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. $\frac{xt}{y}$
• B. $\frac{x+t}{xy}$
• C. $\frac{xyt}{x+y}$
• D. $\frac{x+y+t}{xy}$
• E. $\frac{y+t}{x}-\frac{t}{y}$

Answer 1

The correct option is C $\frac{xyt}{x+y}$

Let $j$ be the number of hours Aaron spends jogging; then let $t-j$ be the total number of hours he spends walking. It can be stated that Aaron jogs a distance of $xj$ miles and walks a distance of $y(t-j)$ miles. Because Aaron travels the same route, the miles jogged must equal the miles walked, and they can be set equal.

$xj=y(t-j)$ set number of miles equal to each other

$xj=yt-jy$ distribute the $y$

$xj+jy=yt$ add $jy$ to both sides to get all terms with $j$ to one side

$j(x+y)=yt$ factor out the $j$

$j=\frac{yt}{x+y}$ divide both sides by $x+y$

So, the number of hours Aaron spends jogging is $j=\frac{yt}{x+y}$

The number of miles he can jog is $xj$ or, by substitution of this value of $j,x\left(\frac{yt}{x+y}\right)=\frac{xyt}{x+y}$

The correct answer is C.