Question

To travel 105 miles, it takes Sue, riding a moped, 3 hours less time than it takes Doreen to travel 64 miles riding a bicycle. Sue travels 13 miles per hour faster than Doreen. Find the times and rates of both girls.

Answer 1

Let t=time Doreen travels; let r=rate of Doreen

                RATE * TIME = DIST.
        Sue | r+13      t-3       105
   Doreen |   r           t          64
 
Now solve the system:
(r+13)(t-3)=105
r*t=64
 
$r= \frac{64}{t} \\ \\(\frac{64}{t} +13)(t-3)=105 \\ \\64- \frac{192} {t}+13t-39-105=0 \\ \\13t- \frac{192} {t}-80=0 \\ \\ 13t^2-80t-192=0 \\ \\t_{1,2}= \frac{80\pm \sqrt{(-80)^2-4\times13(-192)} }{2\times13}= \frac{80\pm128}{26} \\ \\t=\frac{80+128}{26} =8 \\ \\r=\frac{64}{t} =\frac{64}{8} =8$
             
Doreen: rate = 8 mph, time = 8 hours
Sue:  rate = 21 mph, time = 5 hours