Sum of percentages not equal to original sum

Question

For example, let's say I have a sum of money equal to $150,246.76, of which 85%, 10% and 5% need to add up to the original amount, exactly. Calculating each percentage of the total separately does not work. Each percentage must take into account the other percentages otherwise the result is higher than the original amount.

$$0.85x + 0.1x + 0.05x\gt x $$

How do I find the individual values associated with each percentage, so that the total is equal to the original sum?

Worth mentioning, I'm using Excel for calculations.

Answer 1

In fact,

$$0.85x+0.1x+0.05x= (0.85+0.1+0.05)x= 1x=x.$$

On your example we have $x=150246.76$ Thus:

$$0.85 x =127709.746 $$ $$0.1 x =15024.676$$ $$0.05 x = 7512.338$$

Summing these values we obtain the desired result (done with the standard windows calculator).

Answer 2

This is related to the apportionment problem, which bedevils the U.S. every ten years: the exact fractions are not integer multiples of a penny, so they need to be rounded in some way. There is no unique "fair" way of doing so. Using the numbers from Ariel Serranoni's answer, it may seem clear that $7512.338$ should be rounded up to $7512.34$, and that one of $15024.676$ and $127709.746$ should be rounded up and the other rounded down, but it's not obvious how to make that choice. The theory of apportionment is surprisingly nuanced (and the politics is historically contentious).