A can complete a work in 20 days and B can complete the same work in 16 days. C's work efficiency is `33(1)/(3)%` of the work efficiency of A and B together. In how many day A, B and C can complete the work if they work alternatively starting from A followed by B and C?

Options

A. `18(4)/(5)`days
B. 18 days
C. 14 days
D. `19(4)/(5)`days

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Calculate the work done by A and B
- A can complete the work in 20 days. Therefore, A's work efficiency is:
\[ \text{Efficiency of A} = \frac{1}{20} \text{ (work/day)} \]
- B can complete the work in 16 days. Therefore, B's work efficiency is:
\[ \text{Efficiency of B} = \frac{1}{16} \text{ (work/day)} \]

### Step 2: Calculate the total efficiency of A and B
- To find the total efficiency of A and B together, we need to find a common denominator:
\[ \text{Efficiency of A} + \text{Efficiency of B} = \frac{1}{20} + \frac{1}{16} \]
- The LCM of 20 and 16 is 80. Converting the efficiencies:
\[ \frac{1}{20} = \frac{4}{80}, \quad \frac{1}{16} = \frac{5}{80} \]
- Therefore, the total efficiency of A and B is:
\[ \text{Total Efficiency of A and B} = \frac{4}{80} + \frac{5}{80} = \frac{9}{80} \text{ (work/day)} \]

### Step 3: Calculate C's efficiency
- C's efficiency is given as \(33 \frac{1}{3}\%\) of the total efficiency of A and B:
\[ C's \text{ efficiency} = \frac{33.33}{100} \times \frac{9}{80} = \frac{1}{3} \times \frac{9}{80} = \frac{3}{80} \text{ (work/day)} \]

### Step 4: Calculate the combined efficiency of A, B, and C
- The combined efficiency of A, B, and C working together is:
\[ \text{Total Efficiency of A, B, and C} = \frac{4}{80} + \frac{5}{80} + \frac{3}{80} = \frac{12}{80} = \frac{3}{20} \text{ (work/day)} \]

### Step 5: Determine the work done in one complete cycle
- In one complete cycle (A, B, C), the work done is:
\[ \text{Work done in one cycle} = \text{Efficiency of A} + \text{Efficiency of B} + \text{Efficiency of C} = \frac{4}{80} + \frac{5}{80} + \frac{3}{80} = \frac{12}{80} = \frac{3}{20} \]
- The total work is 1 (the entire work), and in one cycle, they complete \(\frac{3}{20}\) of the work.

### Step 6: Calculate the number of cycles needed to complete the work
- To find how many cycles are needed to complete the work:
\[ \text{Number of cycles} = \frac{1}{\frac{3}{20}} = \frac{20}{3} \text{ cycles} \]

### Step 7: Calculate the total time taken for the cycles
- Each cycle takes 3 days (A, B, C):
\[ \text{Total days for complete cycles} = 3 \times \left\lfloor \frac{20}{3} \right\rfloor = 3 \times 6 = 18 \text{ days} \]

### Step 8: Calculate the remaining work after 18 days
- In 18 days, they complete:
\[ \text{Work done} = 6 \times \frac{3}{20} = \frac{18}{20} = \frac{9}{10} \]
- Remaining work is:
\[ 1 - \frac{9}{10} = \frac{1}{10} \]

### Step 9: Determine who works next and how much time it takes
- On the 19th day, A works and completes:
\[ \text{Work done by A on the 19th day} = \frac{1}{20} \]
- Since \(\frac{1}{20} > \frac{1}{10}\), A will finish the remaining work on the 19th day.

### Final Answer
- Therefore, the total time taken by A, B, and C to complete the work is:
\[ 18 \text{ days} + \text{A's work on the 19th day} = 19 \frac{1}{5} \text{ days} \]
- The answer is \(19 \frac{4}{5}\) days.