Round Trips & Time Ratios

🎯 Learning Objective

Solve round-trip time questions and convert time ratios into boat-to-stream speed ratios.

💡 Concept

• Round trip: total time = d/(b + s) + d/(b − s) — compute the two legs separately
• Same distance → time ratio (up : down) is the INVERSE of speed ratio
• If t(up) : t(down) = a : b, then boat : stream = (a + b) : (a − b)
• '4 km downstream in the time of 3 km upstream' → down : up speed = 4 : 3

🧮 Key Formulas

T = d/(b + s) + d/(b − s)

>

t_up : t_down = a : b ⇒ b : s = (a + b) : (a − b)

✏️ Easy Example

Q. A boat rows at 8 km/h in still water; the stream is 2 km/h. Find the time for a round trip to a ghat 15 km away.

1. Down = 15/10 = 1.5 h
2. Up = 15/6 = 2.5 h
3. Total = 1.5 + 2.5

Answer: 4 hours

🇮🇳 Real-Life Example

Kerala backwater ferries publish different up-run and down-run timings for the same route — the canal current adds and subtracts exactly like your exam question.

📝 Exam-Level Example

Q. A man can row at 6 km/h in still water. It takes him twice as long to row up as to row down the river. Find the speed of the stream.

1. Time double → down speed = 2 × up speed
2. 6 + s = 2(6 − s)
3. 6 + s = 12 − 2s → s = 2

Answer: 2 km/h

📝 Exam-Level Example

Q. A man rows to a place 48 km away and comes back in 14 hours. He can row 4 km downstream in the same time as 3 km upstream. Find the stream's speed.

1. Down : Up = 4 : 3 → speeds 4x and 3x
2. 48/4x + 48/3x = 14 → 12/x + 16/x = 14
3. 28/x = 14 → x = 2 → Down 8, Up 6
4. s = (8 − 6)/2 = 1

Answer: 1 km/h

🪄 Memory Trick

'Twice as long upstream' means downstream speed is DOUBLE the upstream — jump straight to b : s = 3 : 1 via (2+1) : (2−1).

⚠️ Common Mistakes

• ❌ Averaging the two leg times for a round trip
• ❌ Forgetting to flip: time ratio is the inverse of speed ratio
• ❌ Using b instead of b + s for the downstream leg

🏆 Exam Tips

• ✅ Write both leg speeds first; the algebra then stays one line long
• ✅ Answers must keep b > s, otherwise upstream motion would be impossible

📌 Summary

• Round trip = sum of two separate legs
• Time ratio = inverse speed ratio
• t up : t down = a : b → b : s = (a+b) : (a−b)
• Always end with b > s check