A can do a work in 42 days and B in 84 days. If they work on it together for 7 days, then what fraction of work is done?

Options

A. `1/3`
B. `1/4`
C. `1/2`
D. `1/5`

Answer

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

### Step 1: Determine the total work
A can complete the work in 42 days, and B can complete it in 84 days. To find a common measure of work, we can take the Least Common Multiple (LCM) of 42 and 84.

- **LCM of 42 and 84**:
- The prime factorization of 42 is \(2 \times 3 \times 7\).
- The prime factorization of 84 is \(2^2 \times 3 \times 7\).
- The LCM is \(2^2 \times 3 \times 7 = 84\).

Thus, the total work is **84 units**.

### Step 2: Calculate the efficiency of A and B
Next, we calculate the work done by A and B in one day.

- **Efficiency of A**:
- A's efficiency = Total work / Days taken by A = \(84 / 42 = 2\) units per day.

- **Efficiency of B**:
- B's efficiency = Total work / Days taken by B = \(84 / 84 = 1\) unit per day.

### Step 3: Calculate the combined efficiency
Now, we can find the combined efficiency of A and B when they work together.

- **Combined efficiency**:
- Combined efficiency of A and B = Efficiency of A + Efficiency of B = \(2 + 1 = 3\) units per day.

### Step 4: Calculate the work done in 7 days
Now, we need to find out how much work A and B can complete together in 7 days.

- **Work done in 7 days**:
- Work done = Combined efficiency × Number of days = \(3 \times 7 = 21\) units.

### Step 5: Determine the fraction of work done
Finally, we need to find out what fraction of the total work (84 units) is completed in 7 days.

- **Fraction of work done**:
- Fraction = Work done / Total work = \(21 / 84 = 1/4\).

Thus, the fraction of work done by A and B together in 7 days is **1/4**.

### Final Answer:
The fraction of work done is **1/4**.

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