A can contains a mixture of two liquids A and B in the ratio 7:5. When 9 litres of mixture are drawn off and the can is filled with B, the ratio of A and B becomes 7: 9. Litres of liquid A contained by the can initially was

Options

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

Answer

To solve the problem step by step, we will follow the information given and use algebra to find the initial quantity of liquid A in the can.

### Step 1: Define the initial quantities of liquids A and B
Let the initial quantities of liquids A and B in the can be represented as:
- Quantity of A = 7x liters
- Quantity of B = 5x liters

Here, x is a common multiplier that represents the parts of the ratio.

### Step 2: Calculate the total initial quantity of the mixture
The total initial quantity of the mixture (A + B) is:
\[ \text{Total mixture} = 7x + 5x = 12x \text{ liters} \]

### Step 3: Determine the quantity of mixture drawn off
When 9 liters of the mixture are drawn off, the quantities of A and B removed can be calculated based on the ratio:
- Quantity of A removed = \( \frac{7}{12} \times 9 = \frac{63}{12} = 5.25 \) liters
- Quantity 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 drawing off 9 liters of the mixture, the remaining quantities of A and B will be:
- Remaining A = \( 7x - 5.25 \)
- Remaining B = \( 5x - 3.75 \)

### Step 5: Fill the can with liquid B
After removing the mixture, the can is filled with 9 liters of liquid B. Therefore, 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, the new ratio of A to B becomes 7:9. Thus, 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 us:
\[ 9(7x - 5.25) = 7(5x + 5.25) \]
Expanding both sides:
\[ 63x - 47.25 = 35x + 36.75 \]

### Step 8: Rearranging the equation
Rearranging the equation to isolate x:
\[ 63x - 35x = 36.75 + 47.25 \]
\[ 28x = 84 \]
\[ x = \frac{84}{28} = 3 \]

### 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:
\[ \text{Quantity of A} = 7x = 7 \times 3 = 21 \text{ liters} \]

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

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