A person travels a total distance of $132 ext{ km}$ in $t$ hours by using a car for some part and a train for the rest. The speed of the train is 20\% higher than the speed of the car. The ratio of the distance covered by car to that covered by train is $4 : 7$. It is also given that the car covers $48 ext{ km}$ at a speed of $24 ext{ km/h}$ in $(t + 0.5)$ hours. What is the approximate speed of the train?

Question

A person travels a total distance of $132 	ext{ km}$ in $t$ hours by using a car for some part and a train for the rest. The speed of the train is 20\% higher than the speed of the car. The ratio of the distance covered by car to that covered by train is $4 : 7$. It is also given that the car covers $48 	ext{ km}$ at a speed of $24 	ext{ km/h}$ in $(t + 0.5)$ hours. What is the approximate speed of the train?

Accepted Answer

**Solution:**

1. Let the speed of the car be $x 	ext{ km/h}$. Then, the speed of the train is $1.2x$ (20\% more than car speed).

2. The ratio of distances covered by car and train is $4 : 7$. Total distance is $132 	ext{ km}$.
   - Distance by car = $\frac{4}{4+7} 	imes 132 = \frac{4}{11} 	imes 132 = 48 	ext{ km}$
   - Distance by train = $132 - 48 = 84 	ext{ km}$

3. Given that the car covers $48 	ext{ km}$ at $24 	ext{ km/h}$ in $(t + 0.5)$ hours.
   - Time taken by car = $\frac{48}{24} = 2$ hours
   - So, $t + 0.5 = 2 \Rightarrow t = 1.5$ hours

4. Total time $t$ is the sum of time taken by car and train.
   - Time by car = $2$ hours (already calculated)
   - Time by train = $t -$ time by car = $1.5 - 2 = -0.5$ hours (This is not possible, so re-check the interpretation.)

Actually, the problem states the car covers $48 	ext{ km}$ in $(t + 0.5)$ hours, so:
   - Time by car = $t + 0.5$
   - Given speed of car = $24 	ext{ km/h}$, distance by car = $48 	ext{ km}$
   - So, time by car = $\frac{48}{24} = 2$ hours
   - Therefore, $t + 0.5 = 2 \Rightarrow t = 1.5$ hours

5. Total time $t = 1.5$ hours is the total time for the entire journey.

6. Time taken by train = total time $t$ minus time taken by car:
   - Time by train = $1.5 - 2 = -0.5$ hours (negative, which is not possible)

This indicates a misinterpretation. The problem states the car covers $48 	ext{ km}$ at $24 	ext{ km/h}$ in $(t + 0.5)$ hours, so the car's time is $(t + 0.5)$ hours, not the total time.

7. Let the time taken by train be $t$ hours.
   - Time by car = $t + 0.5$ hours

8. Using distance = speed $	imes$ time:
   - Distance by car = $24 	imes (t + 0.5) = 48$ km
   - So, $24t + 12 = 48 \Rightarrow 24t = 36 \Rightarrow t = 1.5$ hours

9. Time taken by train = $t = 1.5$ hours

10. Distance by train = $84$ km
    - Speed of train = $\frac{84}{1.5} = 56 	ext{ km/h}$

11. But the speed of train is also $1.2$ times the speed of car:
    - Speed of car = $24 	ext{ km/h}$
    - Speed of train = $1.2 	imes 24 = 28.8 	ext{ km/h}$

This contradicts the previous calculation of $56 	ext{ km/h}$.

12. Re-examining the problem, the ratio of distances is $4 : 7$, so distances are $4k$ and $7k$ respectively.
    - Total distance: $4k + 7k = 11k = 132 \Rightarrow k = 12$
    - Distance by car = $4k = 48$ km (matches given)
    - Distance by train = $7k = 84$ km

13. Let speed of car = $x$ km/h, speed of train = $1.2x$ km/h.

14. Time taken by car = $\frac{48}{x}$
    Time taken by train = $\frac{84}{1.2x} = \frac{84}{1.2x} = \frac{70}{x}$

15. Total time $t = $ time by car + time by train:
    $$t = \frac{48}{x} + \frac{70}{x} = \frac{118}{x}$$

16. Given car covers $48$ km at $24$ km/h in $(t + 0.5)$ hours:
    $$\frac{48}{24} = t + 0.5 \Rightarrow 2 = t + 0.5 \Rightarrow t = 1.5$$

17. Substitute $t = 1.5$ into equation:
    $$1.5 = \frac{118}{x} \Rightarrow x = \frac{118}{1.5} = 78.67 	ext{ km/h}$$

18. Speed of train = $1.2x = 1.2 	imes 78.67 = 94.4 	ext{ km/h}$

**Answer:** Approximately $94 	ext{ km/h}$

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