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# Which of the following pairs of integers have reciprocals whose sum is

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Math Expert
Joined: 02 Sep 2009
Posts: 60647
Which of the following pairs of integers have reciprocals whose sum is  [#permalink]

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10 Dec 2019, 01:45
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Difficulty:

35% (medium)

Question Stats:

65% (01:47) correct 35% (02:02) wrong based on 75 sessions

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Which of the following pairs of integers have reciprocals whose sum is either less than $$\frac{1}{3}$$ or greater than $$\frac{1}{2}$$?

I. 1 and 14
II. 5 and 10
III. 7 and 8

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

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Re: Which of the following pairs of integers have reciprocals whose sum is  [#permalink]

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10 Dec 2019, 09:39
for (I), it is clear that the sum is $$>\frac{1}{2}$$ because the reciprocal of 1 is 1 --> yes

for (II), $$\frac{2}{10}+\frac{1}{10} = \frac{3}{10}$$ $$= 0.3 < 0.3333 = \frac{1}{3}$$ --> yes

for (III), $$\frac{1}{7}+\frac{1}{8} = \frac{7+8}{56} = \frac{15}{56}$$ --> 15 is less than third 56 so $$<\frac{1}{3}$$ --> yes

E
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Re: Which of the following pairs of integers have reciprocals whose sum is  [#permalink]

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16 Dec 2019, 23:37
For this question we start off by taking the fraction equivalents of decimals and see which statements satisfy the conditions given.
We also keep in mind that one of the conditions needs to be satisfied – either less than 1/3 or more than 1/2. Clearly, it cannot be both, right?

1.3 = 0.3333.. and ½ = 0.50. So, the sum of the reciprocals should be less than 0.33 or more than 0.50.

The integers in statement I are 1 and 14. Reciprocals of these integers are 1 and 1/14 respectively. 1/14 = 0.0714… The sum of these values = 1.0714. This is definitely more than 0.50. Statement I has to be a part of the answer. So, answer options B and C can be eliminated.

The integers in statement II are 5 and 10. Reciprocals of these integers are 1/5 and 1/10 respectively. 1/5 = 0.20 and 1/10 = 0.10, which add up to 0.30. This is less than 0.33. Statement II is also true. Answer option A can be eliminated.

The integers in statement III are 7 and 8. Reciprocals of these integers are 1/7 and 1/8. 1/7 = 0.1428 and 1/8 = 0.125. Adding these two will give us a sum less than 0.33. Statement III is also true, based on which answer option D can be eliminated.
The correct answer option is E.

Questions like these test your calculation skills, so it makes sense to work on honing them so that you are quick in answering these sort of questions.

Hope that helps!
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Re: Which of the following pairs of integers have reciprocals whose sum is   [#permalink] 16 Dec 2019, 23:37
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