13.043. The distance between Moscow and Smolensk by railway is 415 km. On this route, there are the cities of Mozhaisk and Vyazma. The distance between Moscow and Mozhaisk is to the distance between Mozhaisk and Vyazma as $7: 9$, and the distance between Mozhaisk and Vyazma is $27 / 35$ of the distance between Vyazma and Smolensk. Find the distances between each pair of neighboring cities.
Solution.

Let $x$ km be the distance between Mozhaisk and Vyazma. Then $\frac{7}{9} x$ km is the distance between Moscow and Mozhaisk; $\frac{35}{27} x$ km between Vyazma and Smolensk. According to the condition, $\frac{7}{9} x + x + \frac{35}{27} x = 415$, from which $x = 135$ km. Between Moscow and Mozhaisk $-\frac{7}{9} \cdot 135 = 105$ km; between Vyazma and Smolensk $-\frac{35}{27} \cdot 135 = 175$ km.

Answer: 105, 135, and 175 km.