## 78. Math Puzzle 11/71

On a $40 \mathrm{~km}$ long stretch, a fast train had a speed $20 \frac{\mathrm{km}}{\mathrm{h}}$ greater than a freight train, which took 20 minutes longer. What were the speeds of the trains?
For the fast train, let $v_{S}$ be the speed and $t_{S}$ the travel time, and for the freight train, let $v_{G}$ and $t_{G}$ be the corresponding values.

It is $v_{S} t_{S}=v_{G} t_{G}=40$ and $v_{S}+v_{G}=20$. Substitution yields $t_{S}=\frac{1}{60} v_{G}$ and

$$
\frac{1}{60} v_{G}^{2}+\frac{1}{3} v_{G}-40=0 \rightarrow v_{G}=40 \frac{\text{km}}{\text{h}} ; v_{S}=60 \frac{\text{km}}{\text{h}}
$$