Finding Speed of Boat and Stream

Video Explanation

Question

A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km downstream. Find the speed of the boat in still water and the speed of the stream.

Solution

Step 1: Let the Variables

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

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

Step 2: Form the Equations

Upstream speed = \(x – y\) Downstream speed = \(x + y\) — First condition:

\[ \frac{30}{x – y} + \frac{44}{x + y} = 10 \quad (1) \]

Second condition:

\[ \frac{40}{x – y} + \frac{55}{x + y} = 13 \quad (2) \]

Step 3: Substitute

Let:

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

Then equations become:

\[ 30a + 44b = 10 \quad (3) \]

\[ 40a + 55b = 13 \quad (4) \]

Step 4: Solve (3) and (4)

Multiply (3) by 4:

\[ 120a + 176b = 40 \quad (5) \]

Multiply (4) by 3:

\[ 120a + 165b = 39 \quad (6) \]

Subtract (6) from (5):

\[ 11b = 1 \]

\[ b = \frac{1}{11} \]

Substitute into (3):

\[ 30a + 44\left(\frac{1}{11}\right) = 10 \]

\[ 30a + 4 = 10 \]

\[ 30a = 6 \]

\[ a = \frac{1}{5} \]

Step 5: Back Substitute

\[ x – y = \frac{1}{a} = 5 \]

\[ x + y = \frac{1}{b} = 11 \]

Step 6: Solve Final Equations

Add:

\[ 2x = 16 \Rightarrow x = 8 \]

Substitute:

\[ 8 + y = 11 \Rightarrow y = 3 \]

Conclusion

Speed of boat in still water:

\[ \boxed{8 \text{ km/h}} \]

Speed of stream:

\[ \boxed{3 \text{ km/h}} \]