A steamer takes one hour more to go 48 km upstream than the time to go 60 km downstream. If the steamer takes equal time to cover 30 km downstream and 18 km upstream then find the ratio of the speed of the boat in still water and the speed of the stream.

Answer

To solve the problem step by step, we will define the variables and use the information given in the question.

### Step 1: Define Variables
Let:
- \( x \) = speed of the boat in still water (in km/h)
- \( y \) = speed of the stream (in km/h)

### Step 2: Write Down the Speeds
- Downstream speed = \( x + y \)
- Upstream speed = \( x - y \)

### Step 3: Set Up the First Condition
According to the problem, the steamer takes one hour more to go 48 km upstream than to go 60 km downstream. We can express this as:

\[ \text{Time upstream} = \frac{48}{x - y} \]
\[ \text{Time downstream} = \frac{60}{x + y} \]

According to the problem statement:
\[ \frac{48}{x - y} = \frac{60}{x + y} + 1 \]

### Step 4: Simplify the Equation
Rearranging the equation gives us:
\[ \frac{48}{x - y} - \frac{60}{x + y} = 1 \]

To combine the fractions, we find a common denominator:
\[ \frac{48(x + y) - 60(x - y)}{(x - y)(x + y)} = 1 \]

Expanding the numerator:
\[ 48x + 48y - 60x + 60y = 12y - 12x \]
So we have:
\[ \frac{-12x + 108y}{(x - y)(x + y)} = 1 \]

Cross-multiplying gives:
\[ -12x + 108y = (x - y)(x + y) \]

### Step 5: Set Up the Second Condition
It is given that the steamer takes equal time to cover 30 km downstream and 18 km upstream:
\[ \frac{30}{x + y} = \frac{18}{x - y} \]

Cross-multiplying gives:
\[ 30(x - y) = 18(x + y) \]

Expanding this:
\[ 30x - 30y = 18x + 18y \]

Rearranging gives:
\[ 12x = 48y \implies x = 4y \]

### Step 6: Substitute Back to Find the Ratio
Now, substituting \( x = 4y \) into the first equation:
\[ -12(4y) + 108y = (4y - y)(4y + y) \]
This simplifies to:
\[ -48y + 108y = 3y(5y) \]
\[ 60y = 15y^2 \]
Dividing both sides by \( y \) (assuming \( y \neq 0 \)):
\[ 15 = y \]

### Step 7: Find \( x \)
Substituting \( y = 15 \) back into \( x = 4y \):
\[ x = 4(15) = 60 \]

### Step 8: Calculate the Ratio
The ratio of the speed of the boat in still water to the speed of the stream is:
\[ \frac{x}{y} = \frac{60}{15} = 4 \]

Thus, the final answer is:
\[ \text{Ratio of the speed of the boat in still water to the speed of the stream} = 4:1 \]