Question - Calculating the Required Volume of a Concentrated Acid Solution to Achieve a Desired Concentration

Let \( V \) be the volume (in litres) of the 30% acid solution needed.

The amount of pure acid in the original solution is \( \frac{12}{100} \times 600 \) litres.

The amount of pure acid in the added solution is \( \frac{30}{100} \times V \) litres.

The total volume of the new solution is \( 600 + V \) litres.

The concentration of the acid in the new solution must be more than 15% but less than 18%:

\[ \frac{15}{100} < \frac{\frac{12}{100} \times 600 + \frac{30}{100} \times V}{600 + V} < \frac{18}{100} \]

Multiply through by \( 100(600 + V) \) to clear the denominators:

\[ 15(600 + V) < 12 \times 600 + 30V < 18(600 + V) \]

Solve each inequality for \( V \):

For the lower bound \( 15(600 + V) > 12 \times 600 \):

\[ 9000 + 15V > 7200 \]

\[ 15V > -1800 \]

\[ V > -120 \] (Which is always true since volume cannot be negative)

For the upper bound \( 18(600 + V) > 12 \times 600 + 30V \):

\[ 10800 + 18V > 7200 + 30V \]

\[ 12V < 3600 \]

\[ V < 300 \]

Therefore, the volume of the 30% acid solution to be added must be less than 300 litres.