A container contains a mixture of two liquid A and B in the ration of `7 : 5`. When 9L of mixture is drawn off and the container is filled with B, the ratio of A and B becomes `7 : 9`. Howe many litres of liquid A was contained by the container initially?

Options

A. 10
B. 20
C. 21
D. 25

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Understand the initial ratio of liquids A and B
The initial ratio of liquids A and B in the container 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 initial volume of the mixture
Let the total initial volume of the mixture in the container be \( x \) liters.

### Step 3: Calculate the volumes of A and B
From the ratio, 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 4: Draw off 9 liters of the mixture
When 9 liters of the mixture is drawn off, the ratio of A to B in the drawn-off mixture remains the same (7:5). Therefore, the volumes of A and B in the 9 liters drawn off can be calculated as follows:
- Volume of A drawn off = \( \frac{7}{12} \times 9 = \frac{63}{12} = 5.25 \) liters
- Volume of B drawn off = \( \frac{5}{12} \times 9 = \frac{45}{12} = 3.75 \) liters

### Step 5: Calculate the remaining volumes of A and B after drawing off
After drawing off 9 liters, the remaining volumes of A and B in the container are:
- Remaining volume of A = \( \frac{7}{12}x - 5.25 \)
- Remaining volume of B = \( \frac{5}{12}x - 3.75 \)

### Step 6: Fill the container with 9 liters of liquid B
After the 9 liters of liquid B is added back to the container, 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 7: Set up the new ratio of A to B
After adding the 9 liters of B, the new ratio of A to B becomes 7:9. Therefore, we can set up the equation:
\[ \frac{\frac{7}{12}x - 5.25}{\frac{5}{12}x + 5.25} = \frac{7}{9} \]

### Step 8: 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 9: Rearranging the equation
Rearranging gives:
\[ \frac{63}{12}x - \frac{35}{12}x = 36.75 + 47.25 \]
\[ \frac{28}{12}x = 84 \]
\[ x = 84 \times \frac{12}{28} = 36 \text{ liters} \]

### Step 10: Calculate the initial volume of liquid A
Now, we can find the initial volume of liquid A:
\[ \text{Volume of A} = \frac{7}{12} \times 36 = 21 \text{ liters} \]

Thus, the initial volume of liquid A contained in the container was **21 liters**.

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