ROn Point: Approximating Square Roots on the GMAT

During your preparation for the GMAT, you will learn myriad techniques, shortcuts, rules, exceptions and strategies. Unfortunately, even the best of us tend to draw a blank once or twice under test day pressure, so sometimes you may have to solve questions using deduction and strategic thinking more than with known mathematical identities and theorems. Consider the following question:

5 + 5√5
————-         = ?
10 + √500

A) ½
B) 2
C) 1 + √5
D) 1 + 5√5
E) 5 + √5

This is as straight-forward as quantitative questions get on the GMAT. There’s no need to interpret meaning, read through a dozen lines of text to discern the underlying question, or wonder which unknown needs to be solved. However, if you don’t remember the rules of algebra for square roots, you might not know where to start and spend a few seconds (or more) staring blankly at the question. For the sake of those of you who love algebra, let’s solve this the algebraic way, and then discuss workarounds for dealing with square roots.

A very good technique to get started on square root questions is to remember that √100 can be rewritten as √10 * √10. That is to say that the number under the square root can be broken down to a product of numbers under square roots. This is the distribution rule for exponents, and square roots are simply shorthand symbols for x^½. If you want to demonstrate this property quickly for square roots without using irrational numbers, turn √100 into √25 * √4. This quickly becomes 10 = 5 * 2. Keeping that in mind, it is always advantageous to take numbers out of square roots that are perfect squares, so that unwieldy numbers can become much more straightforward integers whenever possible.

In the question above, you can transform √500 into √100 * √5. Once you know to make that conversion, the problem becomes the more obvious form below:

5 + 5√5
————
10 + 10√5

However even this is not trivial without more algebra. If you take out 5/10 from the equation above, you get:

5 (1 + √5)
————–
10 (1 + √5)

This expression then simplifies to 5/10 or the more succinct ½, answer choice A.

Now that we’ve done the algebraic solution, what happens if we don’t remember the rules or we get a memory lapse (possibly because the test-taker in the corner keeps tapping their foot to the beat of Justin Bieber’s newest hit)? One simple solution is to approximate the equation. This may look daunting but is actually deceptively simple if you know your perfect squares, which is highly recommended at least until 15^2 and ideally even 20^2.

If the equation were 5 + 5*2, you could simply rewrite it as 15 (PEMDAS ensures that the multiplication gets done first). The equation above is almost exactly this number. √5 won’t give a round integer, but it is located between 4 and 9, which are 2^2 and 3^2 respectively. This guarantees you that √5 is between 2 and 3. Even with no more deduction than this, you can assume it is ~2.5 and write out the equation of 5 + 5√5 –> 5 + 5*~2.5 –> ~17.5. While this is not very accurate, the approximation does put us within ~10% of the actual answer, with no algebra required.

Looking at the denominator, the same technique can more or less be applied, but with a larger number as the GMAT is prone to use. √500 may not be as obvious as √5, but if you use the sign posts in a similar way as √5, you probably know that 20 * 20 = 400, much like 2 * 2 = 4 with a couple of extra 0s  Similarly, 25 * 25 = 625, which you might also recognize as 5^4 by virtue of being 5^2 * 5^2. It is not necessary to memorize these numbers, they can be deduced rather quickly should the need arise. This puts √500 roughly halfway in between these numbers, so you can approximate it to ~22.5 with confidence. This gives us a denominator of 10 + ~22.5 = ~32.5.

Using this approximation technique, a problem that relied heavily on algebraic manipulation can be made into a pure arithmetic question. We now have ~17.5 over ~32.5, which is just over ½. Using the same approximation for the answer choices:

A) ½                   –> ½                          = ½
B) 2                    –> 2                           = 2
C) 1 + √5           –> 1 + ~2.5              = ~3.5
D) 1 + 5√5        –> 1 + 5*(~2.5)        = ~13.5
E) 5 + √5           –> 5 + ~2.5              = ~7.5

The only one that’s even close is answer choice A. This will often be the case on GMAT answer choices, whereby a close enough approximation will yield the correct answer choice, or worst case a tossup between two choices. If you get good at this technique, you may actually solve questions faster than using algebra. Either way it’s another tool in your arsenal that can be deployed when necessary.

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Ron Awad is a GMAT instructor for Veritas Prep based in Montreal, bringing you occasional tips and tricks for success on your exam.  After graduating from McGill and receiving his MBA from Concordia, Ron started teaching GMAT prep and his Veritas Prep students have given him rave reviews ever since.

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