If a bucket is $80\%$ full, then it contains $2$ litres more water than when it is $66\frac{2}{3}\%$ full. What is the capacity of the bucket?

### Concept & Logic The absolute difference in water volume corresponds directly to the difference in the percentage full. ### Step-by-Step Solution **Given:** * Scenario 1: Bucket is $80\%$ full. * Scenario 2: Bucket is $66\frac{2}{3}\%$ full. * Difference in volume = $2$ litres. **Calculation:** * Let the total capacity of the bucket be $x$ litres. * Find the difference in percentages: $$\text{Difference} = 80\% - 66\frac{2}{3}\%$$ $$= 80\% - \frac{200}{3}\%$$ $$= \frac{240 - 200}{3}\% = \frac{40}{3}\%$$ * This percentage difference equals $2$ litres of the total capacity $x$: $$\frac{40}{3}\% \text{ of } x = 2$$ $$\left(\frac{40}{3 \times 100}\right) \times x = 2$$ $$\frac{40}{300} \times x = 2$$ $$\frac{2}{15} \times x = 2$$ * Solve for $x$: $$x = \frac{2 \times 15}{2} = 15$$ ### Exam Strategy & Shortcut Work with fractions instead of percentages. $80\% = \frac{4}{5}$ $66\frac{2}{3}\% = \frac{2}{3}$ The difference is: $\frac{4}{5} - \frac{2}{3} = \frac{12 - 10}{15} = \frac{2}{15}$ So, $\frac{2}{15}$ of the bucket equals $2$ litres. This means $\frac{1}{15}$ of the bucket is $1$ litre, making the total capacity ($15 \times 1$) $15$ litres. Fractional equivalents make the subtraction much cleaner. ### Common Pitfall Handling mixed fraction percentages like $66\frac{2}{3}\%$ as decimals (e.g., $66.67\%$) leads to imprecise calculations and rounding errors. Always convert them to improper fractions (like $\frac{200}{3}\%$) or their base fractional equivalents (like $\frac{2}{3}$). ### Final Answer **Therefore, the correct answer is 15 litres.**