Finding Speed of Boat and Stream

Video Explanation

Question

A motor boat can travel 30 km upstream and 28 km downstream in 7 hours. It can travel 21 km upstream and return (downstream) in 5 hours. 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

\[ \frac{30}{x – y} + \frac{28}{x + y} = 7 \quad (1) \]

\[ \frac{21}{x – y} + \frac{21}{x + y} = 5 \quad (2) \]

Step 4: Convert into Linear Form

Let:

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

Then:

\[ 30a + 28b = 7 \quad (3) \]

\[ 21a + 21b = 5 \quad (4) \]

Step 5: Solve Linear Equations

Divide (4) by 21:

\[ a + b = \frac{5}{21} \quad (5) \]

Multiply (5) by 21:

\[ 21a + 21b = 5 \]

Now solve with (3): Multiply (5) by 30:

\[ 30a + 30b = \frac{150}{21} \quad (6) \]

Subtract (3) from (6):

\[ 2b = \frac{150}{21} – 7 \]

\[ 2b = \frac{150 – 147}{21} = \frac{3}{21} = \frac{1}{7} \]

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

Substitute into (5):

\[ a + \frac{1}{14} = \frac{5}{21} \]

\[ a = \frac{5}{21} – \frac{1}{14} \]

\[ a = \frac{10 – 3}{42} = \frac{7}{42} = \frac{1}{6} \]

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 = 6 km/h, Downstream speed = 14 km/h

Check 1: \(30/6 + 28/14 = 5 + 2 = 7\) ✔

Check 2: \(21/6 + 21/14 = 3.5 + 1.5 = 5\) ✔