Jim drove 252 miles. Jim's brother made the same trip at a speed $21.6$ miles per hour slower than Jim did and required an extra $1 \frac { 1 } { 2 }$ hours. Find Jim's rate and time.
A) $\quad 72 \mathrm { mph } , 2.5 \mathrm { hr }$
B) $\quad 66 \mathrm { mph } , 3.5 \mathrm { hr }$
C) $\quad 72 \mathrm { mph } , 3.5 \mathrm { hr }$
D) $\quad 66 \mathrm { mph } , 2.5 \mathrm { hr }$
E) not possible

Question

Quizplus · Accepted Answer

Let Jim's rate be $x$ mph and the time he took be $t$ hours. Then we have $xt=252$. Let Jim's brother's rate be $x-21.6$ mph and the time he took be $t+\frac{3}{2}$ hours, since he took an extra $\frac{3}{2}$ hours. We have $(x-21.6)(t+\frac{3}{2})=252$, which expands to $xt+ \frac{3}{2}x-21.6t-32.4=252$. Substituting $xt=252$ simplifies this to $\frac{3}{2}x-21.6t=84$. We now have two equations with two variables: $$\begin{cases} xt=252 \ \frac{3}{2}x-21.6t=84 \end{cases}$$ Solving for $x$ and $t$ gives us $x=72$ mph and $t=3.5$ hours, so our answer is $\boxed{\textbf{(C)}}$.

Jim Drove 252 Miles 21.621.621.6 Miles Per Hour Slower Than Jim Did and Required an at a Speed

Jim Drove 252 Miles $21.6$ Miles Per Hour Slower Than Jim Did and Required an at a Speed