Today’s GMAT challenge question comes from our friends at ManhattanGMAT. To help you with your GMAT studying, try to solve the problem on your own, and then read on for the explanation of its solution:
Problem
What is the 99th digit after the decimal point in the decimal expansion of 2/9 + 3/11?
(A) 1
(B) 2
(C) 4
(D) 7
(E) 9
Solution
There are two ways to approach this problem. We’ll go through them in turn.
(1) Figure out the decimal expansions of the two fractions separately (using long division), then add.
2/9 = 0.2222…
3/11 = 0.2727…
Thus, the sum is 0.4949… The first digit is a 4, as is the third digit, the fifth digit, and every digit after that in an odd-numbered position. (The digits in even-numbered positions are all 9’s.) Thus, the 99th digit must also be a 4.
(2) Add the fractions together, then figure out the decimal expansion.
2/9 + 3/11 = 22/99 + 27/99 = 49/99
Now, you can either figure out that the decimal expansion of 49/99 is 0.4949…, or you can simply know a shortcut: any two-digit number divided by 99 becomes a decimal with that two-digit number repeating. For instance, 17/99 = 0.1717…, 91/99 = 0.9191…, and so on. (This pattern generalizes to any number of digits, as long as the denominator is composed of only 9’s. For instance, 125/999 = 0.125125…)
The final analysis is the same: every digit in an odd-numbered position is a 4, so the 99th digit after the decimal point is a 4 as well.
The correct answer is (C).
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Read the full article: GMAT Tip: Decimating Decimal