Finding Speed of Boat and Stream

Video Explanation

Question

A boat goes 24 km upstream and 28 km downstream in 6 hours. It goes 30 km upstream and 21 km downstream in \(6\frac{1}{2}\) hours. Find the speed of the boat in still water and the speed of the stream.

Solution

Step 1: Concept Used

Time = Distance / Speed

Step 2: Let the 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 the Equations

\[ \frac{24}{x – y} + \frac{28}{x + y} = 6 \quad (1) \]

\[ \frac{30}{x – y} + \frac{21}{x + y} = 6.5 \quad (2) \]

Step 4: Substitute

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

\[ 24a + 28b = 6 \quad (3) \]

\[ 30a + 21b = 6.5 \quad (4) \]

Step 5: Solve Linear Equations

Multiply (3) by 3:

\[ 72a + 84b = 18 \quad (5) \]

Multiply (4) by 4:

\[ 120a + 84b = 26 \quad (6) \]

Subtract (5) from (6):

\[ 48a = 8 \]

\[ a = \frac{1}{6} \]

Substitute into (3):

\[ 24\left(\frac{1}{6}\right) + 28b = 6 \]

\[ 4 + 28b = 6 \]

\[ 28b = 2 \]

\[ b = \frac{1}{14} \]

Step 6: Back Substitute

\[ x – y = \frac{1}{a} = 6 \]

\[ x + y = \frac{1}{b} = 14 \]

Step 7: Solve Final Equations

\[ x – y = 6, \quad x + y = 14 \]

Add:

\[ 2x = 20 \Rightarrow x = 10 \]

Substitute:

\[ 10 + y = 14 \Rightarrow y = 4 \]

Conclusion

\[ \text{Boat speed} = 10 \text{ km/h}, \quad \text{Stream speed} = 4 \text{ km/h} \]

Verification

Upstream speed = \(10 – 4 = 6\) km/h

Downstream speed = \(10 + 4 = 14\) km/h

Check 1: \[ \frac{24}{6} + \frac{28}{14} = 4 + 2 = 6 \quad \checkmark \]

Check 2: \[ \frac{30}{6} + \frac{21}{14} = 5 + 1.5 = 6.5 \quad \checkmark \]