PS:Not able to derive final answer

[dhiren8182]

Hi Guru(s),
If K is the sum of reciprocals of the consecutive integers from 43 to 48, inclusive, then K is closest in value to which of the following?

A. 1/12
B. 1/10
C. 1/8
D. 1/6
E. 1/4
OA is A
Not sure how to derive the final answer with approximation
Thanks
Dhiren

[MartyMurray]

Hi Guru(s),
If K is the sum of reciprocals of the consecutive integers from 43 to 48, inclusive, then K is closest in value to which of the following?

A. 1/12
B. 1/10
C. 1/8
D. 1/6
E. 1/4
OA is A
Not sure how to derive the final answer with approximation
Thanks
Dhiren

There are six integers in that set, 43, 44, 45, 46, 47 and 48. Now I was looking at the OA you gave and check this out.

The smallest possible reciprocal is 1/48. Even if all of them were 1/48, the total would be 6 x 1/48 = 6/48 = 1/8. Your OA is even less than 1/8. So it must be incorrect.

Anyway, we are on our way to solving this problem.

The answer has to be a little bigger than 1/8, since the other reciprocals, 1/43, 1/44 and so on, are bigger than 1/48. So let's see which answer is closest.

Hmm. I see a nice clean way to get the other end of the range.

If all of the integers were 42, then the reciprocals would all be 1/42 and the total would be 6/42 or 1/7. So the real answer is somewhere between 1/7 and 1/8.

As numbers get higher, the difference between reciprocals of numbers the same distance apart becomes less. The difference between 1/12 and 1/13 is much less than the difference between 1/2 and 1/3. So 1/7 is a little closer to 1/8 than it is to 1/6.

Do we even need to figure this out? Maybe not. Clearly this number is a fair amount smaller than 1/7.

In any case, the sum of the reciprocals is closer to 1/8 than it is to 1/6.

Choose C.


_________________
Marty Murray
GMAT Coach
[email protected]
https://infinitemindprep.com/

[Brent@GMATPrepNow]

If K is the sum of reciprocals of the consecutive integers from 43 to 48, inclusive, then K is closest in value to which of the following?

A. 1/12
B. 1/10
C. 1/8
D. 1/6
E. 1/4

We want the approximate sum of 1/43 + 1/44 + 1/45 + . . . + 1/48

Let's make the following observations about the upper and lower bounds:
Upper bounds: If all 6 fractions were 1/43, the sum would be (6)(1/43) = 6/43 ~ 6/42 = 1/7
Lower bounds: If all 6 fractions were 1/48, the sum would be (6)(1/48) = 6/48 = 1/8

From this we can conclude that 1/8 < K < 1/7

If K is between 1/8 and 1/7, then K must be closer to 1/8 than it is to 1/6

Answer = C

Cheers,
Brent

[GMATGuruNY]

Hi Guru(s),
If K is the sum of reciprocals of the consecutive integers from 43 to 48, inclusive, then K is closest in value to which of the following?

A. 1/12
B. 1/10
C. 1/8
D. 1/6
E. 1/4

I received a PM about this problem.
My approach would be similar to Marty's and Brent's.

Determine the RANGE of the sum.

If all of the fractions were 1/43, the sum = 6(1/43) ≈ 6/42 = 1/7.
If all of the fractions were 1/48, the sum = 6(1/48) = 6/48 = 1/8.
Thus, the sum must be between 1/8 and 1/7.

The correct answer is C.

[ceilidh.erickson]

To add a general rule: if a question asks you what a particular sum is "closest to," it's asking for an approximation. Never attempt to do the actual calculation! That would take far too long. The GMAT is really trying to assess whether you can find a logical pattern through which to approach the problem.

Taking the minimum and maximum of the range, as other experts have suggested, is the best approach. You could also have taken the reciprocal of the median of the range, and multiplied it by 6. In this case, though, the median was 45.5, so that makes for messy math. On a different problem, though, the median of an evenly-spaced set would be a good gage for approximating.