Why does increasing a number by 15% and then decreasing it by 15% fail to produce the original number?

Question

I start with number: $.425$ and I want to add $15\%$ to get a new number. $.425 \times 1.15 = .489$ However, when I reduce $.489$ by $15\%$ I don't arrive back at $.425$. $.489 \times .85 = .416$. I need to reduce $.499$ by $15\%$ in order to to arrive at $.425$. I would love your help to get my head around something this simple!

Answer 1

If you consider what you're doing algebraically, when you increase a number $x$ by a portion of $k$, you are performing the operation $$x+kx$$ where we take the original amount, then add it to $kx$, the portion extra we want. This could be more easily expressed as multiplying $x(1+k)=x+kx$, by the distributive property. Similarly, subtracting a portion of $k$ is the operator $x-kx =x(1-k)$. So, if we increase $x$ by a factor of $k$, we get $x(1+k)$, and then if we decrease it by that same factor, we get $x(1+k)(1-k)$, which can be expanded as: $$x(1-k^2)$$ which is not equal to $x$ - and is in fact, always less than the amount we started with.

So, try to think not "adding 15%" and "subtracting 15%" but "multiply by 115%" and "multiplying by 85%" - where there is less temptation to think you're actually adding or subtracting anything. Namely, to undo this sort of thing, we need the reciprocal (e.g. $\frac{1}{1.15}$), which is not equal to $2-k$, as would be necessary for adding and subtracting a portion to cancel out.