Quickly undoing percentages in your head

I can't remember how exactly it came up, but my son asked me while we were in the car something along the lines of, "if (some number) was something else minus 20 percent, how much was the original number?" Of course, with a calculator, this is trivial to work out — if you took off 20%, you computed (100 - 20) = 80% of the original, so dividing the result by 0.8 will yield the original number. However, we were in the car without a calculator but after thinking about it for a few minutes, it occurred to me that adding back 1/4 would yield the original number and, in general, if you reduced a number by 1/n, you could recover the original by adding back 1/n-1 of the result. I imagine somebody tried to teach me this simple rule in primary school, but I didn't pay close enough attention, so I had to work it out again for myself. Just to make sure I wasn't imagining things, I decided to work out the algebra to double-check this identity once I had access to a pencil and paper. As it turns out, demonstrating this algebraically turns out to be somewhat interesting:

Figure 1: Reverse a fractional reduction

More intuitively, if you reduce x by 20% = 1/5, you're computing y = (4/5)x (of course, you wouldn't do it that way in your head, you'd compute 20% and subtract). To undo this, you multiply by the reciprocal 5/4, or 1 + 1/4. Of course, this is actually easier done than said.

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Bala Tamilmani, 2019-01-06
Interesting, being a Maths graduate I enjoyed understanding it :)
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