X started from point A towards point B. After 2 hours, Y started from point B towards A. By the time X had covered one-fifth of the total distance, Y had also covered the same distance. Y's speed is three times the speed of X.

Compare the following:

Quantity I: Difference in time (in hours) taken by X and Y to reach their respective destinations.

Quantity II: 12 hours

Question

X started from point A towards point B. After 2 hours, Y started from point B towards A. By the time X had covered one-fifth of the total distance, Y had also covered the same distance. Y's speed is three times the speed of X.

Compare the following:

Quantity I: Difference in time (in hours) taken by X and Y to reach their respective destinations.

Quantity II: 12 hours

Accepted Answer

**Solution:**

1. Let the speed of X be *x* km/h.

2. Distance covered by X in 2 hours = $2x$ km.

3. Suppose X takes *t* hours to cover one-fifth of the total distance AB.

4. Since Y starts 2 hours later and covers the same one-fifth distance in $(t - 2)$ hours, and Y's speed is thrice that of X, we have:

$$\frac{t - 2}{t} = \frac{1}{3}$$

5. Solving for *t*:

$$3(t - 2) = t \Rightarrow 3t - 6 = t \Rightarrow 2t = 6 \Rightarrow t = 3 	ext{ hours}$$

6. One-fifth of the distance AB is covered by X in 3 hours, so:

$$	ext{Distance} = 	ext{speed} 	imes 	ext{time} = x 	imes 3 = 3x 	ext{ km}$$

7. Therefore, total distance AB is:

$$5 	imes 3x = 15x 	ext{ km}$$

8. Time taken by X to cover AB:

$$\frac{15x}{x} = 15 	ext{ hours}$$

9. Time taken by Y to cover AB (speed = $3x$):

$$\frac{15x}{3x} = 5 	ext{ hours}$$

10. Difference in time taken by X and Y:

$$15 - 5 = 10 	ext{ hours}$$

11. Comparing quantities:

- Quantity I = 10 hours
    - Quantity II = 12 hours

Hence, Quantity I < Quantity II.

**Answer:** (d) Quantity I < Quantity II

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