Which of the following is closest to (1)/(1.001)?

Back solving seems the best way. Since you can eliminate D and E from the get go start with C. Back-solving should be easy because multiplications are with 1.

Another approach could be a little bit of factoring:


1/1.001 = 1000/1001 = (1001-1)/1001 = 1 - 1/1001 ~ 1 - 0.001 ~ 0.999. Since there is approximation inherent in the last two steps of this approach, I would be cautious.

We straight away ignore the options D and E as they are greater than 1, because in the above fraction the denominator is greater the numerator. Of the remaining, the answer should be C

Since the question is asking that which is closest to 1/1.001. We can't just eliminate D & E. Though we can eliminate E in comparison to D. But there might be a possibility that 1/1.001 may be closest to 1 also..
So we will have to check it by solving or other methods.

We can convert 1.001 to a fraction and we have:

1/[(1001/1000)] = 1000/1001

It is obvious that this number is just a little smaller than 1 (which makes it just a little greater than 0.999), so the answer is either D or C. We must decide which. We will compare the difference between 1 and 1000/1001 to the difference between 1000/1001 and 0.999 = 999/1000.

1 - 1000/1001 = 1000/1000 - 1000/1001

= 1001000/1001000 - 1000000/1001000

= 1000/1001000

= 1/1001

1000/1001 - 999/1000 = 1000000/1001000 - 999999/1001000

= 1/1001000

Since the difference between 1000/1001 and 999/1000 is much smaller than the difference between 1 and 1000/1001, 1000/1001 is closer to 999/1000 = 0.999 than any other choice.

Answer: C
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1000/1001 > 999/1000

Hence C

\(\frac{1}{1.001} = \frac{1000}{1001}\)
which is slightly less than 1.

\(.999 = \frac{999}{1000}\)
which is slightly less than 1 too. When we add 1 to both the numerator and denominator of 999/1000, we get 1000/1001. This number will be closer to 1 than 999/1000.

So on the number line, we have .997, .998, .999, 1000/1001, 1.000, 1.001 in that order. Closest to 1000/1001 is either .999 or 1.
From 999/1000, we go to 1000/1001 by adding 1 to both numerator and denominator.
But from 1000/1001 to 1, we go by adding 1 to numerator alone. So 1 will be much greater than 1000/1001 (relatively speaking).

Hence closest to 1000/1001 is .999.

Answer (C)

Check this post for this concept: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2011/0 ... round-one/
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Which of the following is closest to \(\frac{1}{1.001}\)?

A. 0.997
B. 0.998
C. 0.999
D. 1.000
E. 1.001


\(\frac{1}{1.001} = \frac{1}{1 + 0.001}\).

Multiply the above by \(\frac{1 - 0.001}{1 - 0.001}\):

\(\frac{1 - 0.001}{(1 + 0.001)(1 - 0.001)} =\frac{1 - 0.001}{1 - 0.001^2} \).

Now, because \(0.001^2\) is very small, \(1 - 0.001^2\) is very close to 1. Therefore, \(\frac{1 - 0.001}{1 - 0.001^2} \approx \frac{1 - 0.001}{1} = 1 - 0.001 = 0.999\).


Answer: C
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