A & B can do a piece of work together in 24 days and C's efficiency is half of A's efficiency. If all three work together, they can complete the same work in 20 days, then find how many days are required to complete the work, if B and C work together?

Options

A. A)45 days
B. B)42 days
C. C)30 days
D. D)48 days

Answer

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

### Step 1: Determine the work done by A and B together.
A and B can complete the work together in 24 days. This means their combined work rate (efficiency) is:
\[ \text{Efficiency of A and B} = \frac{1 \text{ work}}{24 \text{ days}} = \frac{1}{24} \text{ work/day} \]

### Step 2: Calculate the total work in terms of units.
If we consider the total work to be 120 units (a common multiple of 24 and 20 for easier calculations), then:
\[ \text{Total Work} = 120 \text{ units} \]

### Step 3: Find the efficiency of A and B.
Since A and B can complete 120 units of work in 24 days, their combined efficiency in units is:
\[ \text{Efficiency of A and B} = \frac{120 \text{ units}}{24 \text{ days}} = 5 \text{ units/day} \]

### Step 4: Determine the efficiency of A, B, and C together.
A, B, and C can complete the same work in 20 days, so their combined efficiency is:
\[ \text{Efficiency of A, B, and C} = \frac{120 \text{ units}}{20 \text{ days}} = 6 \text{ units/day} \]

### Step 5: Calculate the efficiency of C.
From the efficiencies calculated:
\[ \text{Efficiency of C} = \text{Efficiency of A, B, and C} - \text{Efficiency of A and B} = 6 - 5 = 1 \text{ unit/day} \]

### Step 6: Relate C's efficiency to A's efficiency.
According to the problem, C's efficiency is half of A's efficiency. Let A's efficiency be \( x \):
\[ \text{Efficiency of C} = \frac{x}{2} \]
We found that C's efficiency is 1 unit/day, so:
\[ \frac{x}{2} = 1 \implies x = 2 \text{ units/day} \]
Thus, A's efficiency is 2 units/day.

### Step 7: Find B's efficiency.
Since A and B together have an efficiency of 5 units/day:
\[ \text{Efficiency of B} = 5 - \text{Efficiency of A} = 5 - 2 = 3 \text{ units/day} \]

### Step 8: Calculate the efficiency of B and C together.
Now we can find the combined efficiency of B and C:
\[ \text{Efficiency of B and C} = \text{Efficiency of B} + \text{Efficiency of C} = 3 + 1 = 4 \text{ units/day} \]

### Step 9: Calculate the time taken by B and C to complete the work.
The time taken by B and C to complete 120 units of work is:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Efficiency of B and C}} = \frac{120 \text{ units}}{4 \text{ units/day}} = 30 \text{ days} \]

### Final Answer:
B and C together will take **30 days** to complete the work.

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