How to get to the solution of the problem on percentage?

Question

If a value is increased by n/d then to get the same number p from the resultant value we have to decrease the increased value by (n/d+n). Please explain this in details. I am unable to comprehend this completely.

Answer 1

If the original value is $p$, and we increase it by $n/d$, this means we are multiplying $p$ by $n/d$ and adding that to $p$. Algebraically, the resultant value is $$r:=p+p\left(\frac nd\right)=p\left(1+\frac nd\right)=p\left(\frac{d+n}d\right).\tag1$$ To return to the original value $p$, we rearrange (1) to solve for $p$ in terms of $r$:

$$p = r\left(\frac d{d+n}\right)=r\left(1-\frac n {d+n}\right)=r-r\left(\frac n{d+n}\right).\tag2$$ In other words, to regain $p$ we decrease the resultant value $r$ by $n/(d+n)$.

Examples:

• If you raise the price of an item by 25%, so that $n/d=25/100$, you get back to the original price by lowering the new price by 20%, since $n/(d+n)=25/(100+25)=0.2$.
• If you double the price of an item, so that $n/d=100/100$, you get back to the original price by lowering the new price by 50%, since $n/(d+n)=100/(100+100)$.