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?

GMAT Panda · Accepted Answer

The time is 10nm/3n + 2m. We write Car A’s speed as distance divided by time and do the same for Car B. v_A = 2/n v_B = 3/m We write Car C’s speed as the arithmetic mean of these two speeds using the phrase "average (arithmetic mean) of those rates" from the question. v_C = v_A + v_B/2 We relate Car C’s travel time for 5 miles to its speed. t_C = 5/v_C We replace the speed formula with the average formula and then invert and multiply to find Car C’s time. t_C = 5v_A + v_B/2 = 5 2/v_A + v_B We write the sum of the two rates as a single fraction and simplify the result. t_C = 102m + 3n/nm = 10nm/3n + 2m

Explanation	Calculations	Help
$We write Car A’s speed as distance divided by time and do the same for Car B.$	$v_{A} = \frac{2}{n} v_{B} = \frac{3}{m}$
$We write Car C’s speed as the arithmetic mean of these two speeds using the phrase "average (arithmetic mean) of those rates" from the question.$	$v_{C} = \frac{v _{A} + v _{B}}{2}$
$We relate Car C’s travel time for 5 miles to its speed.$	$t_{C} = \frac{5}{v _{C}}$
$We replace the speed formula with the average formula and then invert and multiply to find Car C’s time.$	$t_{C} = \frac{5}{\frac{v _{A} + v _{B}}{2}} = \frac{5 \times 2}{v _{A} + v _{B}}$
$We write the sum of the two rates as a single fraction and simplify the result.$	$t_{C} = \frac{10}{\frac{2 m + 3 n}{nm}} = \frac{10 nm}{3 n + 2 m}$

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