A shop reduces its prices by 20 percent of the original price. Some days later it wants the original prices back. By what percent must the discounted prices be raised to return to the original price?

Step-by-step solution

1. 1

   Understand why the answer is not $20\%$. A percentage is always a percentage of something. The $20\%$ decrease is measured against the original price; the increase that undoes it is measured against the smaller sale price. Different bases mean different percentages, and that is the entire content of this problem.

2. 2

   Pick a convenient original price. Percentages are scale-free, so let the original price be $100$ units. Any other starting value gives the same percentage answer.

3. 3

   Apply the discount. A $20\%$ reduction removes $20$ units:

   $100-0.20\cdot 100=80.$

4. 4

   Find the increase as a fraction of the new base. To get from $80$ back to $100$ we must add $20$ units, and that gain is measured relative to $80$:

   $\frac{100-80}{80}=\frac{20}{80}=0.25=25\%.$

5. 5

   Confirm with the general formula. Cutting by a fraction $p$ and then raising by a fraction $q$ returns to the start when $(1-p)(1+q)=1$, so

   $q=\frac{1}{1-p}-1=\frac{p}{1-p}.$

   With $p=0.20$ this gives $q=\dfrac{0.20}{0.80}=0.25$. Check: $80\times 1.25=100$. The required increase is $25\%$.