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%?

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> [!TLDR] > This is a weighted average problem where we mix two acid solutions. Adding $x$ litres of 30% acid to 600 litres of 12% acid gives a final concentration between 15% and 18%. Solving the inequalities yields $120 < x < 300$ litres. We have two liquids being mixed: a known quantity of a weaker solution and an unknown quantity of a stronger one. The final concentration is a weighted average of the two concentrations, where the weights are the volumes. This is the core idea — the final percentage is not simply the average of 12% and 30%, because the volumes are different. Let $x$ be the number of litres of 30% acid solution to be added. **Step 1: Find the total volume of the mixture.** The manufacturer starts with 600 litres. Adding $x$ litres gives a total volume of $600 + x$ litres. **Step 2: Find the total amount of pure acid in the mixture.** - From the 12% solution: $600 \times 0.12 = 72$ litres of pure acid. - From the 30% solution: $x \times 0.30 = 0.3x$ litres of pure acid. - Total pure acid: $72 + 0.3x$ litres. **Step 3: Express the final concentration as a fraction.** The concentration (as a decimal) is: $$ \frac{72 + 0.3x}{600 + x} $$ **Step 4: Set up the inequality for "more than 15% but less than 18%".** We want: $$ 0.15 < \frac{72 + 0.3x}{600 + x} < 0.18 $$ > [!WARNING] > A common mistake is to multiply only one side of the inequality by $(600+x)$ without considering that $600+x$ is positive (since $x \ge 0$). Because the denominator is always positive, we can multiply all three parts by it without flipping inequality signs. **Step 5: Solve the left inequality ($> 15\%$).** $$ \frac{72 + 0.3x}{600 + x} > 0.15 $$ Multiply both sides by $(600+x)$: $$ 72 + 0.3x > 0.15(600 + x) $$ $$ 72 + 0.3x > 90 + 0.15x $$ Subtract $0.15x$ from both sides: $$ 72 + 0.15x > 90 $$ Subtract 72: $$ 0.15x > 18 $$ Divide by 0.15: $$ x > \frac{18}{0.15} = 120 $$ So $x > 120$ litres. …

Q.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%?

Concept understanding — Mixture Concentration

Mixture Concentration: The Intuition First

The Core Idea

The Two Most Common Ways to Express Concentration

1. Mass by Mass (or Weight by Weight) — $\% w/w$

2. Volume by Volume — $\% v/v$

Why This Matters for Exams

Q.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%?

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