90% acid solution (90% pure acid and 10% water) and 97% acid solution are mixed to obtain 21 litres of 95% acid solution. How many litres of each solution are mixed.

Answer

To solve the problem of mixing a 90% acid solution and a 97% acid solution to obtain 21 liters of a 95% acid solution, we can follow these steps:

### Step-by-Step Solution:

1. **Define Variables**:
Let \( x \) be the quantity (in liters) of the 90% acid solution, and \( y \) be the quantity (in liters) of the 97% acid solution.

2. **Set Up the First Equation**:
According to the problem, the total volume of the mixed solution is 21 liters. Therefore, we can write the first equation as:
\[ x + y = 21 \quad \text{(Equation 1)} \]

3. **Set Up the Second Equation**:
The total amount of pure acid in the mixture can be expressed as follows:
- The amount of pure acid from the 90% solution is \( 0.90x \).
- The amount of pure acid from the 97% solution is \( 0.97y \).
- The total amount of pure acid in the 95% solution is \( 0.95 \times 21 = 19.95 \).

Therefore, we can write the second equation as:
\[ 0.90x + 0.97y = 19.95 \quad \text{(Equation 2)} \]

4. **Multiply Equation 2 to Eliminate Decimals**:
To simplify calculations, multiply the entire Equation 2 by 100 to eliminate decimals:
\[ 90x + 97y = 1995 \quad \text{(Equation 3)} \]

5. **Solve the System of Equations**:
Now we have two equations:
- Equation 1: \( x + y = 21 \)
- Equation 3: \( 90x + 97y = 1995 \)

From Equation 1, we can express \( y \) in terms of \( x \):
\[ y = 21 - x \]

Substitute \( y \) in Equation 3:
\[ 90x + 97(21 - x) = 1995 \]
Expanding this gives:
\[ 90x + 2037 - 97x = 1995 \]
Combine like terms:
\[ -7x + 2037 = 1995 \]
Rearranging gives:
\[ -7x = 1995 - 2037 \]
\[ -7x = -42 \]
Dividing by -7:
\[ x = 6 \]

6. **Find the Value of \( y \)**:
Substitute \( x = 6 \) back into Equation 1:
\[ 6 + y = 21 \]
\[ y = 21 - 6 = 15 \]

7. **Conclusion**:
The quantities of each solution 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