GMAT Word Problems: If Car A took n hours to travel 2 miles…

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?

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

(10nm)/(3n+2m) (A)

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