How to get original number from percentages

Question

When I calculated $15\%$ of $150$ it's $22,5$. And than I do $150 - 22,5 = 127,5$. Now I have number $127,5$ and I have $15\%$.

How can I get from these two numbers ($127,5$ and $15\%$) back original number $150$?

Thank you a lot.

Answer 1

This is what you did: $$150-150\cdot 0.15=127.5$$ If you factorise, you obtain: $$150(1-0.15)=127.5$$ Rearranging: $$150=\frac{127.5}{1-0.15}=\frac{127.5}{0.85}$$ Hence, to obtain $150$, you must divide $127.5$ by $1-0.15$ ($100\text{%}-15\text{%}$), or $0.85$ ($85\text{%}$).

Answer 2

Percentages can be confusing. For example, reduce 100 by 10% and you get 90. Add 10% to 90 and you get 99. You have not returned to where you started. The reason is that although we talk of adding and subtracting percentages, we are really multiplying. Subtracting 10% actually means multiply by 90/100 = 0.9. Adding 10% means multiply by 110/100 = 1.1. Now, it is more obvious that the net effect of subtracting and adding 10% is to multiply by 0.99.

So, to reverse a reduction by 10% you need to remember that it really means multiply by 0.9 so the reverse is to divide by 0.9 which is adding a little over 11%.

Answer 3

Well, let's try to write the problem down more abstractly. To do so, instead of: $$150-22.5=127.5$$ let us write $$150-(15\%) \cdot 150=127.5$$ Now if we convert percentages to decimals, we get: $$150-(0.15) \cdot 150=127.5$$ Next up, we're going to do some algebraic trickery to the left side of the equation. In specific, we're going to divide by $150$ and multiply the sum we get by $150$ again. Let's make it less confusing in formula form: $$150-(0.15)\cdot150\rightarrow(1-0.15)\rightarrow 150\cdot(1-0.15)$$ We can do this because $$150-(0.15)\cdot150=150\cdot(1-0.15)$$

So now we've got: $$150\cdot(1-0.15)=127.5$$ Therefore $$150=\frac{127.5}{(1-0.15)}$$

So if we buy an item in a shop that costs $\$35$ after a discount of $25\%$ has been applied, we can now calculate its original price (which we'll call $x$): $$x=\frac{$35}{1-0.25}=\frac{$35}{0.75}=$46.67$$