Finding Speed of Boat and Stream

Video Explanation

Question

A boat goes 12 km upstream and 40 km downstream in 8 hours. It can go 16 km upstream and 32 km downstream in the same time. Find the speed of the boat in still water and the speed of the stream.

Solution

Step 1: Concept

Time = Distance / Speed

Step 2: Let Variables

Let speed of boat in still water = \(x\) km/h

Let speed of stream = \(y\) km/h

Upstream speed = \(x – y\), Downstream speed = \(x + y\)

Step 3: Form Equations

\[ \frac{12}{x – y} + \frac{40}{x + y} = 8 \quad (1) \]

\[ \frac{16}{x – y} + \frac{32}{x + y} = 8 \quad (2) \]

Step 4: Convert into Linear Form

Let:

\[ a = \frac{1}{x – y}, \quad b = \frac{1}{x + y} \]

Then:

\[ 12a + 40b = 8 \quad (3) \]

\[ 16a + 32b = 8 \quad (4) \]

Step 5: Solve Linear Equations

Multiply (3) by 4:

\[ 48a + 160b = 32 \quad (5) \]

Multiply (4) by 3:

\[ 48a + 96b = 24 \quad (6) \]

Subtract (6) from (5):

\[ 64b = 8 \]

\[ b = \frac{1}{8} \]

Substitute into (3):

\[ 12a + 40\left(\frac{1}{8}\right) = 8 \]

\[ 12a + 5 = 8 \]

\[ 12a = 3 \]

\[ a = \frac{1}{4} \]

Step 6: Back Substitute

\[ x – y = \frac{1}{a} = 4 \]

\[ x + y = \frac{1}{b} = 8 \]

Step 7: Solve Final Equations

\[ x – y = 4, \quad x + y = 8 \]

Add:

\[ 2x = 12 \Rightarrow x = 6 \]

Substitute:

\[ 6 + y = 8 \Rightarrow y = 2 \]

Conclusion

\[ \text{Boat speed} = 6 \text{ km/h}, \quad \text{Stream speed} = 2 \text{ km/h} \]

Verification

Upstream speed = 4 km/h, Downstream speed = 8 km/h

Check 1: \(12/4 + 40/8 = 3 + 5 = 8\) ✔

Check 2: \(16/4 + 32/8 = 4 + 4 = 8\) ✔