Question
A motorboat covers 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can cover 40 km upstream and 55 km downstream. Find the speed of the boat in still water and the speed of the stream.
Solution — Step by Step
Speed-distance problems become linear equations once you set up the right variables. The trick is recognising that upstream and downstream speeds are combinations of the boat's speed and the stream's speed.
Step 1: Define variables.
Let:
- Speed of boat in still water = x km/h
- Speed of stream = y km/h
Then:
- Upstream speed = x − y km/h (against the current)
- Downstream speed = x + y km/h (with the current)
Step 2: Use Time = Distance / Speed to form equations.
First condition (10 hours total): Time upstream + Time downstream = 10
30/(x − y) + 44/(x + y) = 10 ... (1)
Second condition (13 hours total): 40/(x − y) + 55/(x + y) = 13 ... (2)
Step 3: Make a substitution to convert to linear form.
Let:
- a = 1/(x − y)
- b = 1/(x + y)
Equations become: 30a + 44b = 10 ... (1) 40a + 55b = 13 ... (2)
These are now linear equations in a and b.
Step 4: Solve by elimination.
Multiply equation (1) by 4 and equation (2) by 3: 120a + 176b = 40 120a + 165b = 39
Subtract: 11b = 1 → b = 1/11
Substitute into equation (1): 30a + 44(1/11) = 10 30a + 4 = 10 30a = 6 → a = 1/5
Step 5: Back-convert to find x and y.
a = 1/(x − y) = 1/5 → x − y = 5 ... (3) b = 1/(x + y) = 1/11 → x + y = 11 ... (4)
Adding (3) and (4): 2x = 16 → x = 8 Subtracting: 2y = 6 → y = 3
Verification:
- 30/5 + 44/11 = 6 + 4 = 10 ✓
- 40/5 + 55/11 = 8 + 5 = 13 ✓
Answer
Speed of boat in still water = 8 km/h
Speed of stream = 3 km/h
Why This Works
The key insight is the substitution step. The original equations have x and y in the denominators — that makes them non-linear. By introducing a = 1/(x−y) and b = 1/(x+y), we convert them to linear equations, which we can solve easily. After solving for a and b, we recover x and y.
This technique — substituting to simplify — appears in many JEE problems too. Recognising when to substitute is a high-value skill.
Alternative: Direct Approach (Only When Numbers Are Simple)
If the upstream and downstream speeds come out as nice numbers from the problem setup, you can sometimes guess and check. But for this problem, the substitution method is necessary.
💡 Expert Tip
The three core speed-distance formulas to remember:
- Distance = Speed × Time
- Speed = Distance / Time
- Time = Distance / Speed
For river problems: upstream speed = (boat − stream), downstream speed = (boat + stream). Always set these up in a table before writing equations.
Common Mistake
⚠️ Common Mistake
Mistake: Setting up equations as (x − y)/(30) instead of (30)/(x − y).
Time = Distance / Speed, not Distance × Speed. So the time to cover 30 km upstream at speed (x − y) is 30/(x − y), not 30(x − y). Inverting the formula gives a completely different equation and a wrong answer.