A can do a work in 21 days and B in 42 days. If they work on it together for 7 days, then what fraction of work is left?

Options

A. `(1)/(3)`
B. `(1)/(4)`
C. `(2)/(3)`
D. `(1)/(2)`

Answer

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

### Step 1: Determine the work done by A and B
- A can complete the work in 21 days.
- B can complete the work in 42 days.

### Step 2: Calculate the rate of work for A and B
- The rate of work is calculated using the formula:
\[ \text{Rate} = \frac{\text{Work}}{\text{Time}} \]
- We can assume the total work to be the least common multiple (LCM) of 21 and 42, which is 42.

#### For A:
- Work = 42
- Time = 21
- Rate of A = \(\frac{42}{21} = 2\)

#### For B:
- Work = 42
- Time = 42
- Rate of B = \(\frac{42}{42} = 1\)

### Step 3: Calculate the combined rate of A and B
- When A and B work together, their rates add up:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} = 2 + 1 = 3 \]

### Step 4: Calculate the work done when A and B work together for 7 days
- Work done in 7 days = Combined Rate × Time
\[ \text{Work done} = 3 \times 7 = 21 \]

### Step 5: Determine the total work
- The total work is assumed to be 42 (as calculated from the LCM).

### Step 6: Calculate the work remaining
- Work remaining = Total Work - Work Done
\[ \text{Work remaining} = 42 - 21 = 21 \]

### Step 7: Calculate the fraction of work left
- Fraction of work left = \(\frac{\text{Work remaining}}{\text{Total Work}}\)
\[ \text{Fraction of work left} = \frac{21}{42} = \frac{1}{2} \]

### Final Answer:
The fraction of work left is \(\frac{1}{2}\).

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