16c. Boats and Streams problems step by step solution procedure and detailed explanation for each problem

 16c.  Boats and Streams problems step by step solution procedure and detailed explanation for each problem 

Home page of Arithmetic Problems 

Boats and Streams:
  • Downstream speed = Boat speed + Stream speed
  • Upstream speed = Boat speed - Stream speed
  • Boat speed = (Downstream speed + Upstream speed) / 2
  • Stream speed = (Downstream speed - Upstream speed) / 2

Section 3: Boats and Streams (10 Problems)
Boats and Streams Problem 1: Find Boat and Stream Speeds
A boat travels 36 km downstream in 3 hours and 24 km upstream in 3 hours. Find the boat’s speed and the stream’s speed.
Step-by-Step Solution:
  1. Identify given values:
    • Downstream: 36 km, 3 hours
    • Upstream: 24 km, 3 hours
  2. Calculate speeds:
    • Downstream speed = 36 / 3 = 12 km/h
    • Upstream speed = 24 / 3 = 8 km/h
  3. Calculate boat and stream speeds:
    • Boat speed = (12 + 8) / 2 = 10 km/h
    • Stream speed = (12 - 8) / 2 = 2 km/h
  4. State result: Boat speed = 10 km/h, Stream speed = 2 km/h.
Detailed Explanation: Downstream and upstream speeds are calculated from distance and time. The boat’s speed is the average of these speeds, and the stream’s speed is half their difference, derived from the formulas for downstream and upstream motion.
Answer: Boat speed = 10 km/h, Stream speed = 2 km/h
Boats and Streams Problem 2: Time Downstream
A boat’s speed is 15 km/h, and the stream’s speed is 3 km/h. How long to travel 54 km downstream?
Step-by-Step Solution:
  1. Identify given values:
    • Boat speed = 15 km/h
    • Stream speed = 3 km/h
    • Distance = 54 km
  2. Calculate downstream speed: 15 + 3 = 18 km/h
  3. Calculate time: Time = 54 / 18 = 3 hours
  4. State result: The time is 3 hours.
Detailed Explanation: Downstream speed is the sum of boat and stream speeds. Dividing the distance by this speed gives the time, a straightforward application of the downstream formula.
Answer: 3 hours
Boats and Streams Problem 3: Time Upstream
A boat’s speed is 12 km/h, and the stream’s speed is 4 km/h. How long to travel 32 km upstream?
Step-by-Step Solution:
  1. Identify given values:
    • Boat speed = 12 km/h
    • Stream speed = 4 km/h
    • Distance = 32 km
  2. Calculate upstream speed: 12 - 4 = 8 km/h
  3. Calculate time: Time = 32 / 8 = 4 hours
  4. State result: The time is 4 hours.
Detailed Explanation: Upstream speed is the boat’s speed minus the stream’s speed. The time is the distance divided by this reduced speed, showing the stream’s opposition increases travel time.
Answer: 4 hours
Boats and Streams Problem 4: Round Trip Time
A boat’s speed is 10 km/h, and the stream’s speed is 2 km/h. How long to travel 20 km downstream and 20 km upstream?
Step-by-Step Solution:
  1. Identify given values:
    • Boat speed = 10 km/h
    • Stream speed = 2 km/h
    • Distance = 20 km each way
  2. Calculate speeds:
    • Downstream speed = 10 + 2 = 12 km/h
    • Upstream speed = 10 - 2 = 8 km/h
  3. Calculate times:
    • Downstream time = 20 / 12 ≈ 1.67 hours
    • Upstream time = 20 / 8 = 2.5 hours
  4. Calculate total time: 1.67 + 2.5 ≈ 4.17 hours
  5. State result: The total time is approximately 4.17 hours.
Detailed Explanation: Each segment’s time is calculated using the appropriate speed (downstream faster, upstream slower). The total time sums these, showing the stream’s effect on round-trip duration.
Answer: ≈ 4.17 hours
Boats and Streams Problem 5: Equal Times
A boat takes 5 hours for 30 km downstream and 30 km upstream. If the stream’s speed is 1 km/h, find the boat’s speed.
Step-by-Step Solution:
  1. Identify given values:
    • Total time = 5 hours
    • Distance = 30 km each way
    • Stream speed = 1 km/h
  2. Let boat speed = x km/h:
    • Downstream speed = x + 1
    • Upstream speed = x - 1
  3. Set up equation: 30 / (x + 1) + 30 / (x - 1) = 5
  4. Solve:
    • Multiply by (x + 1)(x - 1): 30(x - 1) + 30(x + 1) = 5(x² - 1)
    • 60x = 5x² - 5
    • 5x² - 60x - 5 = 0
    • x² - 12x - 1 = 0
    • x = [12 ± √(144 + 4)] / 2 ≈ 12.08 / 2 ≈ 6.04
  5. State result: Boat speed ≈ 6 km/h.
Detailed Explanation: The total time equation leads to a quadratic, solved for the boat’s speed. The positive root is used, and verification (times sum to ~5 hours) confirms the solution, handling combined motion.
Answer: ≈ 6 km/h
Boats and Streams Problem 6: Stream Speed
A boat travels 40 km downstream in 2 hours and 40 km upstream in 4 hours. Find the stream’s speed.
Step-by-Step Solution:
  1. Identify given values:
    • Downstream: 40 km, 2 hours
    • Upstream: 40 km, 4 hours
  2. Calculate speeds:
    • Downstream speed = 40 / 2 = 20 km/h
    • Upstream speed = 40 / 4 = 10 km/h
  3. Calculate stream speed: (20 - 10) / 2 = 5 km/h
  4. State result: The stream’s speed is 5 km/h.
Detailed Explanation: The stream’s speed is half the difference between downstream and upstream speeds, derived from the boat-and-stream formulas, providing a quick calculation.
Answer: 5 km/h
Boats and Streams Problem 7: Boat Speed
A boat travels 48 km downstream in 4 hours and 36 km upstream in 6 hours. Find the boat’s speed.
Step-by-Step Solution:
  1. Identify given values:
    • Downstream: 48 km, 4 hours
    • Upstream: 36 km, 6 hours
  2. Calculate speeds:
    • Downstream speed = 48 / 4 = 12 km/h
    • Upstream speed = 36 / 6 = 6 km/h
  3. Calculate boat speed: (12 + 6) / 2 = 9 km/h
  4. State result: The boat’s speed is 9 km/h.
Detailed Explanation: The boat’s speed is the average of downstream and upstream speeds, a direct application of the formula, showing consistent motion effects.
Answer: 9 km/h
Boats and Streams Problem 8: Distance Ratio
A boat’s downstream speed is twice its upstream speed. If it travels 60 km downstream in 3 hours, find the upstream distance in 3 hours.
Step-by-Step Solution:
  1. Identify given values:
    • Downstream time = 3 hours, Distance = 60 km
    • Downstream speed = 2 × Upstream speed
  2. Calculate downstream speed: 60 / 3 = 20 km/h
  3. Calculate upstream speed: 20 / 2 = 10 km/h
  4. Calculate upstream distance: Distance = 10 × 3 = 30 km
  5. State result: The upstream distance is 30 km.
Detailed Explanation: The downstream speed is given as twice the upstream speed. After finding the downstream speed, the upstream speed is halved, and the upstream distance is calculated for the same time.
Answer: 30 km
Boats and Streams Problem 9: Time with Increased Stream
A boat travels 20 km upstream in 2 hours with a stream speed of 2 km/h. If the stream speed increases to 3 km/h, find the new upstream time.
Step-by-Step Solution:
  1. Identify given values:
    • Distance = 20 km, Time = 2 hours
    • Original stream speed = 2 km/h, New stream speed = 3 km/h
  2. Calculate original upstream speed: 20 / 2 = 10 km/h
  3. Calculate boat speed: Upstream speed = Boat speed - Stream speed
    • 10 = Boat speed - 2, Boat speed = 12 km/h
  4. Calculate new upstream speed: 12 - 3 = 9 km/h
  5. Calculate new time: 20 / 9 ≈ 2.22 hours
  6. State result: The new time is approximately 2.22 hours.
Detailed Explanation: The original upstream speed gives the boat’s speed, adjusted for the new stream speed to find the new upstream speed. The increased stream speed reduces the upstream speed, increasing the time.
Answer: ≈ 2.22 hours
Boats and Streams Problem 10: Total Distance
A boat travels downstream at 14 km/h and upstream at 10 km/h. Find the total distance traveled in 5 hours downstream and 3 hours upstream.
Step-by-Step Solution:
  1. Identify given values:
    • Downstream speed = 14 km/h, Time = 5 hours
    • Upstream speed = 10 km/h, Time = 3 hours
  2. Calculate distances:
    • Downstream distance = 14 × 5 = 70 km
    • Upstream distance = 10 × 3 = 30 km
  3. Calculate total distance: 70 + 30 = 100 km
  4. State result: The total distance is 100 km.
Detailed Explanation: Each segment’s distance is speed times time, summed for the total. The different speeds reflect the stream’s effect, but the calculation is additive for total distance.
Answer: 100 km


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