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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.

wellington
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    Not sure what you were doing, but if you were calculating something using a calculator, you need to be mindful of the roundoff errors. For example, if your calculator can only display three digits, it may say that 12.5 percent of 12.5 is 1.56, which is actually less than the correct value 1.5625. – Batominovski Jul 09 '18 at 18:58
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    I suspect it has to add up exactly because you're later comparing the new value to the old value or something like that. Generally you want to compare real numbers(floating points https://en.wikipedia.org/wiki/Floating-point_arithmetic ) with some error allowed. For example you could say two numbers $x_1$ and $x_2$ are equal if: $abs(x_1 - x_2) \leq 0.00001$ – user1949350 Jul 09 '18 at 19:03
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    What makes you say that they don't add up exactly ? –  Jul 09 '18 at 19:30

2 Answers2

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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).

Ariel Serranoni
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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).

Barry Cipra
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