Townville.


Male smokers: There are X residents, and Y are females. So, the number of males = X – Y.  If half of the males smoke, then the number of male smokers = (1/2)(X – Y) = X/2 – Y/2


Female smokers: There are Y females, and 1/Z of them smoke. So, the number of female smokers = (1/Z)(Y) = Y/Z


So, the total number of smokers = X/2 – Y/2 + Y/Z


The total number of NON-smokers = (total number of residents) – (total number of smokers)

= X – (X/2 – Y/2 + Y/Z)

= 2X/2 – X/2 + Y/2 - Y/Z

= X/2 + Y/2 – Y/Z


So, the correct answer is D



Input-Output Approach


For the Input-Output approach, we’ll arbitrarily select values for the variables and then use those chosen values to answer the question. We’ll then plug those same values into the answer choices and see which one yields the same answer to the question.


So, let’s say there are 10 residents in Townville (X = 10). Let’s also say that there are 6 females (Y = 6), and that 1/3 of them smoke (Z = 3).


We’ll now use these selected values to answer the question.


First, if 6 of the 10 residents are females, then the other 4 residents are males. Since half of the males smoke, there must be 2 male smokers.


Next, since 1/3 of the 6 females are smokers, we can conclude that there must be 2 female smokers.


So, in total, there are 4 smokers in Townville, which means there are 6 residents who do not smoke.


In other words, when we INPUT X = 10, Y = 6 andZ = 3, the OUTPUT is 6 non-smokers.


At this point, we’ll plug the three chosen values into the five answer choices to see which one yields and output of 6.


A) X – 2Y – Z:   10 – 2(6) – 3 =-5

B) X - Y/Z - Y:   10 – 6/3 – 6 = 2

C) 2X – Y – ZX:   2(10) – 6 – (3)(10) = -16

D) X/2 + Y/2 – Y/Z:   10/2 + 6/2 – 6/3 = 6

E) X + 2Y – 4Z:   10 + 2(6) – 4(3) = 10  


Since D is the only answer choice that correctly yields the output 6, it must be the correct answer. 




In this article, we examined two ways to approach VIACs. Most students prefer the Input-Output approach, because it involves little or no algebra, and because it often reveals the correct answer quickly. That said, this approach is not without issues.


In the next article, we’ll examine potential pitfalls of the Input-Output approach and how to mitigate them.

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