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

Step 1: Define variables and initial acid content

Let $x$ be the volume (in litres) of the $30\%$ acid solution to be added. The initial solution has a volume of 600 litres and contains $12\%$ acid. We need to find the range of $x$ such that the final mixture's acid concentration is between $15\%$ and $18\%$.

Step 2: Calculate total acid and total volume in the mixture

First, calculate the amount of acid in the initial 600 litres of $12\%$ solution. Then, express the amount of acid in the $x$ litres of $30\%$ solution. The total acid in the mixture will be the sum of these two amounts, and the total volume will be the sum of the initial volume and $x$.

Step 3: Set up inequalities for the acid concentration

The problem states that the acid content in the resulting mixture must be more than $15\%$ but less than $18\%$. This can be written as an inequality where the ratio of total acid to total volume is between $0.15$ and $0.18$.

Step 4: Solve the first inequality

We solve the first part of the inequality: the acid concentration must be greater than $15\%$. Multiply both sides by $(600 + x)$, distribute, and then isolate $x$ to find its lower bound.

Step 5: Solve the second inequality

Next, we solve the second part of the inequality: the acid concentration must be less than $18\%$. Similar to the previous step, multiply by $(600 + x)$, distribute, and isolate $x$ to find its upper bound.

Step 6: Combine the inequalities

By combining the results from both inequalities, we find the range of $x$ for which the acid content in the resulting mixture will be more than $15\%$ but less than $18\%$.

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