24. Anne, Bronwyn and Carl ran a race. They started at the same time, and their speeds were constant. When Anne finished, Bronwyn had $15 \mathrm{~m}$ to run and Carl had $35 \mathrm{~m}$ to run. When Bronwyn finished, Carl had $22 \mathrm{~m}$ to run. What was the length of the race?
A $135 \mathrm{~m}$
B $140 \mathrm{~m}$
C $150 \mathrm{~m}$
D $165 \mathrm{~m}$
E $175 \mathrm{~m}$
Solution
D

First note that Carl ran $(35-22) \mathrm{m}=13 \mathrm{~m}$ while Bronwen ran $15 \mathrm{~m}$. Let the length of the race be $x \mathrm{~m}$. Since their speeds were constant, the ratio of the distances they ran in any time is also constant. Therefore, since Carl ran $(x-35) \mathrm{m}$ while Bronwen ran $(x-15) \mathrm{m}$, we have $\frac{x-35}{x-15}=\frac{13}{15}$. Therefore $15 x-15 \times 35=13 x-13 \times 15$ and hence $2 x=15 \times 35-13 \times 15$. Therefore $2 x=22 \times 15$ and hence $x=11 \times 15$. Therefore the distance they ran is $165 \mathrm{~m}$.