Question

A can finish a piece of work in 36 days, B can finish it in 54 days, and C can finish it in 72 days. All three start the work together, but A leaves 8 days before the work is completed, and B leaves 12 days before the work is completed. For how many days in all did C work, till the entire work was finished?

• A. 48 days
• B. 24 days
• C. 12 days
• D. None

Hint:

Let the whole job take T days; A works (T-8) days, B works (T-12) days, and C works all T days. Add their shares of the work to equal 1 (or 216 units) and solve for T.

Answer 1

The Correct Option is B

Let's solve the same problem using total "work units" instead of fractions, which avoids fractions until the very end.

Take the total work as 216 units (the LCM of 36, 54 and 72), so that each person's daily output is a whole number.

A completes 216 units in 36 days, so A does \(216/36 = 6\) units a day. B completes 216 units in 54 days, so B does \(216/54 = 4\) units a day. C completes 216 units in 72 days, so C does \(216/72 = 3\) units a day.

Let the whole job take \(T\) days to finish. A works for \((T-8)\) days since A stops 8 days before the finish, and B works for \((T-12)\) days since B stops 12 days before the finish. C works right through, so C puts in \(T\) days, exactly the number we need to find.

The total units of work done by all three must add up to 216:

\[ 6(T-8) + 4(T-12) + 3T = 216 \]

Expand each bracket:

\[ 6T - 48 + 4T - 48 + 3T = 216 \] \[ 13T - 96 = 216 \] \[ 13T = 312 \] \[ T = 24 \]

So the entire job takes 24 days from start to finish, and since C is present for the whole job, C works for all 24 of those days. This confirms option (2). Option (1), 48 days, is exactly double the correct value, and option (3), 12 days, is too small to let A even put in a positive number of working days once we subtract the 8-day early exit, so both are ruled out.

\[ \boxed{24 \text{ days}} \]