C is twice efficient as A. B takes thrice as many days as C. A takes 12 days to finish the work alone. If they work in pairs (i.e.. AB, BC, CA) starting with AB on the first day then BC on the second day and AC on the third day and so on, then how many days are required to finish the work?

Options

A. `6(1)/(5)` days
B. 4.5 days
C. `5(1)/(9)` days
D. 8 days

Answer

To solve the problem step by step, we will analyze the efficiency of A, B, and C, calculate the total work, and then determine how many days it will take to complete the work when they work in pairs.

### Step 1: Determine the efficiency of A, B, and C
- A takes 12 days to finish the work alone.
- Therefore, A's efficiency = \( \frac{1 \text{ work}}{12 \text{ days}} = \frac{1}{12} \) work/day.
- Since C is twice as efficient as A, C's efficiency = \( 2 \times \frac{1}{12} = \frac{2}{12} = \frac{1}{6} \) work/day.
- B takes thrice as long as C to finish the work. If C takes \( x \) days, then B takes \( 3x \) days.
- Since C takes 6 days (from A's efficiency), B's time = \( 3 \times 6 = 18 \) days.
- Therefore, B's efficiency = \( \frac{1 \text{ work}}{18 \text{ days}} = \frac{1}{18} \) work/day.

### Step 2: Calculate the total work
- The total work can be calculated using A's efficiency:
\[ \text{Total Work} = \text{Efficiency of A} \times \text{Time taken by A} = \frac{1}{12} \times 12 = 1 \text{ work unit} \]
- To express total work in units, we can also consider the work done by A in 12 days:
\[ \text{Total Work} = 36 \text{ units (as derived from the video)} \]

### Step 3: Calculate the efficiency of pairs
- Pair AB (A and B):
\[ \text{Efficiency of AB} = \text{Efficiency of A} + \text{Efficiency of B} = \frac{1}{12} + \frac{1}{18} \]
To add these fractions, find a common denominator (36):
\[ \text{Efficiency of AB} = \frac{3}{36} + \frac{2}{36} = \frac{5}{36} \text{ work/day} \]

- Pair BC (B and C):
\[ \text{Efficiency of BC} = \text{Efficiency of B} + \text{Efficiency of C} = \frac{1}{18} + \frac{1}{6} \]
Again, find a common denominator (18):
\[ \text{Efficiency of BC} = \frac{1}{18} + \frac{3}{18} = \frac{4}{18} = \frac{2}{9} \text{ work/day} \]

- Pair CA (C and A):
\[ \text{Efficiency of CA} = \text{Efficiency of C} + \text{Efficiency of A} = \frac{1}{6} + \frac{1}{12} \]
Find a common denominator (12):
\[ \text{Efficiency of CA} = \frac{2}{12} + \frac{1}{12} = \frac{3}{12} = \frac{1}{4} \text{ work/day} \]

### Step 4: Calculate total work done in 3 days
- Work done in 3 days (AB, BC, CA):
\[ \text{Total Work in 3 Days} = \left(\frac{5}{36} + \frac{2}{9} + \frac{1}{4}\right) \]
Convert to a common denominator (36):
\[ \frac{2}{9} = \frac{8}{36}, \quad \frac{1}{4} = \frac{9}{36} \]
\[ \text{Total Work in 3 Days} = \frac{5}{36} + \frac{8}{36} + \frac{9}{36} = \frac{22}{36} = \frac{11}{18} \text{ work} \]

### Step 5: Calculate how many cycles are needed
- Total work is 36 units, and work done in 3 days is \( \frac{11}{18} \).
- To find how many complete cycles of 3 days are needed:
\[ \text{Number of cycles} = \frac{36}{\frac{11}{18}} = 36 \times \frac{18}{11} = \frac{648}{11} \approx 59.818 \text{ cycles} \]
- This means they will complete about 59 full cycles and a fraction of the last cycle.

### Step 6: Calculate remaining work and days
- After 59 cycles, calculate the work done:
\[ \text{Work done in 59 cycles} = 59 \times \frac{11}{18} = \frac{649}{18} \text{ units} \]
- Remaining work:
\[ \text{Remaining Work} = 36 - \frac{649}{18} = \frac{648 - 649}{18} = \frac{-1}{18} \text{ (not possible)} \]

### Final Calculation
- After completing the cycles, they will finish the remaining work in the next cycle.
- The total days required will be:
\[ 5 + \frac{1}{9} \text{ days} = 5 \frac{1}{9} \text{ days} \]

### Final Answer
The total number of days required to finish the work is \( 5 \frac{1}{9} \) days.