GMAT Math : Bounding approximations

Bounding approximations is a fantastic tool to solve some of the hardest GMAT numeric series questions.

First, an example :

Let x = 1/90 + 1/91 + 1/92 + 1/93 + 1/94 + 1/95 + 1/96 + 1/98 + 1/99 + 1/100, then which of the following is true of x?

A. 0.9 < x < 0.1

B. 0.1 < x < 0.12

C. x > 0.13

D. x < 0.05

E. x > 0. 15

Hmm, where to start?

Notice, the terms seems to be bounded by 1/90 and 1/100

1/91 is greater than 1/100 but less than 1/90

1/92 is greater than 1/100 but less than 1/90

1/93 is greater than 1/100 but less than 1/90

…. 1/99 is greater than 1/100 but less than 1/90

Interesting!

Hence, it is fair to say

1/90 + 1/91 + 1/92 + 1/93 + 1/94 + 1/95 + 1/96 + 1/98 + 1/99 + 1/100  is less than 1/90 + 1/90 + 1/90 + 1/90 + 1/90 + 1/90 + 1/90 + 1/90 + 1/90 + 1/90 (notice how each term in the second series is less than each term of the first series save for 1/90 which is the same as the first term)

So, 1/90 + 1/91 + 1/92 + 1/93 + 1/94 + 1/95 + 1/96 + 1/98 + 1/99 + 1/100  < 10 x (1/90) = 1/9 = 0.1111

So, x < 0.111

Similarly, 1/90 + 1/91 + 1/92 + 1/93 + 1/94 + 1/95 + 1/96 + 1/98 + 1/99 + 1/100 > 1/100 + 1/100 + 1/100 + 1/100 + 1/100 + 1/100 + 1/100 + 1/100 + 1/100 + 1/100

Hence, x > 10 (1/100) = 0.1

So 0.1 < x < 0.111

The only option that fits this is (B). Elegant solution, isn’t it J

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