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?

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

Explanation	Calculations	Help
$We translate "S is the sum of the reciprocals of the 10 consecutive integers from 21 to 30" into an equation.$	$S = \frac{1}{21} + \frac{1}{22} + \frac{1}{23} + \frac{1}{24} + \frac{1}{25} + \frac{1}{26} + \frac{1}{27} + \frac{1}{28} + \frac{1}{29} + \frac{1}{30}$	Theory & Tactics Method Card TRANS1-C Click to view full details
$We bound S by noting that each term is at least as large as the smallest reciprocal and at most as large as the largest reciprocal.$	$S > 10 \times \frac{1}{30} S < 10 \times \frac{1}{21}$
$We simplify the lower bound and then check that the upper bound is less than one-half by comparing cross-products.$	$10 \times \frac{1}{30} = \frac{10}{30} = \frac{10 \times 1}{10 \times 3} 10 \times \frac{1}{21} = \frac{10}{21} 2 \times 10 = 20 1 \times 21 = 21 20 < 21 \frac{10}{21} < \frac{1}{2}$	Theory & Tactics Method Card FDPR2-A Click to view full details

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S is between \frac{1}{3} and \frac{1}{2} .

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

Explanation	Calculations	Help
$We translate "S is the sum of the reciprocals of the 10 consecutive integers from 21 to 30" into an equation.$	$S = \frac{1}{21} + \frac{1}{22} + \frac{1}{23} + \frac{1}{24} + \frac{1}{25} + \frac{1}{26} + \frac{1}{27} + \frac{1}{28} + \frac{1}{29} + \frac{1}{30}$	Theory & Tactics Method Card TRANS1-C Click to view full details
$We bound S by noting that each term is at least as large as the smallest reciprocal and at most as large as the largest reciprocal.$	$S > 10 \times \frac{1}{30} S < 10 \times \frac{1}{21}$
$We simplify the lower bound and then check that the upper bound is less than one-half by comparing cross-products.$	$10 \times \frac{1}{30} = \frac{10}{30} = \frac{10 \times 1}{10 \times 3} 10 \times \frac{1}{21} = \frac{10}{21} 2 \times 10 = 20 1 \times 21 = 21 20 < 21 \frac{10}{21} < \frac{1}{2}$	Theory & Tactics Method Card FDPR2-A Click to view full details

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S is between \frac{1}{3} and \frac{1}{2} .