Question

A and B can do a work in 9 days and 12 days respectively. If they work on alternate days starting with A, then in how many days will the work be completed?

• A. \( 36 \text{ days} \)
• B. \( 10 \text{ days} \)
• C. \( 10\frac{1}{4} \text{ days} \)
• D. \( 13 \text{ days} \)

Hint:

\textbf{Time and Work - Alternate Days.} When two people work on alternate days, calculate the work done in a cycle of two days. Then find how many such cycles are needed to complete most of the work. Finally, calculate the time taken by the person starting the work to complete the remaining portion.

Answer 1

The Correct Option is C

A can complete the work in 9 days, so A's work per day \( = \frac{1}{9} \). B can complete the work in 12 days, so B's work per day \( = \frac{1}{12} \). They work on alternate days starting with A. Work done on the 1st day (by A) \( = \frac{1}{9} \) Work done on the 2nd day (by B) \( = \frac{1}{12} \) Work done in the first 2 days \( = \frac{1}{9} + \frac{1}{12} = \frac{4 + 3}{36} = \frac{7}{36} \) In 10 days (5 pairs of days), the work done \( = 5 \times \frac{7}{36} = \frac{35}{36} \) Remaining work \( = 1 - \frac{35}{36} = \frac{1}{36} \) On the 11th day, A will work. A's work per day is \( \frac{1}{9} \). Since the remaining work is \( \frac{1}{36} \), which is less than \( \frac{1}{9} \), A will finish the remaining work on the 11th day. Let the number of days A works on the 11th day be \(d\). Work done by A on the 11th day \( = d \times \frac{1}{9} \) $$ d \times \frac{1}{9} = \frac{1}{36} $$ $$ d = \frac{1}{36} \times 9 = \frac{9}{36} = \frac{1}{4} \text{ day} $$ So, the total time taken to complete the work is 10 full days + \( \frac{1}{4} \) day \( = 10\frac{1}{4} \) days.