What amounts (in litres) of `90% and 97%` pure acid solutions are mixed to obtain 21 L of `95%` pure acid solution?

Options

A. 6 and 15 L
B. 14 and 15 L
C. 12 and 15 L
D. 13 and 12 L

Answer

To solve the problem of mixing two solutions of different concentrations to obtain a desired concentration, we can use the method of alligation or set up a system of equations. Here’s a step-by-step solution:

### Step 1: Define Variables
Let:
- \( x \) = amount of 90% pure acid solution (in litres)
- \( y \) = amount of 97% pure acid solution (in litres)

### Step 2: Set Up the Equations
We know that the total volume of the mixture is 21 litres:
\[ x + y = 21 \]

We also know that the total amount of pure acid in the mixture must equal the amount of pure acid contributed by each solution. The amount of pure acid from each solution can be expressed as:
- From the 90% solution: \( 0.90x \)
- From the 97% solution: \( 0.97y \)

Since we want the final solution to be 95% pure acid, the total amount of pure acid in the 21 litres of 95% solution is:
\[ 0.95 \times 21 = 19.95 \]

Thus, we can set up the second equation:
\[ 0.90x + 0.97y = 19.95 \]

### Step 3: Solve the System of Equations
Now we have the system of equations:
1. \( x + y = 21 \)
2. \( 0.90x + 0.97y = 19.95 \)

From the first equation, we can express \( y \) in terms of \( x \):
\[ y = 21 - x \]

Substituting \( y \) in the second equation:
\[ 0.90x + 0.97(21 - x) = 19.95 \]

### Step 4: Simplify and Solve for \( x \)
Expanding the equation:
\[ 0.90x + 20.37 - 0.97x = 19.95 \]
Combining like terms:
\[ -0.07x + 20.37 = 19.95 \]
Subtracting 20.37 from both sides:
\[ -0.07x = 19.95 - 20.37 \]
\[ -0.07x = -0.42 \]
Dividing both sides by -0.07:
\[ x = \frac{-0.42}{-0.07} = 6 \]

### Step 5: Find \( y \)
Now that we have \( x \), we can find \( y \):
\[ y = 21 - x = 21 - 6 = 15 \]

### Step 6: Conclusion
Thus, the amounts of the solutions needed are:
- \( 6 \) litres of 90% pure acid solution
- \( 15 \) litres of 97% pure acid solution

### Final Answer:
- Amount of 90% pure acid solution: **6 litres**
- Amount of 97% pure acid solution: **15 litres**

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