Question

A bicyclist covered $\frac{5}{7}$ of his route and an additional $40\text{miles}$.

He has yet to cover $118\text{miles}$ less than $0.75$ of his route.

How long is his route?

Answer 1

Find the length of a bicyclist route:

Let $x\text{miles}$ be the total length of the route.

A bicyclist covered $\frac{5}{7}$ of his route and an additional $40\text{miles}$, that is $\left(\frac{5}{7}x+40\right)\text{miles}.$

He has yet to cover $118\text{miles}$ less than $0.75$ of his route, $\left(0.75x-118\right)\text{miles}.$

Since the total length of the route is $x\text{miles}$.

It follows that,

$\left(\frac{5}{7}x+40\right)+\left(0.75x-118\right)=x$

In order to determine the total length, solve the above equation:

$$\begin{gathered} \left(\frac{5}{7}x+40\right)+\left(0.75x-118\right)=x \\ \Rightarrow \frac{5}{7}x+40+\frac{75}{100}x-118=x \\ \Rightarrow \frac{5}{7}x+\frac{75}{100}x-78=x \\ \Rightarrow \frac{5}{7}x+\frac{75}{100}x-x=78 \\ \Rightarrow \frac{5\left(100\right)x+75\left(7\right)x-x\left(700\right)}{700}=78 \\ \Rightarrow \frac{500x+525x-700x}{700}=78 \\ \Rightarrow 500x+525x-700x=54600 \\ \Rightarrow 325x=54600 \\ \Rightarrow x=\frac{54600}{325} \\ \Rightarrow x=168\text{miles} \end{gathered}$$

Therefore, the total length of the route is $\mathbf{168}\mathbf{}\mathbf{\text{miles}}$.