GMAT Practice Question: If Car A took n hours to travel 2 miles and Car B took m...
Question
If Car A took n hours to travel 2 miles and Car B took m hours to travel 3 miles, which of the following expresses the time it would take Car C, traveling at the average (arithmetic mean) of those rates, to travel 5 miles?
- \frac{10 n m}{3 n+2 m}
- \frac{3 n+2 m}{10(n+m)}
- \frac{2 n+3 m}{5 n m}
- \frac{10(n+m)}{2 n+3 m}
- \frac{5(n+m)}{2 n+3 m}
Topics: word problems, rates
Solution
Step 1
We write Car A’s speed as distance divided by time and do the same for Car B.
v_A = \frac{2}{n} v_B = \frac{3}{m}
Step 2
We write Car C’s speed as the arithmetic mean of these two speeds using the phrase \textbf{\text{"average (arithmetic mean) of those rates"}} from the question.
v_C = \frac{v_A + v_B}{2}
Step 3
We relate Car C’s travel time for 5 miles to its speed.
t_C = \frac{5}{v_C}
Step 4
We replace the speed formula with the average formula and then invert and multiply to find Car C’s time.
t_C = \frac{5}{\frac{v_A + v_B}{2}} = \frac{5 \times 2}{v_A + v_B}
Step 5
We write the sum of the two rates as a single fraction and simplify the result.
t_C = \frac{10}{\frac{2m + 3n}{nm}} = \frac{10nm}{3n + 2m}
Answer
The time is \frac{10nm}{3n + 2m}.
