Problem Solving

This is from Manhattan Gmat challenge problem of the week:
https://www.beatthegmat.com/mba/2012/08/ ... ugust-2012


Rounded to four decimal places, the square root of the square root of 0.9984 is approximately

A. 0.9990
B. 0.9992
C. 0.9994
D. 0.9996
E. 0.9998




"Now we have the insight we need. Since the square of 1 - x is approximately 1 - 2x (doubling the gap between the number and 1) if x is very small, then we can go in the opposite direction: the square root of 1 - 2x is approximately 1 - x. In other words, you cut the gap between the number and 1 in half."

More specifically, above is the part of the explanation that I don't understand (not good considering this is the 'insight we need'). I don't understand how you get square root of (1-2x) roughly equals (1-x).




Answer

First, notice that the number you have to take the first square root of, 0.9984, is just a little less than 1, meaning that you could represent it as 1 - (something small).

Now, it's hard to deal with square roots algebraically. But we can deal with their opposites - that is, squares. What would the square of 1 - (something small) be? Let's write that as 1 - x, where we know that x is a small number, much less than 1.

(1-x)^2 = 1 - 2x - x^2

Now, since x is much less than 1, the term is much much less than 1. (To see why, imagine that x = 1/1,000. Then = 1/1,000,000.) Since we are rounding in this problem, we can make an approximation, dropping the term:

(1-x)^2 = 1- 2x - x^2 ≈ 1 - 2x

Now we have the insight we need. Since the square of 1 - x is approximately 1 - 2x (doubling the gap between the number and 1) if x is very small, then we can go in the opposite direction: the square root of 1 - 2x is approximately 1 - x. In other words, you cut the gap between the number and 1 in half.

Write 0.9984 as 1 - 0.0016. In this case, 2x = 0.0016, so x = 0.0008.

The square root of 1 - 0.0016 is approximately 1 - 0.0008, or 0.9992.

Take the final step. The square root of 1 - 0.0008 is approximately 1 - 0.0004, or 0.9996.

You could also get to the answer by working backwards from the answer choices: the square of the square of the right answer must be approximately 0.9984. It will take longer, but brute force will get you there, eventually.

The correct answer is D.