The speeds of trains A ($S_a$) and B ($S_b$) in km/hr, with $S_a S_b$, are the roots of the equation:

$$x^2 - 130x + 4200 = 0$$

Let $T_a$ and $T_b$ be the usual times (in hours) taken by trains A and B respectively to cover a distance $D$ km. If their speeds are interchanged for the same distance $D$ (i.e., train A travels at speed $S_b$ and train B at speed $S_a$), then train A takes $(T_a + 2)$ hours and train B takes $(T_b - 2)$ hours.

Find the sum of the original times $T_a + T_b$.

Question

The speeds of trains A ($S_a$) and B ($S_b$) in km/hr, with $S_a > S_b$, are the roots of the equation:

$$x^2 - 130x + 4200 = 0$$

Let $T_a$ and $T_b$ be the usual times (in hours) taken by trains A and B respectively to cover a distance $D$ km. If their speeds are interchanged for the same distance $D$ (i.e., train A travels at speed $S_b$ and train B at speed $S_a$), then train A takes $(T_a + 2)$ hours and train B takes $(T_b - 2)$ hours.

Find the sum of the original times $T_a + T_b$.

Accepted Answer

**Solution:**

1. The speeds $S_a$ and $S_b$ are roots of the quadratic equation:
   $$x^2 - 130x + 4200 = 0$$

2. Using the quadratic formula or factoring, find the roots:
   - Factors of 4200 that add up to 130 are 70 and 60.
   - So, $S_a = 70$ km/hr and $S_b = 60$ km/hr (given $S_a > S_b$).

3. Let the distance be $D$ km. Then,
   $$D = S_a 	imes T_a = 70 	imes T_a$$
   $$D = S_b 	imes T_b = 60 	imes T_b$$

4. When speeds are swapped:
   - Train A travels at $S_b = 60$ km/hr and takes $(T_a + 2)$ hours:
     $$D = 60 	imes (T_a + 2)$$
   - Train B travels at $S_a = 70$ km/hr and takes $(T_b - 2)$ hours:
     $$D = 70 	imes (T_b - 2)$$

5. Equate the distances:
   - From train A:
     $$70 T_a = 60 (T_a + 2)$$
     $$70 T_a = 60 T_a + 120$$
     $$10 T_a = 120 ightarrow T_a = 12 	ext{ hours}$$

- From train B:
     $$60 T_b = 70 (T_b - 2)$$
     $$60 T_b = 70 T_b - 140$$
     $$-10 T_b = -140 ightarrow T_b = 14 	ext{ hours}$$

6. Sum of original times:
   $$T_a + T_b = 12 + 14 = 26 	ext{ hours}$$

**Answer:** 26

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