5. Grandfather, father, and grandson ran the distance from home to the stadium and back. The grandson ran there and back at the same speed. Grandfather ran there twice as fast as the grandson, and back three times slower. Father ran there twice as slow as the grandson, and back three times faster. In what order will they return home?
Solution. Let the grandson's speed be denoted by $x$, and the distance by $S$. Then the times for the grandson, father, and grandfather to complete their runs are respectively:

$$
t_{1}=\frac{S}{x}+\frac{S}{x}, t_{2}=\frac{S}{\frac{1}{2} x}+\frac{S}{3 x}, t_{3}=\frac{S}{2 x}+\frac{S}{\frac{1}{3} x}
$$

Thus, we need to compare the following numbers:

$$
a_{1}=2, a_{2}=2+\frac{1}{3}, a_{3}=\frac{1}{2}+3
$$

Therefore, the grandson came in first, the father second, and the grandfather last.

Answer: The grandson came in first, the father second, and the grandfather last.