If $\frac{4}{9}$th of a bucket is filled in 1 minute, the rest of it will be filled in (Hotel Management, 2010)

### Concept & Work and Time Fractions When a fraction of a task is completed in a certain time, the time required for the remaining task is directly proportional to the remaining fraction. $$ \frac{\text{Time}_1}{\text{Work}_1} = \frac{\text{Time}_2}{\text{Work}_2} $$ ### Step-by-Step Solution * Given: $\frac{4}{9}$ of the bucket is filled in 1 minute. * Calculate the remaining part to be filled: $1 - \frac{4}{9} = \frac{5}{9}$ of the bucket. * Let $x$ be the time taken to fill the remaining part. * Set up the proportion: $\frac{1 \text{ min}}{4/9} = \frac{x \text{ min}}{5/9}$. * Multiply both sides by $\frac{5}{9}$: $x = 1 \times \frac{5/9}{4/9} = \frac{5}{4}$. * The rest will be filled in $\frac{5}{4}$ minutes. ### Exam Strategy & Shortcut Think in terms of "units" of work. If 4 units take 1 minute, then 1 unit takes $\frac{1}{4}$ minute. The remaining work is $9 - 4 = 5$ units. Therefore, 5 units will take $5 \times \frac{1}{4} = \frac{5}{4}$ minutes. ### Common Pitfall Calculating the total time to fill the entire bucket ($\frac{9}{4}$ minutes) instead of just the time for the *remaining* part, which leads to choosing option (b). ### Final Answer Therefore, the correct answer is **$\frac{5}{4}$ min**.