Rounding a percentage to the nearest multiple of $\frac{1}{n}$

Question

If I take a percentage like $60\%$ I can easily round it to a multiple of $\frac{1}{n}$ where $n=2$ like this...

$$60\%\doteq50\%$$ $$50\%=\frac{1}{2}$$

...or where $n=3$ like this.

$$60\%\doteq 66\%$$ $$66\%\doteq \frac{2}{3}$$

But what if $n$ was a not-so-friendly number, like 43? How do I round $60\%$ to the nearest multiple of a fraction like $\frac{1}{43}$ without doing so much guessing and checking?

Is there a consistent method for rounding $k\%$ to the nearest multiple of $\frac{1}{n}$ with minimal use of the guess and check method?

Answer 1

Let $m = \text{round}\left(\dfrac{k \cdot n}{100}\right)$, i.e. compute $\dfrac{k \cdot n}{100}$ and round it to the nearest integer.

Then, the nearest multiple of $\dfrac{1}{n}$ to $k\%$ is $\dfrac{m}{n}$.

This works since the following statements are equivalent:

$\dfrac{m}{n}$ is the nearest multiple of $\dfrac{1}{n}$ to $k\%$

$\dfrac{m-\tfrac{1}{2}}{n} < \dfrac{k}{100} \le \dfrac{m+\tfrac{1}{2}}{n}$

$m-\dfrac{1}{2} < \dfrac{kn}{100} \le m+\dfrac{1}{2}$

$m$ is the nearest integer to $\dfrac{kn}{100}$

Answer 2

Round $kn\%$ to the nearest integer and that's your numerator!

eg $60\%*43=25.8$ so $60\%≈\frac{26}{43}$.