Question

To travel 147 miles, it takes Sue, riding a moped, 2 hours less time than it takes Doreen to travel 72 miles riding a bicycle. Sue travels 13 miles per hour faster than Doreen. Find the times and rates of both girls.

Answer 1

let 'vs' be the velocity of Sue
'vd' be the velocity of Doreen
\[t - 2 = \frac{147}{v_s} ----equation 1\]
\[t = \frac{72}{v_d} -----equation 2\]
\[v_s = v_d + 13 -----equation 3\]
from equation 1
\[t - 2 = \frac{147}{v_s}\]
\[t = \frac{147}{v_s} + 2 ----equation 1'\]
equate equation 1' and 2
\[ \frac{147}{v_s} + 2 = \frac{72}{v_d} \]
substitute equation 3
\[\frac{147}{v_d+13}+2 = \frac{72}{v_d}\]
rearranging you'll have:
\[2{v_d}^2 + 101v_d = 936\]
\[v_d = 8mph\]
substitute to equation 3
\[v_s = v_d + 13\]
\[v_s = 8 + 13\]
\[v_s = 21mph\]

Answer 2

for the time:
using equation 1'
\[t = \frac{147}{v_s} \]
\[t = \frac{147mi}{21mph}\]
\[t = 7 hours ---Sue's time\]
\[t = \frac{72}{v_d}\]
\[t = \frac{72mi}{8mph}\]
\[t = 9 hours ----Doreen's time\]