To increase a number by a percentage I was taught the formula no x 1.percentage, so for example:
100 increased by 15%:
100 x 1.15 = 115
But today, I learned the hard way that doing the inverse of the above (i.e. no / 1.percentage), isn't the correct way of decreasing a number by a percentage (aka "applying a discount"). E.g:
100 decreased by 15%:
100 / 1.15 = 86.95652 <- WRONG
100 - (100 x 0.15) = 85 <- RIGHT
My Question: Why doesn't the formula no / 1.percentage work, when decreasing a number by a percentage?
Side Question: Is there a simpler formula to decrease a number by a percentage than no - (no x 0.percent)?
Note: My math knowledge is super basic, so maybe pretend you're explaining this to a twelve year old when answering. But if you want to give sophisticated answers for future readers, that's fine too, I might just not understand it though.
Best Answer
The method of increasing a value $V$ by $p\%$ is actually adding $V\times\frac p{100}$: $$V_\text{new} = V + V\times\frac p{100} = V\times\left(1 + \frac p{100}\right)$$ For $p=15$ you have a nice multiplier $$\left(1 + \frac p{100}\right) = 1 + 0.15 = 1.15$$
Remember, however, that '1.p' is a shortcut or mnemonic, not a method. It wouldn't even work for $p$ greater than 99, say for $+120\%$. And it certainly won't work for $p <0$.
For decreasing you need to apply the method, which is: $$V_\text{new}=V - V\times\frac p{100} = V\times\left(1 - \frac p{100}\right)$$ so for 15-percent decrement you get: $$V_\text{new}=V - V\times\frac {15}{100} = V\times(1 - 0.15) = V\times 0.85$$
EDIT
To answer your main question directly, forget percentages. Suppose you need to rise a value from 100 to 125. The new value is $$125 = 100\times\frac 54=100\times(1+\tfrac 14) $$ hence an increase by 25%. Now, if you want to reduce it back to 100, you get $$100=125\times \frac 45=120\times(1-\tfrac 15)$$ which is 20% decrease. Why was it 1/4 before, and 1/5 now? Because the same difference $\pm 25$ was taken relative to 100 in the former case and to 125 in the latter one. When we added a fourth part, we got five fourths of the initial value, so we needed to take away one of those five, i.e. a fifth part to get back.
And here you have the difference in percentages: 1/4=25%, while 1/5=20%.
So, a division like $value/1.p$ reduces the value by p% of the resulting value, not of the original one.