University of South Carolina High School Math Contest/1993 Exam/Problem 6
Problem
After a price reduction, what increase does it take to restore the original price?
![$\mathrm{(A) \ }p\% \qquad \mathrm{(B) \ }\frac p{1-p}\% \qquad \mathrm{(C) \ } (100-p)\% \qquad \mathrm{(D) \ } \frac{100p}{100+p}\% \qquad \mathrm{(E) \ } \frac{100p}{100-p}\%$](http://latex.artofproblemsolving.com/b/3/9/b3913d5fd3abbcbc40bdcd74cd4b700fa906e2c9.png)
Solution
Let the unknown be . Initially, we have something of price
. We reduce the price by
to
. We now increase this price by
to get
We can cancel
from both sides to get
so
and
and
, so our answer is
.
Alternatively, select a particular value for such that the five answer choices all have different values. For instance, let
. Thus if
dollars was the original price, after the price reduction, we have
dollars. We need
dollars. Thus,
and
. This only matches up with answer
when we plug in
.