Please see these fractions:

(A) $\frac{33}{128}$ (B) $\frac{45}{138}$ (C) $\frac{53}{216}$ (D) $\frac{83}{324}$ (E) $\frac{15}{59}$.

I need to find out quickly (in about a minute) the smallest of these fractions. I am not allowed to use a calculator, though a little rough calculation is allowed.

As I see the problem, without a calculator, converting these to decimal values is not an option. Moreover, since the numbers involved are fairly large, finding the least common denominator and changing each fraction to make their denominators the same as the least common denominator is again next to impossible without a calculator. For the same reason, we can not try the method of making the numerators identical.

If we try the approximate method of changing the numerators and denominators to easy numbers, we can get something like this:

(A) $\frac{30}{120}$ (B) $\frac{45}{135}$ (C) $\frac{50}{200}$ (D) $\frac{80}{320}$ (E) $\frac{15}{60}$.

Unfortunately, this makes all of the fractions approximately equivalent to $\frac{1}{4}$ except B, which becomes $\frac{1}{3}$.

This is the point where I can not proceed any further.