In a mixture, the ratio between milk and water is 8: 1. If 18 litres of mixtures is removed and 4 lires of water is added, then the quantity of milk is 24 litres more than that of water. Find the initial quantity of milk in the mixture.

We are given the following information about a mixture of milk and water:

- The initial ratio of milk to water is \( 8:1 \).
- 18 litres of the mixture is removed, and 4 litres of water is added.
- After this, the quantity of milk is 24 litres more than the quantity of water.

Let's denote:
- \( x \) as the initial total amount of the mixture in litres.

### Step 1: Express initial quantities of milk and water
From the ratio \( 8:1 \), the total quantity of milk is \( \frac{8}{9}x \) and the total quantity of water is \( \frac{1}{9}x \).

### Step 2: Removal of 18 litres of the mixture
When 18 litres of the mixture is removed, both milk and water are removed in the same ratio \( 8:1 \). Therefore:
- The quantity of milk removed is \( \frac{8}{9} \times 18 = 16 \) litres.
- The quantity of water removed is \( \frac{1}{9} \times 18 = 2 \) litres.

### Step 3: New quantities after removal
After removing 18 litres, the new quantities of milk and water are:
- Milk: \( \frac{8}{9}x - 16 \).
- Water: \( \frac{1}{9}x - 2 \).

### Step 4: Adding 4 litres of water
After adding 4 litres of water, the new quantity of water becomes:
- Water: \( \frac{1}{9}x - 2 + 4 = \frac{1}{9}x + 2 \).

### Step 5: Condition for milk being 24 litres more than water
We are told that after the changes, the quantity of milk is 24 litres more than the quantity of water:
\[
\frac{8}{9}x - 16 = \left( \frac{1}{9}x + 2 \right) + 24
\]

### Step 6: Solve the equation
Simplify the equation:
\[
\frac{8}{9}x - 16 = \frac{1}{9}x + 26
\]
Subtract \( \frac{1}{9}x \) from both sides:
\[
\frac{7}{9}x - 16 = 26
\]
Add 16 to both sides:
\[
\frac{7}{9}x = 42
\]
Multiply both sides by 9:
\[
7x = 378
\]
Divide by 7:
\[
x = 54
\]

### Step 7: Calculate the initial quantity of milk
Now, we can calculate the initial quantity of milk:
\[
\text{Milk} = \frac{8}{9} \times 54 = 48 \, \text{litres}.
\]

### Final Answer:
The initial quantity of milk in the mixture was **48 litres**.

---

Would you like a more detailed explanation of any part? Here are a few related questions for further practice:

1. What would the new quantity of milk be if instead of adding water, 4 litres of milk were added to the mixture?
2. How would the final amounts of milk and water change if 10 litres of the mixture were removed instead of 18?
3. If the initial ratio was 5:3, how would the quantities change after the same removal and addition process?
4. How would the equation change if the final condition was that the milk quantity is 10 litres more than the water quantity?
5. What would happen to the final quantities if the same amount of milk and water were removed (e.g., 10 litres instead of 18)?

**Tip:** Always double-check the relationships between different quantities when dealing with mixtures and ratios.

In a mixture, the ratio between milk and water is 8: 1. If 18 litres of mixtures is removed and 4 litres of water is added, then the quantity of milk is 24 litres more than that of water. Find the initial quantity of milk in the mixture.

Learn how to solve mixture problems involving ratios, with a detailed solution to the question where the ratio of milk to water is 8:1. This problem involves removing part of the mixture and adding water, leading to a scenario where the quantity of milk is 24 litres more than water. A step-by-step breakdown suitable for Grades 8-10.

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