If \(S\) is the sum of the reciprocals of the consecutive

AAPL wrote:GMAT Prep

If \(S\) is the sum of the reciprocals of the consecutive integers from 91 to 100, inclusive, which of the following is less than \(S\)?

I. \(1/8\)
II. \(1/9\)
III. \(1/10\)

A. None
B. I only
C. III only
D. II and III only
E. I, II, and III

The OA is C

So, we have S = 1/91 + 1/92 + ... + 1/100. There are 10 terms.

Among 1/91, 1/92, ..., 1/100, the greatest is 1/91 and the smallest is 1/100. Thus,

=> 10/100 < S < 10/91

1/10 < S < 1/9.1

Let's see the options.

I. \(1/8\): Since 1/9.1 < 1/8, this is not the correct answer.
II. \(1/9\): Since 1/9.1 < 1/9, this is not the correct answer.
III. \(1/10\): Since 1/10 < S, this is the correct answer.

The correct answer: C

Hope this helps!

-Jay
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AAPL wrote:GMAT Prep

If \(S\) is the sum of the reciprocals of the consecutive integers from 91 to 100, inclusive, which of the following is less than \(S\)?

I. \(1/8\)
II. \(1/9\)
III. \(1/10\)

A. None
B. I only
C. III only
D. II and III only
E. I, II, and III

The OA is C

We want the sum 1/91 + 1/92 + 1/93 + . . . + 1/100
Of these 10 fractions, 1/91 has the GREATEST value, and 100 has the SMALLEST value

So, let's examine some EXTREMES

If all of the 10 fractions were 1/91, then the sum would equal 1/91 + 1/91 + 1/91 + .... + 1/91
= 10/91
Of course most of the fractions are less than 1/91, so we can conclude that S < 10/91

If all of the 10 fractions were 1/100, then the sum would equal 1/100 + 1/100 + 1/100 + . . . + 1/100
= 10/100 = 1/10
Of course most of the fractions are greater than 1/100, so we can conclude that S > 1/10

So, we know that 1/10 < S < 10/91

Since 1/10 < S, we know that statement III works

What about S < 10/91 . What does this tell us?
First of all, 1/9 = 10/90
Second, 10/91 < 10/90, so we can conclude that 10/91 < 1/9
So, we know that S < 10/91 < 1/9
Since 1/9 is greater than S, we know that statement II does NOT work

Since 1/8 is greater than 1/9, we know that statement I does NOT work

Answer: C

Cheers,
Brent

Hi All,

Based on the wording of the prompt, you might think that you should add up the fractions 1/91 + 1/92 + .... 1/100, but the GMAT would NEVER require that you do that math. Instead, lets do some real basic estimation of what that sum would be LESS than and GREATER than....

There are 10 total fractions and 9 of them are GREATER than 1/100. So, at the 'lower end', let's just say that all 10 fractions are equal to 1/100....

(10)(1/100) = 10/100 = 1/10

Thus, we know that the sum of those 10 fractions will be GREATER than 1/10.

Similarly, we know that all 10 of those fractions are LESS than 1/90. So, at the 'higher end', let's just say that all 10 fractions are equal to 1/90...

(10)(1/90) = 10/90 = 1/9

Thus, we know that the sum of those 10 fractions will be LESS than 1/9.

With those two deductions, there's only one answer that 'fits'...

Final Answer: C

GMAT assassins aren't born, they're made,
Rich