M24-07

Official Solution:

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:

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:

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


Answer: C