Discussion
A completes \(\frac{2}{3}\) of a certain job in 6 days. B can complete \(\frac{1}{3}\) of the same job in 8 days and C can complete \(\frac{3}{4}\) of the job in 12 days. All of them work together for 4 days and then A and C quit. How long will it take for B to complete the remaining work alone ?
Answer: B
B will take about 3.33 days.
Turn each rate into a whole-job time: A does 2/3 in 6 days, so the whole in 9; B does 1/3 in 8 days, so the whole in 24; C does 3/4 in 12 days, so the whole in 16.
\[ \begin{aligned} \text{daily rate} &= \tfrac{1}{9} + \tfrac{1}{24} + \tfrac{1}{16} \\ &= \tfrac{16 + 6 + 9}{144} = \tfrac{31}{144} \\ \text{4 days} &= \tfrac{124}{144} \\ \text{left} &= \tfrac{20}{144} = \tfrac{5}{36} \end{aligned} \]
\[ \begin{aligned} \text{B alone} &= \tfrac{5}{36} \times 24 \\ &= \tfrac{10}{3} \approx 3.33 \text{ days} \end{aligned} \]
The trap is to use the given fractions as if they were whole jobs (for example treating B as finishing in 8 days). 2·67, 4 and 4·67 days do not match the remaining 5/36 of the work.
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