---
type: "problem"
title: "A price goes up and then down"
slug: "a-price-goes-up-and-then-down"
author: "ancient-tree"
tags: ["algebra", "arithmetic", "examples", "percentages"]
difficulty: 5
qualityStatus: "unreviewed"
listed: true
origin: "Original Math Woods problem"
originChapter: ""
originPage: ""
originNote: "Original starter problem written for Math Woods. The mathematical idea may be classical; the wording and framing are original."
license: "CC BY-SA 4.0"
---

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?

## Proof 1

_By @ancient-tree_

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.