A motorcyclist's speed is 40 km/h greater than a cyclist's, so the motorcyclist covers 30 km in one hour less than the cyclist. How long did the cyclist take? Choose from 2 hours, 1.5 hours, 1 hour or 3 hours.

Step-by-step solution

1. 1

   Name the speed, not the time. Let $v$ be the cyclist's speed in km/h; the motorcyclist's is $v+40$. Choosing speed as the unknown keeps both times as simple quotients of the same distance:

   $t_{\text{cyclist}}=\frac{30}{v},\qquad t_{\text{motor}}=\frac{30}{v+40}.$

2. 2

   Turn 'one hour less' into an equation. The slower rider takes one hour more, so subtract in that order:

   $\frac{30}{v}-\frac{30}{v+40}=1.$

3. 3

   Clear the denominators. Multiply through by $v(v+40)$, which is non-zero since $v>0$:

   $30(v+40)-30v=v(v+40)\ \Longrightarrow\ 1200=v^{2}+40v.$

   The $30v$ terms cancel, which is why the arithmetic stays clean.

4. 4

   Solve the quadratic and discard the negative root.

   $v^{2}+40v-1200=0\ \Longrightarrow\ v=\frac{-40\pm\sqrt{1600+4800}}{2}=\frac{-40\pm 80}{2},$

   giving $v=20$ or $v=-60$. A speed cannot be negative, so $v=20\ \text{km/h}$.

5. 5

   Answer the question that was asked. The question wants the cyclist's time, not the speed:

   $t=\frac{30}{20}=1.5\ \text{h},$

   which is option 2.

6. 6

   Check against the story. The motorcyclist rides at $60\ \text{km/h}$ and takes $30/60=0.5$ h; the cyclist takes $1.5$ h. The difference is exactly one hour, and the speeds differ by $40\ \text{km/h}$ ✓.