A can contains a mixture of the liquids A and B in the ratio 7:5. When 9 liters of mixture is drawn off and the can is filled with B, the ratio of A and B becomes 7:9. How many liters of liquid A were contained by the can initially?

Options

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

Answer

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

### Step 1: Understand the initial ratio
We are given that the initial ratio of liquids A and B in the can is 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 in the can be \( V \) liters. Then, the quantities of A and B can be expressed as:
- Quantity of A = \( \frac{7}{12} V \)
- Quantity of B = \( \frac{5}{12} V \)

### Step 3: Calculate the quantities of A and B drawn off
When 9 liters of the mixture is drawn off, the amounts of A and B removed can be calculated based on the initial ratio:
- Amount of A drawn off = \( 9 \times \frac{7}{12} = \frac{63}{12} = 5.25 \) liters
- Amount of B drawn off = \( 9 \times \frac{5}{12} = \frac{45}{12} = 3.75 \) liters

### Step 4: Calculate the remaining quantities of A and B
After removing 9 liters of the mixture, the remaining quantities in the can will be:
- Remaining A = \( \frac{7}{12} V - 5.25 \)
- Remaining B = \( \frac{5}{12} V - 3.75 \)

### Step 5: Add 9 liters of liquid B
After drawing off the mixture, the can is filled with 9 liters of liquid B. Therefore, the new quantity of B becomes:
- New quantity of B = \( \left(\frac{5}{12} V - 3.75\right) + 9 = \frac{5}{12} V + 5.25 \)

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

### Step 7: Cross-multiply to solve for V
Cross-multiplying gives:
\[ 9\left(\frac{7}{12} V - 5.25\right) = 7\left(\frac{5}{12} V + 5.25\right) \]

Expanding both sides:
\[ \frac{63}{12} V - 47.25 = \frac{35}{12} V + 36.75 \]

### Step 8: Rearranging the equation
Rearranging gives:
\[ \frac{63}{12} V - \frac{35}{12} V = 36.75 + 47.25 \]
\[ \frac{28}{12} V = 84 \]

### Step 9: Solve for V
Multiplying both sides by \( \frac{12}{28} \):
\[ V = 84 \times \frac{12}{28} = 36 \]

### Step 10: Calculate the initial quantity of A
Now that we have \( V = 36 \) liters, we can find the initial quantity of A:
\[ \text{Quantity of A} = \frac{7}{12} \times 36 = 21 \text{ liters} \]

### Final Answer
The initial quantity of liquid A contained by the can is **21 liters**.

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