Question

Aaron will jog from home at x miles per hour and then walk back home by the same route at y miles per hour. How many miles from home can Aaron jog so that he spends a total of t hours jogging and walking?
(a) $\dfrac{xt}{y}$
(b) $\dfrac{x+t}{xy}$
(c) $\dfrac{xyt}{x+y}$
(d) $\dfrac{x+y+t}{xy}$
(e) $\dfrac{y+t}{x}-\dfrac{t}{y}$

Answer 1

Hint: We can clearly see that we are given the total time and the speeds in the both directions at which Aaron travels the distance so we need to find the distance that Aaron jogs and it will be the distance between the starting point i.e. home and the ending point i.e. from where he heads back to home. So we will assume the time taken to travel when he travels at x miles per hour as ${{t}_{1}}$ and, ${{t}_{2}}$when he travels at y miles per hour. So we will equate the ${{t}_{1}}+{{t}_{2}}$ with the total time t and assume the total distance as s and find the value of s by putting values of ${{t}_{1}}\,and\,{{t}_{2}}$ using the formula $\operatorname{distance}\,=\,speed\,\times \,time$.

Complete step by step answer:
We are given that Aaron jog from home at x miles per hour and then walk back home at y miles per hour on the same route, and the total time he took for that is T hours
And we have to find the number of miles that Aaron jogs.
So first we will assume that Aaron takes ${{t}_{1}}$ time for jogging and ${{t}_{2}}$ time for walking back to home,
So according to the question, we have
${{t}_{1}}+{{t}_{2}}=t\,\,.....\left( 1 \right)$
Now we will assume the distance for one side as s i.e. he covers s distance while jogging and again s while coming back to home,
And we know that Distance = speed $\times $ time so, we have
For jogging, we have
$\begin{align}
  & \Rightarrow s=x\times {{t}_{1}} \\
 & \Rightarrow {{t}_{1}}=\dfrac{s}{x}\,\,\,\,\,....\left( 2 \right) \\
\end{align}$
For walking back home we have,
$\begin{align}
  &\Rightarrow s=y\times {{t}_{2}} \\
 &\Rightarrow {{t}_{2}}=\dfrac{s}{y}\,\,\,\,...\left( 3 \right) \\
\end{align}$
Now putting the values of equations 2 and 3 in equation 1, we get
$\begin{align}
  & \Rightarrow t={{t}_{1}}+{{t}_{2}} \\
 & \Rightarrow t=\dfrac{s}{x}+\dfrac{s}{y} \\
\end{align}$
Taking LCM of RHS we get,
$\Rightarrow t=\dfrac{sx+sy}{xy}$
Cross multiplying we get
$ \begin{align}
  & \Rightarrow txy=sx+sy \\
 &\Rightarrow txy=s\left( x+y \right) \\
 & \Rightarrow s=\dfrac{xyt}{x+y} \\
\end{align}$
So Aaron jogs a distance of $\dfrac{xyt}{x+y}$ miles.

So, the correct answer is “Option C”.

Note: To solve these kinds of problems we first need to analyse the problem carefully. What variable do we have to manipulate here to solve it easily. For example in this question that variable was time, we manipulated it in such a way to find the value of distance s. Some students might calculate the total distance that Aaron covered i.e. 2s but it is wrong as we have to calculate the distance of jogging only and not walking.