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The Quest for 700: Weekly GMAT Challenge (Answer)

Yesterday, Manhattan GMAT posted a GMAT question on our blog. Today, they have followed up with the answer:

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.

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