90% and 97% pure acid solutions are mixed to obtain 21 litres of 95% pure acid solution. Find the quantity of each type of acids to be mixed to from the mixture.

Explanation

To solve the problem of mixing two acid solutions to obtain a specific concentration, we can follow these steps:

### Step-by-Step Solution:

1. **Define Variables**:
Let \( X \) be the quantity (in liters) of the 90% acid solution.
Let \( Y \) be the quantity (in liters) of the 97% acid solution.

2. **Set Up the Equations**:
We know that the total volume of the mixture is 21 liters. Therefore, we can write the first equation:
\[ X + Y = 21 \quad \text{(1)} \]

3. **Set Up the Concentration Equation**:
The total amount of pure acid in the mixture can be expressed as:
- From the 90% solution: \( 0.90X \)
- From the 97% solution: \( 0.97Y \)

The total amount of pure acid in the final mixture (which is 95% of 21 liters) is:
\[ 0.95 \times 21 = 19.95 \text{ liters} \]
Thus, we can write the second equation:
\[ 0.90X + 0.97Y = 19.95 \quad \text{(2)} \]

4. **Solve the System of Equations**:
We can solve equations (1) and (2) simultaneously. From equation (1), we can express \( Y \) in terms of \( X \):
\[ Y = 21 - X \quad \text{(3)} \]

5. **Substitute Equation (3) into Equation (2)**:
Substitute \( Y \) in equation (2):
\[ 0.90X + 0.97(21 - X) = 19.95 \]
Expanding this gives:
\[ 0.90X + 20.37 - 0.97X = 19.95 \]
Combine like terms:
\[ -0.07X + 20.37 = 19.95 \]
Subtract 20.37 from both sides:
\[ -0.07X = 19.95 - 20.37 \]
\[ -0.07X = -0.42 \]
Divide both sides by -0.07:
\[ X = \frac{-0.42}{-0.07} = 6 \]

6. **Find \( Y \)**:
Substitute \( X = 6 \) back into equation (3):
\[ Y = 21 - 6 = 15 \]

7. **Conclusion**:
The quantities of the acid solutions to be mixed are:
- \( X = 6 \) liters of the 90% acid solution
- \( Y = 15 \) liters of the 97% acid solution

### Final Answer:
- 6 liters of 90% acid solution
- 15 liters of 97% acid solution