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To solve this question, let's go through it step-by-step.

The question provides us with:
- The upstream speed of the boat: \(4x - 2y\) km/hr.
- The downstream speed of the boat: \(8x - 4y\) km/hr.

Let's denote:
- The speed of the boat in still water as \(B\).
- The speed of the stream as \(S\).

In general, the relationships for upstream and downstream speeds are:
- **Upstream speed** = \(B - S\)
- **Downstream speed** = \(B + S\)

Now, let's set up the equations.

### Step 1: Set Up Equations
1. For upstream speed:
   \[
   B - S = 4x - 2y
   \]

2. For downstream speed:
   \[
   B + S = 8x - 4y
   \]

### Step 2: Solve for \(B\) and \(S\)
Add the two equations to eliminate \(S\):
\[
(B + S) + (B - S) = (8x - 4y) + (4x - 2y)
\]
\[
2B = 12x - 6y
\]
\[
B = 6x - 3y
\]

Now, subtract the upstream equation from the downstream equation to solve for \(S\):
\[
(B + S) - (B - S) = (8x - 4y) - (4x - 2y)
\]
\[
2S = 4x - 2y
\]
\[
S = 2x - y
\]

### Step 3: Find the Ratio \(\frac{B}{S}\)
Now we have:
- \(B = 6x - 3y\)
- \(S = 2x - y\)

The required ratio of the speed of the boat in still water to the speed of the stream is:
\[
\frac{B}{S} = \frac{6x - 3y}{2x - y}
\]

Divide both the numerator and the denominator by \(2x - y\):
\[
\frac{B}{S} = \frac{6x - 3y}{2x - y} = 3:1
\]

### Final Answer:
The correct answer is (a) \(3:1\).

---

Do you need further explanation on any part of this solution? 

Here are five related questions for additional practice:
1. If the upstream and downstream speeds are 15 km/hr and 25 km/hr, find the speed of the boat in still water and the speed of the stream.
2. A boat travels 12 km upstream in 2 hours and the same distance downstream in 1 hour. Find the speed of the boat in still water and the speed of the stream.
3. If the speed of a boat in still water is 10 km/hr and the speed of the stream is 3 km/hr, what is the downstream speed?
4. If the speed ratio of a boat in still water to the speed of the stream is 4:1, and the downstream speed is 25 km/hr, find the upstream speed.
5. A boat takes 3 hours to travel 24 km downstream and 4 hours to return upstream. Calculate the speed of the boat in still water and the speed of the stream.

**Tip:** For boat and stream problems, remember that downstream speed is the sum of the boat's speed in still water and the stream's speed, while upstream speed is the difference. This can simplify problem-solving.

A boat travels in upstream with speed (4x - 2y) km/hr and travels in downstream with speed (8x - 4y) km/hr. Find the ratio between speed of boat in still water to speed of stream.

Learn how to solve a boat and stream problem involving upstream and downstream speeds. The problem requires finding the ratio between the speed of a boat in still water and the speed of the stream, given specific speed expressions. Suitable for Grades 9-10.

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