Math CentralQuandaries & Queries


Question from Lorrinn, a parent:

I process gifts for a nonprofit agency. For some gifts I calculate administrative fees. I understand calculating a straight percentage of a gift (i.e. 10% of $100.00 = $10.00). That's a fee "off the top". What I would like to better understand is the theory or the "mathematical reasoning" behind calculating percentages "on the inside". For example, to calculate 10% "on the inside" of $100.00, I use the following formula: $100.00/1.10*10.0%. The answer here is $9.09. I know how to do this, but I would like to better understand the "why". Thank you.

Hi Lorrinn,

Percent means "per 100" so 10% is 10/100 = 0.10. When you say 10% of $100 is $10 you are multiplying

10% of $100 is 0.10 × $100 = $10.00

When you say "on the inside" you don't know the amount (the $100 above), instead you know that whatever this number is, adding 10% gives a result of $100. The beauty of algebra lets me take the awkward previous sentence and write it more clearly. If you let x be the unknown amount then

x plus 10% of x is 100

In other words

x + 0.10x = 100


1.10 x = 100

Dividing both sides by 1.10 gives

x = 100/1.10 = 90.91

That is part of the formula you use. You then take 10% of this number to get

10% of 90.91 = 0.10 × 90.91 = 9.09

I hope this helps,

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