Question

Three workers A, B, and C are working on a piece of work. A and B can complete the work in 18 days, B and C in 24 days, and C and A in 36 days. If A alone works on the job, how many days will it take for A to complete the job?

• A. \(40\)
• B. \(45\)
• C. \(48\)
• D. \(50\)

Hint:

When efficiencies of pairs of workers are given, first add all pair equations. This gives the combined efficiency of all workers and helps determine individual efficiencies easily.

Answer 1

The Correct Option is C

Concept: If the work rates of A, B and C are represented by \(a\), \(b\), and \(c\) respectively, then: \[ a+b=\frac1{18}, \qquad b+c=\frac1{24}, \qquad c+a=\frac1{36} \] Adding all three equations allows us to determine the combined rate of all three workers.

Step 1: Add the three given equations. \[ (a+b)+(b+c)+(c+a) = \frac1{18}+\frac1{24}+\frac1{36} \] Taking LCM \(=72\), \[ 2(a+b+c) = \frac4{72}+\frac3{72}+\frac2{72} = \frac9{72} = \frac18 \] Therefore, \[ a+b+c=\frac1{16} \]

Step 2: Find A's individual work rate. \[ a=(a+b+c)-(b+c) \] \[ a=\frac1{16}-\frac1{24} \] Taking LCM \(48\), \[ a=\frac3{48}-\frac2{48} = \frac1{48} \]

Step 3: Find the number of days required by A alone. Since A completes \[ \frac1{48} \] of the work per day, A alone requires \[ 48 \] days to complete the entire work. \[ \boxed{48} \]