Answer
$$\text{C}$$
Work Step by Step
Let $x$ = rate (in miles per hour) Mike rode his bike
Since Mike rode $5$ miles per hour faster than Jane then, Jane's rate is $x-5$.
Recall:
$\text{Distance} = \text{(rate)} \times \text{(time)}$
Mike and Jane went opposite directions so the distance between them can be found by adding the distance they traveled.
In the given problem, we have:
time = $4$ hours
Mike's rate = $x$
Jane's rate =$x-5$
Distance between Mike and Jane after $4$ hours = $124$ miles
Thus, we have:
\begin{align*}
124& = \text{distance Mike traveled} + \text{distance Jane traveled}\\
124&= 4(x) + 4(x-5)\\
124&= 4x + 4x-20\\
124&= 8x-20\\
124+20&=8x-20+20\\
144&=8x\\
\frac{144}{8} &= \frac{8x}{8}\\
18&=x
\end{align*}
Thus, Mike rode his bike at $18$ miles per hour.
The answer is Option $|\text{C}$.
