A is 50% as efficient as B. C does half of the work done by A and B together. If C alone does the work in 20 days, then A, B and C together can do the work in

Options

A. `5(2)/(3)` days
B. `6(2)/(3) ` days
C. 6 days
D. 7 days

Answer

To solve the problem step by step, let's break it down:

### Step 1: Define the efficiency of A and B
Let B's efficiency be represented as 1 unit of work per day. Since A is 50% as efficient as B, A's efficiency will be:
\[ \text{Efficiency of A} = 0.5 \times \text{Efficiency of B} = 0.5 \times 1 = 0.5 \text{ units of work per day} \]

### Step 2: Calculate the combined efficiency of A and B
The combined efficiency of A and B together is:
\[ \text{Efficiency of A + B} = \text{Efficiency of A} + \text{Efficiency of B} = 0.5 + 1 = 1.5 \text{ units of work per day} \]

### Step 3: Determine C's efficiency
According to the problem, C does half of the work done by A and B together. Therefore, C's efficiency is:
\[ \text{Efficiency of C} = \frac{1}{2} \times \text{Efficiency of (A + B)} = \frac{1}{2} \times 1.5 = 0.75 \text{ units of work per day} \]

### Step 4: Calculate the total work done by C in 20 days
If C alone can complete the work in 20 days, the total work can be expressed as:
\[ \text{Total Work} = \text{Efficiency of C} \times \text{Time taken by C} = 0.75 \times 20 = 15 \text{ units of work} \]

### Step 5: Calculate the total efficiency of A, B, and C together
Now, we need to find the combined efficiency of A, B, and C:
\[ \text{Efficiency of (A + B + C)} = \text{Efficiency of A} + \text{Efficiency of B} + \text{Efficiency of C} = 0.5 + 1 + 0.75 = 2.25 \text{ units of work per day} \]

### Step 6: Calculate the time taken by A, B, and C together to complete the work
To find out how many days it will take for A, B, and C to complete the total work of 15 units, we use the formula:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Combined Efficiency}} = \frac{15}{2.25} \]
Calculating this gives:
\[ \text{Time} = \frac{15}{2.25} = \frac{15 \times 100}{225} = \frac{1500}{225} = \frac{60}{9} = \frac{20}{3} \text{ days} \]

### Final Answer
Thus, A, B, and C together can complete the work in:
\[ \frac{20}{3} \text{ days} \text{ or } 6 \frac{2}{3} \text{ days} \]