If x is the sum of the reciprocals of the five consecutive even

Question

If x is the sum of the reciprocals of the five consecutive even integers starting with 40, which of the following is true?

(A) 0.00 <x<0.05
(B) 0.05 <x<0.10
(C) 0.10 <x<0.15
(D) 0.15 <x<0.20
(E) 0.20 <x <0.25

This is a  PS Butler Question

Check the links to other Butler Projects:

Data Sufficiency Butler  
  Critical Reasoning Butler

Bunuel · Accepted Answer

GMATNinja Video Solution:

https://www.youtube.com/watch?v=L8Qy5OexFag

modiassumenda · Answer

The five consecutive even integers starting with 40 are:
40, 42, 44, 46, 48
So,
Each reciprocal is a little bigger than 1/50 = 0.02.
There are 5 terms, so x is a little bigger than
5 × 0.02 = 0.10
Also each term is less than 
1/40 = 0.025, so
x < 5 × 0.025 
= 0.125
Thus,
0.10 < x < 0.125
which immediately gives:
 (C) 0.10 < x < 0.15.

vasu1104 · Answer

consecutive even integers: 40,42,44,46,48

sum of reciprocal of these numbers: 1/40 + 1/42 + 1/44 + 1/46 + 1/48

the options are giving us the range. so we can use that in our advantage and find the approximate range.

consider two extreme in here. first and last number.
assume all are 40.

so that sum would be, 5/40=0.125

now consider the extreme and that's 48.
sum would be, 5/48.

now 5/48 is between 5/45 and 5/50. but its more closer to 5/50.
so its close to 0.10

so our range is from 0.10 to 0.125

this range fits into choice C.

harjas2222 · Answer

if someone can tell me how to do it in short I will be grateful

Here i bit the bullet

5/2 (1/48 + 1/40)

5/2 (11/240) = 11/96 = .104x (x is the numbers coming after 4)

.104x> 0.1 Thus answer C but it took me around 2 minutes 30 seconds which will waste my time in the exams

Thanks and have a great day!

AllisonBell · Answer

Great question!

Estimation  is a helpful tool here. I started by noticing that these numbers are all pretty close to, but greater than, 1/50.

1/50 = 2/100 = .02. If all the numbers were equal to 1/50, we could find the sum by multiplying .02*5 = .1. Since the numbers are all greater than 1/50, the sum must be greater than .1. We can eliminate choices A and B.

If we were really looking to save time, we could take an educated guess and select choice C, since it’s the only option with .1 as the lower boundary of the range.

However, we can also do a quick evaluation of the upper boundary in choice C. If we divide .15 by 5, we would get a maximum average value of the terms of .03 or 3/100. If we divide the numerator and denominator by 3, this reduces to approximately 1/33, which is a larger value than any of the numbers in the set. So this works as an upper boundary.

For more practice problems and efficient solutions, check out ManhattanPrep’s   Free GMAT QBank and Starter Kit  .

Best,

egmat · Answer

Hi harjas2222,

Your math is perfectly correct, and the averaging trick ( 5/2 × (1/40 + 1/48) ) is actually a clean idea. The reason it ate  2.5  minutes is that you computed it  exactly  - finding  11/240 , then  11/96 , then converting to a decimal. The answer choices never asked for that precision.

Notice how wide the buckets are.  Each option spans  0.05 . That's a huge window. When the ranges are that loose, you don't need the real value of  x  - you just need to trap it between two quick numbers.

The fast route: bound it in seconds

The five terms are  1/40 ,  1/42 ,  1/44 ,  1/46 ,  1/48  - all very close together.

-  Biggest possible:  pretend all five are the largest,  1/40 = 0.025 . Then  5 × 0.025 = 0.125 .
-  Smallest possible:  pretend all five are the smallest,  1/48 ≈ 0.0208 . Then  5 × 0.0208 ≈ 0.104 .

So  x  is squeezed between  0.104  and  0.125  - both sit inside  0.10-0.15 , so the answer is  C . No exact fractions, no common denominators.

Even faster - one middle term

Since the five numbers are nearly identical, just take the middle one:  1/44 ≈ 0.0227 , times  5  ≈  0.114 . Done in about  15  seconds.

The habit to build:  before grinding exact arithmetic, glance at how far apart the answer choices are. Wide ranges = estimate and bound. Save the precise computation for when the choices are tight.

Answer: C

If x is the sum of the reciprocals of the five consecutive even

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

If x is the sum of the reciprocals of the five consecutive even

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards