# Difference between revisions of "University of South Carolina High School Math Contest/1993 Exam/Problem 6"

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== Problem == | == Problem == | ||

− | After a <math>p%</math> price reduction, what increase does it take to restore the original price? | + | After a <math>p\%</math> price reduction, what increase does it take to restore the original price? |

− | <center><math> \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}% | + | <center><math> \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}\% </math></center> |

== Solution == | == Solution == |

## Revision as of 21:04, 5 July 2017

## Problem

After a price reduction, what increase does it take to restore the original price?

## Solution

Let the unknown be . Initially, we have something of price . We reduce the price by $p%$ (Error compiling LaTeX. ! Missing $ inserted.) to $Q - Q\cdot p% = Q - Q\frac p{100} = Q\cdot\frac{100 - p}{100}$ (Error compiling LaTeX. ! Missing $ inserted.). We now increase this price by $x%$ (Error compiling LaTeX. ! Missing $ inserted.) to get $\left(Q\cdot\frac{100 - p}{100}\right) + \left(Q\cdot\frac{100 - p}{100}\right)\cdot x% = \left(Q\cdot\frac{100 - p}{100}\right)\cdot(1 + x%) = Q$ (Error compiling LaTeX. ! Missing $ inserted.) We can cancel from both sides to get $\frac{100 - p}{100}\cdot\left(1 + x%\right) = 1$ (Error compiling LaTeX. ! Missing $ inserted.) so $1 + x% = \frac{100}{100 - p}$ (Error compiling LaTeX. ! Missing $ inserted.) and $x% = \frac{p}{100 - p}$ (Error compiling LaTeX. ! Missing $ inserted.) 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, $90(1+x%)=100 \Longrightarrow x%=\frac{10}{90}$ (Error compiling LaTeX. ! Missing $ inserted.) and . This only matches up with answer when we plug in .