GMAT Practice Question: 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?
- \frac{1}{3} and \frac{1}{2}
- \frac{1}{4} and \frac{1}{3}
- \frac{1}{5} and \frac{1}{4}
- \frac{1}{6} and \frac{1}{5}
- \frac{1}{7} and \frac{1}{6}
Topics: fractions, consecutive integers, approximations, inequalities
Solution
Step 1
We translate \textbf{"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}
Step 2
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}
Step 3
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{\cancel{10}\times1}{\cancel{10}\times3} 10 \times \frac{1}{21} = \frac{10}{21} 2 \times 10 = 20 1 \times 21 = 21 20 < 21 \frac{10}{21} < \frac{1}{2}
Answer
S is between \frac{1}{3} and \frac{1}{2}.
