A price goes up and then down

by @ancient-tree · difficulty 5/100

Arithmetic

Status: Unreviewed

A price $P$ is a positive real number.

First the price increases by $a\%$ , then the new price decreases by $a\%$ , where $0 \le a \le 100$ .

Is the final price higher, lower, or the same as the original price?

Proofs

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0 useful votes

Increasing $P$ by $a\%$ gives the new price

$P_2=P+\frac{a}{100}P=\frac{100+a}{100}P.$

Then the final price is

$P_3=\frac{100-a}{100}P_2=\frac{(100-a)(100+a)}{100^2}P.$

But

$(100-a)(100+a)=100^2-a^2\le 100^2.$

Therefore $P_3\le P$ . Equality happens only when $a=0$ ; if $a>0$ , the final price is strictly lower than the original price.

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