Ratio of speed of boat in still water to the speed of stream are respectively 8.1 A boat covers 36 km distance in downstream and 42 km distance in upstream takes total 5 hours. Then find in how much time it covers 90 km distance in downstream ?

Options

A. 3 hr
B. 5 hr
C. 4 hr
D. None of these

Answer

To solve the problem step by step, we will follow the given information and apply the concepts of speed, distance, and time.

### Step 1: Define Variables
Let:
- Speed of the boat in still water = 8x (where x is a common multiplier)
- Speed of the stream = x

### Step 2: Calculate Downstream and Upstream Speeds
- Downstream speed (with the flow of water) = Speed of boat + Speed of stream = 8x + x = 9x
- Upstream speed (against the flow of water) = Speed of boat - Speed of stream = 8x - x = 7x

### Step 3: Set Up the Equation for Total Time
The boat covers:
- 36 km downstream
- 42 km upstream

The total time taken for both journeys is 5 hours. We can express this as:
\[ \text{Time downstream} + \text{Time upstream} = 5 \text{ hours} \]
This can be expressed mathematically as:
\[ \frac{36}{9x} + \frac{42}{7x} = 5 \]

### Step 4: Simplify the Equation
To simplify, we will find a common denominator:
\[ \frac{36}{9x} = \frac{4}{x} \quad \text{and} \quad \frac{42}{7x} = \frac{6}{x} \]
So the equation becomes:
\[ \frac{4}{x} + \frac{6}{x} = 5 \]
Combining the fractions gives:
\[ \frac{10}{x} = 5 \]

### Step 5: Solve for x
To find x, we can cross-multiply:
\[ 10 = 5x \implies x = 2 \]

### Step 6: Calculate the Speeds
Now that we have x, we can calculate the speeds:
- Speed of the boat in still water = 8x = 8 * 2 = 16 km/h
- Speed of the stream = x = 2 km/h
- Downstream speed = 9x = 9 * 2 = 18 km/h
- Upstream speed = 7x = 7 * 2 = 14 km/h

### Step 7: Find Time to Cover 90 km Downstream
Now we need to find the time taken to cover 90 km downstream at the speed of 18 km/h:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{90}{18} = 5 \text{ hours} \]

### Final Answer
The time taken to cover 90 km downstream is **5 hours**.

---