A container contains two liquids A and B in the ratio 8:5.When 13 liters of mixture is drawn off and is completely replaced with liquid B, then the ratio of A and B in the container becomes 1: 1. How many liter of liquid A was in the container initially?

Options

A. A)`128//3` liter
B. B)117 liter
C. C)`134//3` liter
D. D)`121//3` liter

Answer

To solve the problem step by step, we will break down the information given and apply the concepts of ratios and mixtures.

### 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 8:5. This means that for every 8 parts of liquid A, there are 5 parts of liquid B.

### Step 2: Define the total parts of the mixture
Let the total parts of the mixture be \(8 + 5 = 13\) parts. This means the total volume of the mixture can be expressed in terms of these parts.

### Step 3: Draw off 13 liters of the mixture
When 13 liters of the mixture is drawn off, we need to determine how much of liquid A and liquid B is removed. Since the ratio is maintained, we can calculate the amounts of A and B removed:
- Amount of liquid A drawn off = \(\frac{8}{13} \times 13 = 8\) liters
- Amount of liquid B drawn off = \(\frac{5}{13} \times 13 = 5\) liters

### Step 4: Determine the remaining amounts of A and B after drawing off
Let the initial amounts of A and B in the container be \(8x\) and \(5x\) respectively (where \(x\) is a common multiplier representing the total volume of the mixture).
After drawing off 13 liters:
- Remaining amount of liquid A = \(8x - 8\) liters
- Remaining amount of liquid B = \(5x - 5\) liters

### Step 5: Add 13 liters of liquid B to the mixture
After removing the 13 liters, we replace it with 13 liters of liquid B. Therefore, the new amount of liquid B becomes:
- New amount of liquid B = \((5x - 5) + 13 = 5x + 8\) liters

### Step 6: Set up the equation for the final ratio
According to the problem, after the replacement, the ratio of A to B becomes 1:1. Therefore, we can set up the equation:
\[ 8x - 8 = 5x + 8 \]

### Step 7: Solve the equation
Now, we will solve for \(x\):
\[ 8x - 5x = 8 + 8 \]
\[ 3x = 16 \]
\[ x = \frac{16}{3} \]

### Step 8: Calculate the initial amount of liquid A
Now that we have \(x\), we can find the initial amount of liquid A:
\[ \text{Initial amount of liquid A} = 8x = 8 \times \frac{16}{3} = \frac{128}{3} \text{ liters} \]

### Conclusion
Thus, the initial amount of liquid A in the container was \(\frac{128}{3}\) liters.

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