A and B together can do a job in 2 days; B and C can do it in 4 days; A and C in $2 \frac{2}{5}$ days. The number of days required for A to do the job alone is

### Concept & Formula / Logic When given the times taken by pairs to complete a job, adding the pair rates gives twice the total combined rate. From the total rate, you can isolate an individual's rate by subtracting the rate of the pair that doesn't include them. $$ 2(A + B + C) = (A + B) + (B + C) + (A + C) $$ $$ A = (A + B + C) - (B + C) $$ ### Step-by-Step Solution 1. **Identify the given 1-day work rates:** (A + B)'s 1-day work = $\frac{1}{2}$ (B + C)'s 1-day work = $\frac{1}{4}$ (A + C) take $2 \frac{2}{5}$ days. Convert to improper fraction: $\frac{12}{5}$ days. So, (A + C)'s 1-day work = $\frac{5}{12}$ 2. **Find the total combined 1-day rate:** Sum the 1-day work of all pairs: $2(A + B + C) = \frac{1}{2} + \frac{1}{4} + \frac{5}{12}$ Find the LCM of 2, 4, and 12, which is 12. $2(A + B + C) = \frac{6}{12} + \frac{3}{12} + \frac{5}{12} = \frac{14}{12} = \frac{7}{6}$ Therefore, $(A + B + C)'s$ 1-day work = $\frac{7}{6 \times 2} = \frac{7}{12}$ 3. **Calculate A's 1-day work:** Subtract (B + C)'s rate from the total combined rate. A = (A + B + C) - (B + C) $A = \frac{7}{12} - \frac{1}{4} = \frac{7}{12} - \frac{3}{12} = \frac{4}{12} = \frac{1}{3}$ Since A's 1-day work is $\frac{1}{3}$, A alone can finish the job in 3 days. ### Exam Strategy & Shortcut Use the LCM (Total Work) method. Let total work = LCM of (2, 4, and 12/5). To find LCM of fractions, use LCM of numerators = LCM(2, 4, 12) = 12. Efficiency of $(A+B) = \frac{12}{2} = 6$. Efficiency of $(B+C) = \frac{12}{4} = 3$. Efficiency of $(A+C) = \frac{12}{12/5} = 5$. $2(A+B+C) = 6 + 3 + 5 = 14 \Rightarrow (A+B+C) = 7$. Efficiency of A = $(A+B+C) - (B+C) = 7 - 3 = 4$. Days taken by A = $\frac{\text{Total Work}}{\text{A's Efficiency}} = \frac{12}{4} = 3$ days. ### Common Pitfall A common mistake when adding the fractions is forgetting to divide the sum by 2, leading to an inflated combined work rate for $A+B+C$. ### Final Answer Therefore, the correct answer is **3**.