Jar A contains 1 litre juice while jar B contains 1 litre water. First, 200 ml of juice is transferred from A to B. Next, 200 ml of mixture is transferred from B to A. If the final ratio of juice to water in jars Aand B be m and n respectively, then what is the value of m x n?

Options

A. 1
B. `1/5`
C. `1/25`
D. `2/5`

Answer

To solve the problem step by step, let's break it down:

### Step 1: Initial Setup
- Jar A contains 1 liter (1000 ml) of juice.
- Jar B contains 1 liter (1000 ml) of water.

### Step 2: Transfer Juice from A to B
- Transfer 200 ml of juice from Jar A to Jar B.
- After this transfer:
- Jar A now has \(1000 - 200 = 800\) ml of juice.
- Jar B now has \(1000\) ml of water + \(200\) ml of juice = \(1000 + 200 = 1200\) ml of mixture (which consists of 1000 ml of water and 200 ml of juice).

### Step 3: Calculate the Ratio in Jar B
- The mixture in Jar B consists of:
- Juice: 200 ml
- Water: 1000 ml
- The total mixture in Jar B is \(1200\) ml.
- The ratio of juice to water in Jar B is:
- Juice proportion = \( \frac{200}{1200} = \frac{1}{6} \)
- Water proportion = \( \frac{1000}{1200} = \frac{5}{6} \)

### Step 4: Transfer Mixture from B to A
- Now, we transfer 200 ml of this mixture back to Jar A.
- The amount of juice transferred from B to A:
- Juice in 200 ml of mixture = \(200 \times \frac{1}{6} = \frac{200}{6} = \frac{100}{3}\) ml
- The amount of water transferred from B to A:
- Water in 200 ml of mixture = \(200 \times \frac{5}{6} = \frac{1000}{6} = \frac{500}{3}\) ml

### Step 5: Update Contents of Jar A
- After transferring back to Jar A:
- Juice in Jar A = \(800 + \frac{100}{3}\) ml
- Water in Jar A = \(0 + \frac{500}{3}\) ml
- Thus:
- Juice in Jar A = \(800 + \frac{100}{3} = \frac{2400}{3} + \frac{100}{3} = \frac{2500}{3}\) ml
- Water in Jar A = \(\frac{500}{3}\) ml

### Step 6: Update Contents of Jar B
- After transferring to Jar A, Jar B will have:
- Juice = \(200 - \frac{100}{3} = \frac{600}{3} - \frac{100}{3} = \frac{500}{3}\) ml
- Water = \(1000 - \frac{500}{3} = \frac{3000}{3} - \frac{500}{3} = \frac{2500}{3}\) ml

### Step 7: Calculate Ratios
- **In Jar A**:
- Juice = \(\frac{2500}{3}\) ml
- Water = \(\frac{500}{3}\) ml
- Ratio of Juice to Water in A = \(\frac{\frac{2500}{3}}{\frac{500}{3}} = \frac{2500}{500} = 5\)
- Let this ratio be \(m = 5\).

- **In Jar B**:
- Juice = \(\frac{500}{3}\) ml
- Water = \(\frac{2500}{3}\) ml
- Ratio of Juice to Water in B = \(\frac{\frac{500}{3}}{\frac{2500}{3}} = \frac{500}{2500} = \frac{1}{5}\)
- Let this ratio be \(n = \frac{1}{5}\).

### Step 8: Calculate \(m \times n\)
- Now, we need to calculate \(m \times n\):
- \(m \times n = 5 \times \frac{1}{5} = 1\).

### Final Answer
The value of \(m \times n\) is **1**.

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