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= (64)/(t) \\ \\((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)= (80\pm √((-80)^2-4*13(-192)) )/(2*13)= (80\pm128)/(26) \\ \\t=(80+128)/(26) =8 \\ \\r=(64)/(t) =(64)/(8) =8`

Doreen: rate = 8 mph, time = 8 hours
Sue: rate = 21 mph, time = 5 hours