A and B Work Rates Calculation
First, calculate the time each person takes to complete the entire task and their respective daily work rates.
- Person A: Completes $\frac{2}{5}$ of the task in 12 days.
- Time for A to complete the whole task = $12 \div \frac{2}{5} = 12 \times \frac{5}{2} = 30$ days.
- A's daily work rate = $\frac{1}{30}$ of the task per day.
- Person B: Completes $\frac{3}{4}$ of the task in 25 days.
- Time for B to complete the whole task = $25 \div \frac{3}{4} = 25 \times \frac{4}{3} = \frac{100}{3}$ days.
- B's daily work rate = $\frac{1}{\frac{100}{3}} = \frac{3}{100}$ of the task per day.
Combined Work Rate Calculation
Determine their combined daily work rate by adding their individual rates.
- Combined daily work rate = A's rate + B's rate
- Combined rate = $\frac{1}{30} + \frac{3}{100}$
- Find a common denominator (LCM of 30 and 100 is 300).
- Combined rate = $\frac{1 \times 10}{30 \times 10} + \frac{3 \times 3}{100 \times 3} = \frac{10}{300} + \frac{9}{300} = \frac{19}{300}$ of the task per day.
Task Completion Time Together
Calculate the total time needed to complete the task when they work together. This is the reciprocal of their combined daily work rate.
- Time together = $\frac{1}{\text{Combined rate}}$
- Time together = $\frac{1}{\frac{19}{300}} = \frac{300}{19}$ days.
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