A's rate = 1/12, B's rate = 1/18.

Combined rate = 1/12 + 1/18 = 3/36 + 2/36 = 5/36.

Time = 36/5 = 7.2 days

Combined rate = 1/10 + 1/15 + 1/30 = 3/30 + 2/30 + 1/30 = 6/30 = 1/5.

Time = 5 days

A completes 1/3 work in 5 days, so rate = 1/15 per day.

Remaining work = 2/3.

Days needed = (2/3)/(1/15) = (2/3) × 15 = 10 days

Combined rate = 1/8, A's rate = 1/12. B's rate = 1/8 - 1/12 = 3/24 - 2/24 = 1/24. B takes 24 days

A's rate = 1/12, B's rate = 1/18. Combined rate = 5/36. In 6 hours, they fill 30/36 = 5/6. Remaining = 1/6. Time for A alone = (1/6)/(1/12) = 2 hours. Total = 6+2 = 8 hours. But with both for 6 hrs: (5/36)×6 = 5/6 filled. Remaining 1/6 by A: 2 hours. Check: Total effective = 4 hours additional work needed.

Item 1: CP=500, SP=600 (20% gain). Item 2: CP=500, SP=425 (15% loss). Total CP=1000, Total SP=1025. Profit=25. Percentage = 2.5%

P's rate = 1/30, Q's rate = 1/40. Combined = 7/120. In 10 days: 10 × 7/120 = 7/12 completed. Remaining = 5/12. Q's time = (5/12)/(1/40) = 5/12 × 40 = 16.67 ≈ 15 days

Combined rate = 1/6 + 1/8 + 1/10 = 20/120 + 15/120 + 12/120 = 47/120. Time = 120/47 ≈ 2.73 hours

Let CP = 100. MP = 140. SP = 140 × 80/100 = 112. Profit = 12%

When two trains move towards each other, their relative speed is the sum of their individual speeds, and they must cover a combined distance equal to the sum of their lengths.

Step 1: Convert speeds to m/s

Speed is given in km/h; convert to m/s by multiplying by \(\frac{5}{18}\):

Step 2: Find relative speed (trains moving towards each other)

When trains move towards each other, relative speed is the sum:

Or more precisely: \(v_{\text{relative}} = \frac{25}{2} + \frac{50}{3} = \frac{75 + 100}{6} = \frac{175}{6} \text{ m/s}\]

Step 3: Find total distance to be covered

For the trains to completely cross each other, the combined distance equals the sum of their lengths:

Step 4: Calculate time to cross

Using \(t = \frac{d}{v}\):

Answer: The trains will take \(12\) seconds to cross each other. (Option B)