# Finding distance using rates of change -- best approach?

The question:

A man drives from state $A$ to state $B$ going $60 \frac{miles}{hour}$. Then he returns from state $B$ to state $A$, driving $45 \frac{miles}{hour}$. His total driving time is $2.5 hour$. How many miles is the distance from state $A$ to state $B$?

Possible solution:

Using $distance = rate \times time$, we observe

$d=60 \frac{miles}{hour} \times p (2.5 hour)$,

where $p$ denotes the proportion of time driving from $A$ to $B$, and

$d=45 \frac{miles}{hour} \times (1-p) (2.5 hour)$.

Solving these two equations,

\begin{align*} 60 \frac{miles}{hour} \times p (2.5 hour)&=45 \frac{miles}{hour} \times (1-p) (2.5 hour) \\ 150p miles &= (112.5-112.5p) miles \\ 262.5p miles &= 112.5 miles \\ p&= \frac{112.5}{262.5} \end{align*}

Now we solve

$d=60 \frac{miles}{hour} \times p (2.5 hour) \approx 64 miles$.

It works, but maybe isn't the most time-efficient approach during a GRE exam where you should spend less than a minute on a given question. When practicing and timing myself, this is the approach I started with, decided to skip it, and never had time to come back to it until after my time was up. Is this how you would solve this question, or is there a more straightforward and faster approach?

P.S. I notice we're talking about rates of change and determining the total change -- could this problem be expressed differently and solved using calculus in a faster way?

Let the one way distance be $d$. Then ${ d \over 60} + {d \over 45} = { 5 \over 2}$.
Then $d = { { 5 \over 2} \over { 1 \over 60} + {1 \over 45}} = { 450 \over 7} \approx 64.29$.
• Ah. This is simply $t_1+t_2=2.5$. Then, using $d=r_1\times t_1$ and $d=r_2 \times t_2$, the first equation can be rewritten as $\frac{d}{r_1}+\frac{d}{r_2}=2.5$, or $\frac{d}{60}+\frac{d}{45}=2.5$. Thank you for your help. Commented Aug 17, 2014 at 3:10