A person travelled $750 ext{ km}$ by train, $440 ext{ km}$ by bus, and $210 ext{ km}$ by car. The total time taken for the journey was $25$ hours. If the ratio of the speed of the car to the bus is $7:11$ and the ratio of the speed of the bus to the train is $11:25$, what is the speed of the bus?

Question

A person travelled $750 	ext{ km}$ by train, $440 	ext{ km}$ by bus, and $210 	ext{ km}$ by car. The total time taken for the journey was $25$ hours. If the ratio of the speed of the car to the bus is $7:11$ and the ratio of the speed of the bus to the train is $11:25$, what is the speed of the bus?

Accepted Answer

**Solution:**

1. Let the speeds of car, bus, and train be $7x$, $11x$, and $25y$ respectively, based on the given ratios:
   - Speed of car : bus = $7 : 11$ implies car speed = $7x$, bus speed = $11x$
   - Speed of bus : train = $11 : 25$ implies bus speed = $11y$, train speed = $25y$

2. Since bus speed is common in both ratios, equate $11x = 11y$ which gives $x = y$.

3. Therefore, speeds are:
   - Car = $7x$
   - Bus = $11x$
   - Train = $25x$

4. Calculate the time taken for each mode of travel:
   - Time by train = $\frac{750}{25x} = \frac{30}{x}$ hours
   - Time by bus = $\frac{440}{11x} = \frac{40}{x}$ hours
   - Time by car = $\frac{210}{7x} = \frac{30}{x}$ hours

5. Total time given is 25 hours, so:
   $$\frac{30}{x} + \frac{40}{x} + \frac{30}{x} = 25$$
   $$\frac{100}{x} = 25$$
   $$x = \frac{100}{25} = 4$$

6. Speed of bus = $11x = 11 	imes 4 = 44 	ext{ km/h}$

**Answer:** $44 	ext{ km/h}$

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