Train A takes 2 hours more than train B to cover a certain distance $X$. Train A covers a distance of $(X + 160)$ km in 8 hours. If the speed of train B is 1.5 times the speed of train A, what is the speed of train B?

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**Solution:**

1. Let the speed of train A be $v_A$ km/h and speed of train B be $v_B$ km/h.

2. Given, $v_B = 1.5 	imes v_A$.

3. Train A covers $(X + 160)$ km in 8 hours, so:
   $$v_A = \frac{X + 160}{8}$$

4. Time taken by train A to cover distance $X$ is:
   $$t_A = \frac{X}{v_A}$$

5. Time taken by train B to cover distance $X$ is:
   $$t_B = \frac{X}{v_B} = \frac{X}{1.5 v_A}$$

6. Given train A takes 2 hours more than train B to cover distance $X$:
   $$t_A = t_B + 2$$

Substitute values:
   $$\frac{X}{v_A} = \frac{X}{1.5 v_A} + 2$$

7. Multiply both sides by $v_A$:
   $$X = \frac{X}{1.5} + 2 v_A$$

8. Simplify:
   $$X - \frac{X}{1.5} = 2 v_A$$

$$X \left(1 - \frac{2}{3}ight) = 2 v_A$$

$$X 	imes \frac{1}{3} = 2 v_A$$

$$X = 6 v_A$$

9. Substitute $X = 6 v_A$ into the expression for $v_A$:
   $$v_A = \frac{X + 160}{8} = \frac{6 v_A + 160}{8}$$

10. Multiply both sides by 8:
    $$8 v_A = 6 v_A + 160$$

11. Simplify:
    $$8 v_A - 6 v_A = 160$$

$$2 v_A = 160$$

$$v_A = 80 	ext{ km/h}$$

12. Since $v_B = 1.5 v_A$:
    $$v_B = 1.5 	imes 80 = 120 	ext{ km/h}$$

**Answer:** $120 	ext{ km/h}$

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