2. Alistair, Conrad, Emma, and Salma compete in a three-sport race. They each swim $2 \mathrm{~km}$, then bike $40 \mathrm{~km}$, and finally run $10 \mathrm{~km}$. Also, they each switch instantly from swimming to biking and from biking to running.
(a) Emma has completed $\frac{1}{13}$ of the total distance of the race. How many kilometers has she travelled?
(b) Conrad began the race at 8:00 a.m. and completed the swimming portion in 30 minutes. Conrad biked 12 times as fast as he swam, and ran 3 times as fast as he swam. At what time did he finish the race?
(c) Alistair and Salma also began the race at 8:00 a.m. Alistair finished the swimming portion in 36 minutes, and then biked at $28 \mathrm{~km} / \mathrm{h}$. Salma finished the swimming portion in 30 minutes, and then biked at $24 \mathrm{~km} / \mathrm{h}$. Alistair passed Salma during the bike portion. At what time did Alistair pass Salma?
2. (a) The total length of the race is $2+40+10=52 \mathrm{~km}$.
When Emma has completed $\frac{1}{13}$ of the total distance, she has travelled $\frac{1}{13} \times 52=4 \mathrm{~km}$.
(b) Since Conrad completed the $2 \mathrm{~km}$ swim in 30 minutes (which is half an hour), then his speed was $2 \div \frac{1}{2}=4 \mathrm{~km} / \mathrm{h}$.
Since Conrad biked 12 times as fast as he swam, then he biked at $12 \times 4=48 \mathrm{~km} / \mathrm{h}$.
Since Conrad biked $40 \mathrm{~km}$, then the bike portion took him $\frac{40}{48}=\frac{5}{6}$ hours.
Since 1 hour equals 60 minutes, then the bike portion took him $\frac{5}{6} \times 60=50$ minutes.
Since Conrad ran 3 times as fast as he swam, then he ran at $3 \times 4=12 \mathrm{~km} / \mathrm{h}$.
Since Conrad ran $10 \mathrm{~km}$, then the running portion took him $\frac{10}{12}=\frac{5}{6}$ hours. This is again 50 minutes.
Therefore, the race took him $30+50+50=130$ minutes, or 2 hours and 10 minutes.
Since Conrad began the race at 8:00 a.m., then he completed the race at 10:10 a.m.
(c) Suppose that Alistair passed Salma after $t$ minutes of the race.
Since Alistair swam for 36 minutes, then he had biked for $t-36$ minutes (or $\frac{t-36}{60}$ hours) when they passed.
Since Salma swam for 30 minutes, then she had biked for $t-30$ minutes (or $\frac{t-30}{60}$ hours) when they passed.
When Alistair and Salma passed, they had travelled the same total distance. At this time, Alistair had swum $2 \mathrm{~km}$. Since he bikes at $28 \mathrm{~km} / \mathrm{h}$, he had biked $28 \times \frac{t-36}{60}$ $\mathrm{km}$.
Similarly, Salma had swum $2 \mathrm{~km}$. Since she bikes at $24 \mathrm{~km} / \mathrm{h}$, she had biked $24 \times \frac{t-30}{60}$ $\mathrm{km}$.
Since their total distances are the same, then
$$
\begin{aligned}
2+28 \times \frac{t-36}{60} & =2+24 \times \frac{t-30}{60} \\
28 \times \frac{t-36}{60} & =24 \times \frac{t-30}{60} \\
28(t-36) & =24(t-30) \\
7(t-36) & =6(t-30) \\
7 t-252 & =6 t-180 \\
t & =72
\end{aligned}
$$

Therefore, Alistair passed Salma 72 minutes into the race.
Since the race began at 8:00 a.m., then he passed her at 9:12 a.m.
Alternatively, we could notice that Salma started the bike portion of the race 6 minutes before Alistair.
Since 6 minutes is $\frac{1}{10}$ of an hour, she biked a distance of $\frac{1}{10} \times 24=2.4 \mathrm{~km}$ before Alistair started riding.
In other words, she has a $2.4 \mathrm{~km}$ head start on Alistair.
Since Alistair bikes $4 \mathrm{~km} / \mathrm{h}$ faster than Salma, it will take $\frac{2.4}{4}=0.6$ hours for him to make up this difference.
Since 0.6 hours equals $0.6 \times 60=36$ minutes and he swam for 36 minutes before starting the bicycle portion, he will catch up a total of $36+36=72$ minutes into the race.