If S is the sum of the reciprocals of the 10 consecutive integers from 21 to 30 , then S...

Question

If S is the sum of the reciprocals of the 10 consecutive integers from 21 to 30 , then S is between which of the following two fractions?

GMAT Panda · Accepted Answer

S is between 1/3 and 1/2. We translate "S is the sum of the reciprocals of the 10 consecutive integers from 21 to 30" into an equation. S = 1/21 + 1/22 + 1/23 + 1/24 + 1/25 + 1/26 + 1/27 + 1/28 + 1/29 + 1/30 We bound S by noting that each term is at least as large as the smallest reciprocal and at most as large as the largest reciprocal. S > 10 1/30 S < 10 1/21 We simplify the lower bound and then check that the upper bound is less than one-half by comparing cross-products. 10 1/30 = 10/30 = 101103 10 1/21 = 10/21 2 10 = 20 1 21 = 21 20 < 21 10/21 < 1/2

Explanation	Calculations	Help
$We translate "S is the sum of the reciprocals of the 10 consecutive integers from 21 to 30" into an equation.$	$S = \frac{1}{21} + \frac{1}{22} + \frac{1}{23} + \frac{1}{24} + \frac{1}{25} + \frac{1}{26} + \frac{1}{27} + \frac{1}{28} + \frac{1}{29} + \frac{1}{30}$
$We bound S by noting that each term is at least as large as the smallest reciprocal and at most as large as the largest reciprocal.$	$S > 10 \times \frac{1}{30} S < 10 \times \frac{1}{21}$
$We simplify the lower bound and then check that the upper bound is less than one-half by comparing cross-products.$	$10 \times \frac{1}{30} = \frac{10}{30} = \frac{10 \times 1}{10 \times 3} 10 \times \frac{1}{21} = \frac{10}{21} 2 \times 10 = 20 1 \times 21 = 21 20 < 21 \frac{10}{21} < \frac{1}{2}$

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