A boat going downstream takes 5 hrs to cover a certain distance, while it takes 3 hrs to cover `3/7` of the same distance running upstream. What is the ratio of the speed of the boat to the speed of the stream?

Options

A. `6:1`
B. `1:6`
C. `1:5`
D. `5:1`

Answer

To solve the problem, we need to find the ratio of the speed of the boat (x) to the speed of the stream (y). Here’s a step-by-step breakdown:

### Step 1: Define Variables
Let:
- Speed of the boat in still water = x km/hr
- Speed of the stream = y km/hr

### Step 2: Distance Calculation Downstream
When the boat is going downstream, it takes 5 hours to cover a certain distance (d). The effective speed downstream is (x + y). Therefore, we can express the distance as:
\[ d = (x + y) \times 5 \]
This gives us our first equation:
\[ d = 5(x + y) \] (Equation 1)

### Step 3: Distance Calculation Upstream
When the boat is going upstream, it takes 3 hours to cover \( \frac{3}{7} \) of the same distance (d). The effective speed upstream is (x - y). Therefore, we can express the distance as:
\[ \frac{3}{7}d = (x - y) \times 3 \]
This gives us our second equation:
\[ d = \frac{7}{3}(x - y) \] (Equation 2)

### Step 4: Equate the Two Expressions for Distance
From Equation 1 and Equation 2, we can set them equal to each other:
\[ 5(x + y) = \frac{7}{3}(x - y) \]

### Step 5: Clear the Fraction
To eliminate the fraction, multiply both sides by 3:
\[ 15(x + y) = 7(x - y) \]

### Step 6: Expand Both Sides
Expanding both sides gives:
\[ 15x + 15y = 7x - 7y \]

### Step 7: Rearranging the Equation
Now, rearranging the equation to isolate terms involving x and y:
\[ 15x - 7x = -7y - 15y \]
This simplifies to:
\[ 8x = -22y \]

### Step 8: Solve for the Ratio
Dividing both sides by y gives:
\[ \frac{x}{y} = \frac{-22}{8} \]
This simplifies to:
\[ \frac{x}{y} = \frac{11}{4} \]

### Step 9: Final Ratio
The ratio of the speed of the boat to the speed of the stream is:
\[ x:y = 11:4 \]