Two bottles A and B are filled with dilute sulphuric acid (sulphuric acid + water). In bottle A, water is `40%` of the acid and in bottle B, acid is `60% `of the water. How much mixture should we take from each bottle respectively to make 190 lt dilute sulphuric acid containing half acid and half water?

Answer

To solve the problem of how much mixture to take from each bottle A and B to create 190 liters of dilute sulfuric acid containing equal parts acid and water, we can follow these steps:

### Step 1: Understand the Composition of Each Bottle
- **Bottle A**: Water is 40% of the acid.
- Let the amount of acid in bottle A be \(100A\).
- Therefore, the amount of water in bottle A is \(40\% \text{ of } 100A = 40A\).
- Total solution in bottle A = \(100A + 40A = 140A\).

- **Bottle B**: Acid is 60% of the water.
- Let the amount of water in bottle B be \(100B\).
- Therefore, the amount of acid in bottle B is \(60\% \text{ of } 100B = 60B\).
- Total solution in bottle B = \(100B + 60B = 160B\).

### Step 2: Set Up the Equations
We need to find how much of each bottle (let's say \(x\) liters from bottle A and \(y\) liters from bottle B) is required to make 190 liters of a mixture that is half acid and half water.

1. The total volume equation:
\[ x + y = 190 \quad \text{(1)} \]

2. The acid content equation:
- From bottle A, the acid content is \( \frac{100A}{140A} \cdot x = \frac{100}{140}x = \frac{5}{7}x \).
- From bottle B, the acid content is \( \frac{60B}{160B} \cdot y = \frac{60}{160}y = \frac{3}{8}y \).
- Total acid in the mixture should be \( \frac{190}{2} = 95 \) liters.
\[ \frac{5}{7}x + \frac{3}{8}y = 95 \quad \text{(2)} \]

### Step 3: Solve the Equations
From equation (1), we can express \(y\) in terms of \(x\):
\[ y = 190 - x \quad \text{(3)} \]

Substituting equation (3) into equation (2):
\[ \frac{5}{7}x + \frac{3}{8}(190 - x) = 95 \]

Now, simplify and solve for \(x\):
\[ \frac{5}{7}x + \frac{570}{8} - \frac{3}{8}x = 95 \]
Multiply through by 56 (the least common multiple of 7 and 8) to eliminate fractions:
\[ 40x + 3990 - 21x = 5320 \]
Combine like terms:
\[ 19x = 5320 - 3990 \]
\[ 19x = 1330 \]
\[ x = \frac{1330}{19} = 70 \]

### Step 4: Find \(y\)
Using equation (3):
\[ y = 190 - x = 190 - 70 = 120 \]

### Final Answer
The amount of mixture to take from bottle A is **70 liters**, and from bottle B is **120 liters**.

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