# 2008 iTest Problems/Problem 45

## Problem

In order to save money on gas and use up less fuel, Hannah has a special battery installed in the family van. Before the installation, the van averaged $18$ miles per gallon of gas. After the conversion, the van got $24$ miles per gallon of gas.

Michael notes, "The amount of money we will save on gas over any time period is equal to the amount we would save if we were able to convert the van to go from $24$ miles per gallon to $m$ miles per gallon. It is also the same that we would save if we were able to convert the van to go from m miles per gallon to $n$ miles per gallon."

Assuming Michael is correct, compute $m+n$. In this problem, assume that gas mileage is constant over all speeds and terrain and that the van gets used the same amount regardless of its present state of conversion.

## Solution

Let $d$ be the total distance traveled for the time period. Because the amount saved is the same from the two conversions, the amount of gallons decreased is the same between the two conversions. That means $$\frac{d}{18} - \frac{d}{24} = \frac{d}{24} - \frac{d}{m}$$ $$\frac{d}{72} = \frac{d}{24} - \frac{d}{m}$$ $$\frac{d}{m} = \frac{d}{36}$$ $$m = 36$$ Similarly, $$\frac{d}{24} - \frac{d}{36}= \frac{d}{36} - \frac{d}{n}$$ $$\frac{d}{72} = \frac{d}{36} - \frac{d}{n}$$ $$\frac{d}{n} = \frac{d}{72}$$ $$n = 72$$ Thus, $m+n = \boxed{108}$.