25/0.128 =

Question

25/0.128 =

A. 195
B. 196
C. 197
D. 198

IanStewart · Answer

First, 0.128 is equal to 2^7/10^3. So we have
 \frac{25}{0.128} = \frac{5^2}{\frac{2^7}{(2^3)(5^3)}} = \frac{5^5}{2^4} 
There is no chance of this being an integer, so we can't use divisibility properties here; it's purely coincidental that the closest integer to this expression is a multiple of 5. You can see why there is no reason to expect a similar expression to be close to a multiple of 5 by looking at a fraction with smaller powers; for example, the fraction 5^2/2^2 is equal to 6.25, so the closest integer to 5^2/2^2 is not a multiple of 5.

The question should mention that it's asking for an approximation, since 25/0.128 is not exactly equal to any of the answer choices. If I were to need to estimate its value, I'd first get integers in the numerator and denominator:

\frac{25}{0.128} = \frac{25000}{128} = \frac{3125}{16}

Now we know 3200 = 16*200. 3125 is 75 less than 3200, and 75/16 is just less than 5, so 3125/16 will be just larger than 195 (it's exactly equal to 195 and 5/16ths).

25/0.128 =

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