A manufacturer has 600 liters of 12% solution of acid. How many litres of a 30% acid solution must be added to it so that the acid content in the resulting mixture will be more then 15% but less than 18%?

Explanation

To solve the problem step by step, we need to find how many liters of a 30% acid solution must be added to 600 liters of a 12% acid solution so that the resulting mixture has an acid concentration greater than 15% but less than 18%.

### Step 1: Define the Variables
Let \( x \) be the number of liters of the 30% acid solution to be added.

### Step 2: Calculate the Acid Content
The total volume of the resulting mixture will be:
\[ 600 + x \text{ liters} \]

The amount of acid in the 600 liters of 12% solution is:
\[ \text{Acid from 12% solution} = 0.12 \times 600 = 72 \text{ liters} \]

The amount of acid in \( x \) liters of 30% solution is:
\[ \text{Acid from 30% solution} = 0.30 \times x = 0.3x \text{ liters} \]

### Step 3: Set Up the Inequalities
The total acid in the mixture will be:
\[ \text{Total acid} = 72 + 0.3x \text{ liters} \]

We want the concentration of acid in the mixture to be more than 15% and less than 18%. Therefore, we can set up the following inequalities:

1. For more than 15%:
\[ \frac{72 + 0.3x}{600 + x} > 0.15 \]

2. For less than 18%:
\[ \frac{72 + 0.3x}{600 + x} < 0.18 \]

### Step 4: Solve the First Inequality
Multiply both sides by \( 600 + x \) (assuming \( 600 + x > 0 \)):
\[ 72 + 0.3x > 0.15(600 + x) \]
Expanding the right side:
\[ 72 + 0.3x > 90 + 0.15x \]
Rearranging gives:
\[ 72 - 90 > 0.15x - 0.3x \]
\[ -18 > -0.15x \]
Dividing by -0.15 (remember to reverse the inequality):
\[ x > 120 \]

### Step 5: Solve the Second Inequality
Now, we solve the second inequality:
\[ 72 + 0.3x < 0.18(600 + x) \]
Expanding the right side:
\[ 72 + 0.3x < 108 + 0.18x \]
Rearranging gives:
\[ 72 - 108 < 0.18x - 0.3x \]
\[ -36 < -0.12x \]
Dividing by -0.12 (again, reverse the inequality):
\[ x < 300 \]

### Step 6: Combine the Results
From the two inequalities, we have:
\[ 120 < x < 300 \]

### Conclusion
The manufacturer must add more than 120 liters but less than 300 liters of the 30% acid solution.