To solve this question, we must break down the question into different scenarios.
The speed expression for the first rider is:
[tex]\begin{gathered} s=\frac{d}{t} \\ \text{let us make the distance the first rider covers as y.} \\ d=y\text{ miles} \\ t=3\text{ hours.} \\ s_1=\frac{y}{3} \end{gathered}[/tex]The speed expression for the second cyclist:
[tex]\begin{gathered} s=\frac{d}{t} \\ the\text{ first rider covered a distance of y miles, the remaining distance } \\ \text{left for the second cyclist to cover is:} \\ (108-y)\text{miles at the same time of 3 hours.} \\ s_2=\frac{108-y}{3} \end{gathered}[/tex]Since one cyclist cycles 3 times as fast as the other:
It is expressed thus:
[tex]\begin{gathered} s_1=3\times s_2 \\ s_1=3s_2 \end{gathered}[/tex]Now substitute the values for the speed expression into the expression above, we will have:
[tex]\frac{y}{3}=3\times(\frac{108-y}{3})[/tex]By solving the above expression, we will get the value of y (part of the distance travelled) and we can get the speed of the faster cyclist.
[tex]\begin{gathered} \frac{y}{3}=\frac{324-3y}{3} \\ y=324-3y \\ y+3y=324 \\ \end{gathered}[/tex][tex]\begin{gathered} 4y=324 \\ y=\frac{324}{4} \\ y=81\text{ miles.} \\ \\ So\text{ the speed of the faster cyclist will be:} \\ _{}=\frac{y}{3} \\ =\frac{81\text{ miles}}{3\text{ hours}} \\ =27mi\text{/h} \end{gathered}[/tex]The speed of the faster cyclist is 27 mi/h.