2013 AMC 10B Problem 8

The combined rate means total miles driven divided by total gallons used. The problem does not give the distance, and it will not matter, so choose one that makes the division easy.

1. Suppose each of them drives $40$ miles.
2. Ray gets $40$ miles per gallon, so he uses $40 \div 40 = 1$ gallon.
3. Tom gets $10$ miles per gallon, so he uses $40 \div 10 = 4$ gallons.
4. Together they drive $40 + 40 = 80$ miles on $1 + 4 = 5$ gallons.
5. Combined rate: $80 \div 5 = 16$ miles per gallon.

The answer is $\boxed{\textbf{(B)}\ 16}$.

Miles per gallon is a ratio, and you combine ratios by adding up the miles and adding up the gallons, not by averaging the two rates. Because both drivers go the same distance, the distance you pick cancels out, so any convenient number gives the same answer.

Keep the distance as $d$ miles each. Ray uses $\frac{d}{40}$ gallons and Tom uses $\frac{d}{10} = \frac{4d}{40}$ gallons, so together they use $\frac{5d}{40} = \frac{d}{8}$ gallons. They drive $2d$ miles, so the rate is $2d \div \frac{d}{8} = 16$.

The combined rate is the total distance divided by the total gasoline used. Since the distance is the same for both, pick a convenient one: suppose each drives $40$ miles.

Ray uses $\frac{40}{40} = 1$ gallon. Tom uses $\frac{40}{10} = 4$ gallons.

Together they cover $80$ miles on $5$ gallons, so the combined rate is
$$\frac{80}{5} = 16 \text{ miles per gallon}.$$

The answer is $\boxed{\textbf{(B)}\ 16}$.