sum

[jainrahul1985]

What is the range of (1/101 + 1/102 + 1/103 + ...­+1/150)?
A. 1/150-1/100 B. 1/5-1/4 C. 1/4-1/3 D. 1/3-1/2 C. 1/2-1

OA D

[Brent@GMATPrepNow]

What is the range of (1/101 + 1/102 + 1/103 + ...­+1/150)?
A. 1/150-1/100 B. 1/5-1/4 C. 1/4-1/3 D. 1/3-1/2 C. 1/2-1

OA D

This is a nice question.
First recognize that there are 50 fractions in this sum (the number of integers from x to y inclusive is y-x+1, so the number of integers from 101 to 150 is 150-101+1=50

From here, notice that every fraction in the sum is less than 1/100
So, the sum (1/101 + 1/102 + 1/103 + ...­+1/150) must be less than the sum of fifty 1/100's (i.e., 1/100 + 1/100 + 1/100 + ...+ 1/100)
Since the sum of fifty 1/100's is 1/2, we know that (1/101 + 1/102 + 1/103 + ...­+1/150) < 1/2

On a similar note, notice that every fraction in the sum is greater than or equal to 1/150.
So, the sum (1/101 + 1/102 + 1/103 + ...­+1/150) must be greater than the sum of fifty 1/150's (i.e., 1/150 + 1/150 + 1/150 + ...+ 1/150)
Since the sum of fifty 1/150's is 1/3, we know that (1/101 + 1/102 + 1/103 + ...­+1/150) > 1/3

When we combine the two results, we get 1/3 < 1/101 + 1/102 + 1/103 + ...­+1/150) < 1/2

[Mike G]

Hi Brent,
how do you apply the definition of range to this problem? I understand that range is the difference of the largest and the smalest term in a set. Is there another logical approach?

Thanks

Mike

[Brent@GMATPrepNow]

Hi Brent,
how do you apply the definition of range to this problem? I understand that range is the difference of the largest and the smalest term in a set. Is there another logical approach?

Thanks

Mike

I wouldn't really call this a "range" question, since range questions are related to sets of number. In this instance, there is really just one number (the sum of 1/101 + 1/102 + 1/103 + ...­+1/150) and we need to determine which range of values the sum falls into.

So this is really a version of an "evaluation question" where we need to get an idea of the approximate value of a particular sum.

In this regard, the original question is similar to a question like:
The square root of 407 is between
A) 10 and 11
B) 14 and 15
C) 17 and 18
D) 20 and 21
E) 30 and 31

Here, since root 400 = 20 and root 441 = 21, then we know that root 407 is between 20 and 21.
So even though the answer choices consist of ranges, this question (like the original question) is more of an "evaluation" question

[Ian Stewart]

Hi Brent,
how do you apply the definition of range to this problem? I understand that range is the difference of the largest and the smalest term in a set. Is there another logical approach?

The wording of the question is terrible (though I agree with Brent that the math in the question is worthwhile); it has nothing to do with the concept of 'range' from statistics. The question should be worded differently - it should ask something like

The value of 1/101 + 1/102 + 1/103 + ... 1/150 is between:


On the GMAT, the word 'range' will refer to the difference between the largest and smallest values in a set. They won't misuse the word, as in the question in the post above.