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

Options

A. 21 liters
B. 23 liters
C. 25 liters
D. 27 liters

Answer

To solve the problem step by step, let's break down the information given and the operations we need to perform.

### Step 1: Understand the initial ratio
The initial ratio of liquids A and B is given as \( 7:5 \). This means that for every 12 parts of the mixture, 7 parts are liquid A and 5 parts are liquid B.

### Step 2: Define the total volume of the mixture
Let the total volume of the mixture be \( x \) liters. Then the volumes of A and B can be expressed as:
- Volume of A = \( \frac{7}{12}x \)
- Volume of B = \( \frac{5}{12}x \)

### Step 3: Calculate the volumes after removing 9 liters of the mixture
When we remove 9 liters of the mixture, we are removing both liquids A and B in the same ratio \( 7:5 \). The volume of A removed can be calculated as:
- Volume of A removed = \( \frac{7}{12} \times 9 = \frac{63}{12} = 5.25 \) liters
- Volume of B removed = \( \frac{5}{12} \times 9 = \frac{45}{12} = 3.75 \) liters

### Step 4: Update the volumes of A and B
After removing 9 liters of the mixture, the new volumes of A and B will be:
- New volume of A = \( \frac{7}{12}x - 5.25 \)
- New volume of B = \( \frac{5}{12}x - 3.75 \)

### Step 5: Add 9 liters of liquid B
Now, we add 9 liters of liquid B to the mixture. Therefore, the new volume of B becomes:
- New volume of B = \( \left(\frac{5}{12}x - 3.75\right) + 9 = \frac{5}{12}x + 5.25 \)

### Step 6: Set up the new ratio
According to the problem, after adding 9 liters of B, the new ratio of A to B becomes \( 7:9 \). Thus, we can set up the equation:
\[ \frac{\frac{7}{12}x - 5.25}{\frac{5}{12}x + 5.25} = \frac{7}{9} \]

### Step 7: Cross-multiply to solve for \( x \)
Cross-multiplying gives:
\[ 9\left(\frac{7}{12}x - 5.25\right) = 7\left(\frac{5}{12}x + 5.25\right) \]
Expanding both sides:
\[ \frac{63}{12}x - 47.25 = \frac{35}{12}x + 36.75 \]

### Step 8: Combine like terms
Rearranging gives:
\[ \frac{63}{12}x - \frac{35}{12}x = 47.25 + 36.75 \]
\[ \frac{28}{12}x = 84 \]
\[ x = \frac{84 \times 12}{28} = 36 \text{ liters} \]

### Step 9: Calculate the volume of liquid A
Now that we have the total volume of the mixture, we can find the volume of liquid A:
\[ \text{Volume of A} = \frac{7}{12} \times 36 = 21 \text{ liters} \]

### Final Answer
The amount of liquid A in the bucket is **21 liters**.

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