GRE Reasoning problem time/distance/average speed

Question

David drove to work at an average (arithmetic mean) speed of 45 miles per hour. After work, David drove home at an average speed of 60 miles per hour. If David spent a total of 2 hours commuting to and from work, how many miles does David drive to work?

A) 48

B) $\dfrac{256}{5}$

C) $\dfrac{360}{7}$

D) $\dfrac{105}{2}$

E) $\dfrac{160}{3}$

Solution: (C)

My answer:

$\begin{align} t_1 &=\dfrac{v_1}{d} = \dfrac{45}{d} \\ t_2 &=\dfrac{v_2}{d} = \dfrac{60}{d} \end{align}$

and

$t = t_1 + t_2 =2$

so: $\hspace{5mm} d =\dfrac{105}{2} \hspace{5mm}$ (D)

Apparently, my thinking is wrong, but I don't know where.

Any thoughts? Very much appreciated!

Answer 1

Your formula is slightly off- you should have $v_1\cdot t_1=d$ and $v_2\cdot t_2=d$. You can see this with units of measurement: miles per hour can be expressed as $\frac{\text{miles}}{\text{hours}}$ and time as $hours$ so there product should be distance as $miles$. If you want to verify, your answer should be $C$.

Answer 2

Another way to solve this is by harmonic mean. Since you are given two average speeds over same distance, average speed of total travel is given by $$v = \frac{2}{\frac 1{v_1} + \frac 1{v_2}} = \frac{2}{\frac 1{45} + \frac 1{60}} = \frac{360}{7}\,\text{mph}$$

Since twice the distance from home to work takes $2$ hours at that speed, the distance is precisely $\frac{360}7$ miles.