Data Sufficiency - When to Plug in Values

After reading a few solutions in various Data Sufficiency forums, one notices that sometimes people plug in values to test sufficiency, and sometimes they use a different approach. So, how do we identify those times when the best strategy is to plug in values?

To answer this question, we must first recognize the strengths and limitations of plugging in numbers.

Plugging in values can prove that a statement is not sufficient

Consider this question:

If integer n is greater than 1, is 2^n – 1 prime?

1) n is even  

Let’s plug in some even n-values.

Try n = 2. We get 2^n – 1 = 2^2 – 1 = 3, and 3 is prime.

Try n = 4. We get 2^n – 1 = 2^4 – 1 = 15, and 15 is not prime.

Great, these conflicting answers to the target question tell us that statement 1 is definitely not sufficient. DONE!

Plugging in values cannot (typically) prove that a statement is sufficient

Consider this question:

If integer n is greater than 1, is 2^n – 1 prime?

1) n is prime 

Let’s plug in some prime n-values.

If n = 2, we get 2^n – 1 = 2^2 – 1 = 3, and 3 is prime.

If n = 3, we get 2^n – 1 = 2^3 – 1 = 7, and 7 is prime.

If n = 5, we get 2^n – 1 = 2^5 – 1 = 31, and 31 is prime.

If n = 7, we get 2^n – 1 = 2^7 – 1 = 127, and 127 is prime.

If n = 11, we get 2^n – 1 = 2^11 – 1 = 2047, and 2047 is prime.

So, is statement 1 sufficient?

Well, it looks like 2^n – 1 will always be prime whenever n is prime, but can we be certain? No, cannot be certain. All we can conclude is that the evidence seems to suggest that 2^n – 1 will prime when n is prime.  Since we can’t continue plugging in every possible n-value, we might stop here conclude that statement 1 is probably sufficient.

Well, it turns out that statement 1 is not sufficient. For example, if n = 89, we get 2^89 – 1, which evaluates to some gigantic non-prime number.

So, while plugging in numbers may help to suggest that a statement is sufficient, this strategy typically fails to provide the level of certainty we’d like on test day.

Aside: The two questions we’ve examined so far are well beyond the scope of the GMAT. See http://primes.utm.edu/mersenne/  for more information. I’ve merely used them to demonstrate the strengths and weaknesses of plugging in numbers.

Speed issues

While plugging in values may help prove that a statement is not sufficient, the time required to do so depends on whether or not we plug in the right numbers. If we fail to plug in the right numbers, then we can waste valuable time (or even draw the wrong conclusion). Consider this example: 

Is k^2 – 12k + 35 > 0?

1) k is an even integer

Even if we suspect that statement 1 is not sufficient, it may take some time to find k-values that yield conflicting answers to the target question. In fact, almost all even values of k will yield the same answer to the target question. For example, if we plug in k = 0, 2, -2, 10, -10, 16 and -24, then k^2 – 12k + 35 evaluates to be greater than 0 every time. The only value that yields a different answer to the target question is k = 6. When k = 6, then k^2 – 12k + 35 = -1, and -1 is less than 0, which means statement 1 is not sufficient.

Aside: a faster and more reliable approach is to factor the expression and go from there (I’ll leave that to you).

As you can see, our ability to draw fast conclusions depends largely on our ability to plug in “good” values. For more on choosing good number to plug in, see this free video.

When to plug in values

So, plugging in values allows us to make definitive conclusions about a statement only when that statement is not sufficient. Also, plugging in values can take precious time. Given these potential drawbacks, there are only two situations in which we should consider this strategy. 

1) When we suspect that a statement is not sufficient

2) When we don’t know how to proceed any other way.  

Of course, suspecting that a statement is not sufficient does not mean we should head straight to plugging in values. As with all GMAT quantitative questions, you should consider all of your options before choosing an approach.

Practice question

So, what should we do with this partial question?

If xy 0, is (x+y)/y > 1?

1) x and y are both positive

If we can think of an algebraic approach here, we should probably proceed with that, since plugging in values may take time. 

However, if we don’t see another approach, then we might start plugging in values. Likewise, if we feel that statement 1 is not sufficient, then we might consider plugging in values.

Let’s plug in some x and y values that satisfy the condition that x < y

If x = 2 and y = 1, then (x+y)/y = 3, and 3 is greater than 1.

If x = 0.2 and y = 0.5, then (x+y)/y = 1.4, and 1.4 is greater than 1.

If x = 1 and y = 4, then (x+y)/y = 1.25, and 1.25 is greater than 1.

If x = 80 and y = 20, then (x+y)/y = 50, and 50 is greater than 1.

Is statement 1 sufficient? Well, the evidence suggests that it’s sufficient, so we might make that conclusion and move on. If we did that, we’d be right; statement 1 is, indeed, sufficient.

To reach a more definitive (and faster) conclusion, we’ll use some algebra. First take (x+y)/y and rewrite it as x/y + y/y and then simplify this to get x/y + 1.

So, we can now rephrase the target question as “Is x/y + 1 > 1?

From here, if we subtract 1 from both sides of the inequality, we get

If xy does not equal 0, is x/y > 0?

Perfect. Now that we’ve rephrased the target question, we can handle statement 1 with relative ease. If x and y are both positive, then x/y must be positive, which means x/y is definitely greater than 0. Since we can answer the rephrased target question with certainty, statement 1 is sufficient.

Final words

Plugging in values is a useful strategy to include in our arsenal of Data Sufficiency techniques. So, be on the lookout for situations in which you can use it. For more information about this strategy, watch our free video

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