A bucket contains a mixture of two liquids A and B in the proportion 7: 5. If 9 litres of the mixture is replaced by 9 litres of liquid B, then the ratio of the two liquid becomes 7: 9. How much of the liquid A was there in the bucket ?

Options

A. a. 21 litres
B. b. 15 litres
C. c. 18 litres
D. d. 23 litres

Explanation

To solve the problem step by step, we will follow the given information and derive the required quantities.

### Step 1: Define the initial quantities of liquids A and B
Let the initial quantities of liquids A and B be represented as:
- A = 7x (where x is a common multiplier)
- B = 5x

### Step 2: Calculate the total volume of the mixture
The total volume of the mixture is:
\[ \text{Total Volume} = A + B = 7x + 5x = 12x \]

### Step 3: Determine the amount of liquid A and B in the 9 liters removed
When 9 liters of the mixture is removed, the ratio of A to B in the removed mixture is still 7:5. Therefore, the quantities of A and B removed can be calculated as follows:
- Amount of A removed = \( \frac{7}{12} \times 9 = \frac{63}{12} = 5.25 \) liters
- Amount of B removed = \( \frac{5}{12} \times 9 = \frac{45}{12} = 3.75 \) liters

### Step 4: Calculate the remaining quantities of A and B after removal
After removing 9 liters of the mixture:
- Remaining A = \( 7x - 5.25 \)
- Remaining B = \( 5x - 3.75 \)

### Step 5: Add 9 liters of liquid B to the mixture
After adding 9 liters of liquid B, the new quantity of B becomes:
\[ \text{New B} = (5x - 3.75) + 9 = 5x + 5.25 \]

### Step 6: Set up the new ratio of A to B
According to the problem, after the replacement, the new ratio of A to B becomes 7:9. Therefore, we can set up the equation:
\[ \frac{7x - 5.25}{5x + 5.25} = \frac{7}{9} \]

### Step 7: Cross-multiply to solve for x
Cross-multiplying gives:
\[ 9(7x - 5.25) = 7(5x + 5.25) \]
Expanding both sides:
\[ 63x - 47.25 = 35x + 36.75 \]

### Step 8: Rearranging the equation
Rearranging gives:
\[ 63x - 35x = 36.75 + 47.25 \]
\[ 28x = 84 \]
\[ x = 3 \]

### Step 9: Calculate the initial quantity of liquid A
Now that we have the value of x, we can find the initial quantity of liquid A:
\[ A = 7x = 7 \times 3 = 21 \text{ liters} \]

### Conclusion
The initial quantity of liquid A in the bucket was **21 liters**.

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