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 determine 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 content that is more than 15% but less than 18%.

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

### Step 2: Calculate the Total Volume of the Mixture
The total volume of the mixture after adding \( x \) liters of the 30% solution will be:
\[ 600 + x \text{ liters} \]

### Step 3: Calculate the Acid Content in the Mixture
The acid content from the 12% solution is:
\[ \text{Acid from 12% solution} = 12\% \text{ of } 600 = \frac{12}{100} \times 600 = 72 \text{ liters} \]
The acid content from the 30% solution is:
\[ \text{Acid from 30% solution} = 30\% \text{ of } x = \frac{30}{100} \times x = 0.3x \text{ liters} \]
Thus, the total acid content in the mixture is:
\[ \text{Total acid} = 72 + 0.3x \text{ liters} \]

### Step 4: Set Up the Inequalities
We want the acid content to be more than 15% and less than 18% of the total mixture. This gives us two inequalities:
1. More than 15%:
\[ \frac{72 + 0.3x}{600 + x} > 0.15 \]
2. Less than 18%:
\[ \frac{72 + 0.3x}{600 + x} < 0.18 \]

### Step 5: Solve the First Inequality
Starting with the first inequality:
\[ 72 + 0.3x > 0.15(600 + x) \]
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 (and flipping the inequality):
\[ x > 120 \]

### Step 6: 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 (and flipping the inequality):
\[ x < 300 \]

### Step 7: 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.