# Increases and Decreases

**Problem**

A rectangle has length *L*, width *W*, and area *A*. If *L* is increased by 1/10 and *A* is decreased by 1/100, by what fraction does *W* change?

## Solution

It may be tempting to say that if area decreases by 1/100 while length is increases by 1/10, then width must increase 1/100 + 1/10 = 11/100. But that doesn’t work.

Call the width of the changed triangle *x*.

Let’s see what’s going on:

- Area of original rectangle: .
- Length of changed rectangle: .
- Area of changed rectangle:
**.** - But the area of the changed rectangle also equals the changed length times the changed width: .

Set the last two expressions for the changed area *A* equal to each other and solve for *x: *

The changed width is 9/10 of the original width, so *W* decreases by **1/10**.

Increase one dimension of a rectangle by 1/10, decrease the other by 1/10, and the result is a decrease in area of 1/100. Sound funny? Consider: When you increase the original length by 1/10, you are increasing the area by 1/10 of the **original** area. Then when you decrease the length by 1/10, you are decreasing the area by 1/10 of the **enlarged ** area. So you’ve decreased by just a little more than you increased. Hence, the net change is a very small decrease.

The same sort of thing happens when you buy an item on sale for, say, 5% off and then you have to pay 5% tax on the sale price: The final price will be just a tiny bit less than the original price without tax.