The boat goes 25 km upstream and 33 km downstream in 8 hours. It can also go 40 km upstream and 77 km downstream in 15 hours. Find the speed of the stream and that of boat in still water.

Answer

To solve the problem, we need to find the speed of the boat in still water (U) and the speed of the stream (V). We will use the information given about the distances traveled upstream and downstream along with the total time taken for each journey.

### Step 1: Set up the equations

We know that:
- Speed upstream = U - V
- Speed downstream = U + V

From the problem, we have two scenarios:

1. The boat goes 25 km upstream and 33 km downstream in 8 hours.
2. The boat goes 40 km upstream and 77 km downstream in 15 hours.

Using the formula for time (Time = Distance / Speed), we can set up the following equations:

For the first scenario:
\[ \frac{25}{U - V} + \frac{33}{U + V} = 8 \quad \text{(Equation 1)} \]

For the second scenario:
\[ \frac{40}{U - V} + \frac{77}{U + V} = 15 \quad \text{(Equation 2)} \]

### Step 2: Simplify the equations

Let \( X = \frac{1}{U - V} \) and \( Y = \frac{1}{U + V} \). Then we can rewrite the equations:

1. \( 25X + 33Y = 8 \) (Equation 3)
2. \( 40X + 77Y = 15 \) (Equation 4)

### Step 3: Solve the equations

Now we will solve Equations 3 and 4 simultaneously.

First, we can multiply Equation 3 by 5 and Equation 4 by 8 to eliminate \( X \):

\[ 125X + 165Y = 40 \quad \text{(Equation 5)} \]
\[ 320X + 616Y = 120 \quad \text{(Equation 6)} \]

Now, we will subtract Equation 5 from Equation 6:

\[ (320X + 616Y) - (125X + 165Y) = 120 - 40 \]
\[ 195X + 451Y = 80 \]

### Step 4: Isolate one variable

Now we can isolate \( Y \):

From Equation 5:
\[ 125X + 165Y = 40 \implies 165Y = 40 - 125X \implies Y = \frac{40 - 125X}{165} \]

Substituting \( Y \) in the equation \( 195X + 451Y = 80 \):

\[ 195X + 451\left(\frac{40 - 125X}{165}\right) = 80 \]

### Step 5: Solve for \( X \)

Multiply through by 165 to eliminate the fraction:

\[ 195 \times 165X + 451(40 - 125X) = 80 \times 165 \]
\[ 32175X + 18040 - 56375X = 13200 \]
\[ -24100X = 13200 - 18040 \]
\[ -24100X = -4830 \implies X = \frac{4830}{24100} = \frac{1}{5} \]

### Step 6: Solve for \( Y \)

Now substitute \( X \) back into Equation 3 to find \( Y \):

\[ 25\left(\frac{1}{5}\right) + 33Y = 8 \]
\[ 5 + 33Y = 8 \implies 33Y = 3 \implies Y = \frac{3}{33} = \frac{1}{11} \]

### Step 7: Find \( U \) and \( V \)

Recall:
\[ X = \frac{1}{U - V} \implies U - V = 5 \]
\[ Y = \frac{1}{U + V} \implies U + V = 11 \]

Now we can solve these two equations:

1. \( U - V = 5 \) (Equation 7)
2. \( U + V = 11 \) (Equation 8)

Adding Equation 7 and Equation 8:

\[ (U - V) + (U + V) = 5 + 11 \]
\[ 2U = 16 \implies U = 8 \]

Substituting \( U \) back into Equation 8:

\[ 8 + V = 11 \implies V = 3 \]

### Final Answer

- Speed of the boat in still water (U) = 8 km/h
- Speed of the stream (V) = 3 km/h