A, B and C can do a piece of work in 36, 54 and 72 days respectively. They started the work but A left 8 days before the completion of the work while B left 12 days before the completion. The number of days for which C worked is

### Concept & Work Completion from the End When workers leave a specific number of days *before* the work is completed, a powerful technique is to "force" them to work until the end by adding their imaginary uncompleted work to the total work pool, then dividing by the combined efficiency of everyone. ### Step-by-Step Solution * **Determine Total Work and Efficiencies:** * Let Total Work = LCM of 36, 54, 72 = 216 units. * A's efficiency = $216 \div 36 = 6 \text{ units/day}$. * B's efficiency = $216 \div 54 = 4 \text{ units/day}$. * C's efficiency = $216 \div 72 = 3 \text{ units/day}$. * **Set Up the Algebraic Equation:** * Let the total number of days to complete the work be $x$. * Since C worked from start to finish, C worked for $x$ days. * A left 8 days early, so A worked for $(x - 8)$ days. * B left 12 days early, so B worked for $(x - 12)$ days. * Total Work = A's work + B's work + C's work $$ 6(x - 8) + 4(x - 12) + 3x = 216 $$ * **Solve for x:** $$ 6x - 48 + 4x - 48 + 3x = 216 $$ $$ 13x - 96 = 216 $$ $$ 13x = 312 $$ $$ x = 24 $$ * C worked for the entire duration, which is 24 days. ### Exam Strategy & Shortcut Use the "Add Back" method. If A had stayed those 8 days, A would have done $6 \times 8 = 48$ extra units. If B had stayed those 12 days, B would have done $4 \times 12 = 48$ extra units. The new "artificial" total work if everyone worked the whole time = $216 + 48 + 48 = 312$. The combined efficiency of all three is $6+4+3=13$. The total days the project lasted (which is how long C worked) = $312 \div 13 = 24$. ### Common Pitfall A common mistake is attempting to calculate the work from the start sequentially, which becomes extremely complicated because you don't know the total duration upfront. Framing the timeline from the unknown end date $x$ is required. ### Final Answer Therefore, the correct answer is **24**.