Which of the following is closest to 4/2.001?

Which of the following is closest to \(\frac{4}{2.001}\)?

A. 1.997
B. 1.998
C. 1.999
D. 2.000
E. 2.001


The answer options are extremely close to each other, necessitating a very precise calculation. We can't simply round off 2.001 to 2 and assert that \(\frac{4}{2.001}\) is closest to 2 from the options given, due to this small difference.

Let's express 2.001 as 2 + 0.001 and multiply both the numerator and the denominator by the conjugate, 2 - 0.001:

\(\frac{4}{2.001}=\)

\(=\frac{4}{2+0.001}=\)

\(=\frac{4(2-0.001)}{(2+0.001)(2-0.001)}=\)

\(=\frac{4(2-0.001)}{4-0.001^2}\).

Now, \(0.001^2=0.000001\) is a very small number. Compared to 4, this value is practically negligible (do note that while we couldn't ignore 0.001 earlier, we can overlook 0.000001 now because it is 1000 times smaller than 0.001). Hence, we can approximate the equation as:

\(\frac{4(2-0.001)}{4-0.001^2} \approx \)

\(\approx \frac{4(2-0.001)}{4}=\)

\(= 2-0.001 =\)

\(=1.999\).

Therefore, \(\frac{4}{2.001}\) is closest to 1.999 among the provided options.

Answer: C

hi, you cannot assume that here since all vaues are very very close...

Hi Bunuel,

Thanks for the solution. I understood the way to approach it and why the answer is C.

I have a small query - while solving as you mentioned that 4 - 0.002^2 is equal to 4 only because 0.002^2 is negligible here....However in the main question where 2 - 0.002 is written 0.002 is not negligible.
Is it not negligible only because it is not sufficiently smaller no.? If it were 0.002^2 in the main question instead of 0.002 then could that have been negligible?

Please share your valuable inputs.

hi fireinbelly,
if i may try to explain you..
look at the answer values they are varying to .001(for eg 2.000 or 1.999 or 1.998)... so yes .001 is important here....
but .001^2=.000001, which is way to small as compared to .001 .... so can be left..

ok...thank you chetan2u!

Dear Bunuel, how the denominator (2+0.001) become equal to (2+0.001)(2-0.001)

Dear Bunuel, how the denominator (2+0.001) become equal to (2+0.001)(2-0.001)[/quote]

hi,
both denominator and numerator are multiplied by (2-.001) to basically simploiify the equation

Responding to a pm:

I think you got your answer. Whether something is negligible depends on with what we are comparing it. When working with 1.999 and 2.001, then .001 is not negligible. But .000001 is. Similarly, if the options were far apart such as 2, 3 etc then we know that we can ignore .001.

Thanks Karishma....it really helps!

Re: Which of the following is closest to 4/2.001?

I used Numbers concept: Not sure if its the right approach though.

Multiply both Numerator and denominator by 1000

4000/2001= Since the denominator is not of the form 2^m * 5^n The decimal will not terminate.

Looking at the options Only C is Non terminating