There are two solutions of Sulphuric acid (acid + water) with concentration of 50% and 80% respectively. They are mixed in a certain ratio to get a 62% sulphuric acid solution. This solution is mixed with 6 liters of water to get back 50% solution. How much of the 80% solution has been used in the entire process?

Options

A. 15 liters
B. 12 liters
C. 10 litres
D. None of these

Answer

To solve the problem, we need to determine how much of the 80% sulphuric acid solution is used in the entire process of mixing to achieve a 62% solution and then diluting it to a 50% solution.

### Step 1: Set up the equations for mixing the two solutions.

Let:
- \( x \) = the amount of 50% solution used (in liters)
- \( y \) = the amount of 80% solution used (in liters)

The resulting mixture has a concentration of 62%. Therefore, we can set up the equation based on the concentration of sulphuric acid:

\[ 0.5x + 0.8y = 0.62(x + y) \]

### Step 2: Simplify the equation.

Expanding the right side gives:

\[ 0.5x + 0.8y = 0.62x + 0.62y \]

Rearranging the terms leads to:

\[ 0.5x - 0.62x + 0.8y - 0.62y = 0 \]

This simplifies to:

\[ -0.12x + 0.18y = 0 \]

### Step 3: Rearranging the equation.

We can express \( y \) in terms of \( x \):

\[ 0.18y = 0.12x \implies y = \frac{0.12}{0.18}x = \frac{2}{3}x \]

### Step 4: Determine the total volume of the mixture before dilution.

The total volume of the mixture before adding water is:

\[ x + y = x + \frac{2}{3}x = \frac{5}{3}x \]

### Step 5: Add water and set up the equation for the final concentration.

After mixing, 6 liters of water is added to the mixture, and we want the final concentration to be 50%. The total volume after adding water becomes:

\[ \frac{5}{3}x + 6 \]

The total amount of sulphuric acid in the mixture is:

\[ 0.62 \left( \frac{5}{3}x \right) \]

Setting up the equation for the final concentration:

\[ \frac{0.62 \left( \frac{5}{3}x \right)}{\frac{5}{3}x + 6} = 0.5 \]

### Step 6: Cross-multiply and simplify.

Cross-multiplying gives:

\[ 0.62 \left( \frac{5}{3}x \right) = 0.5 \left( \frac{5}{3}x + 6 \right) \]

Expanding both sides:

\[ 0.62 \cdot \frac{5}{3}x = \frac{5}{6}x + 3 \]

### Step 7: Solve for \( x \).

Multiplying through by 6 to eliminate the fraction:

\[ 6 \cdot 0.62 \cdot \frac{5}{3}x = 5x + 18 \]

Calculating \( 6 \cdot 0.62 \cdot \frac{5}{3} \):

\[ 12.4x = 5x + 18 \]

Rearranging gives:

\[ 12.4x - 5x = 18 \implies 7.4x = 18 \implies x = \frac{18}{7.4} \approx 2.43 \text{ liters} \]

### Step 8: Calculate \( y \).

Using the relation \( y = \frac{2}{3}x \):

\[ y = \frac{2}{3} \cdot 2.43 \approx 1.62 \text{ liters} \]

### Conclusion:

The amount of the 80% solution used is approximately **1.62 liters**.

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