A can do a piece of work in 14 days while B can do it in 21 days. In how many days, working together they will complete the whole work?

Options

A. 10.5
B. 8
C. 8.4
D. 9

Answer

To solve the problem of how many days A and B will take to complete the work together, we can follow these steps:

### Step 1: Determine the work done by A in one day
A can complete the work in 14 days. Therefore, the work done by A in one day is:
\[ \text{A's one day work} = \frac{1}{14} \]

### Step 2: Determine the work done by B in one day
B can complete the work in 21 days. Therefore, the work done by B in one day is:
\[ \text{B's one day work} = \frac{1}{21} \]

### Step 3: Calculate the combined work done by A and B in one day
To find the total work done by A and B together in one day, we add their individual work rates:
\[ \text{Combined one day work} = \text{A's one day work} + \text{B's one day work} = \frac{1}{14} + \frac{1}{21} \]

### Step 4: Find a common denominator
The least common multiple (LCM) of 14 and 21 is 42. We will convert both fractions to have this common denominator:
\[ \frac{1}{14} = \frac{3}{42} \quad \text{(since } 14 \times 3 = 42\text{)} \]
\[ \frac{1}{21} = \frac{2}{42} \quad \text{(since } 21 \times 2 = 42\text{)} \]

### Step 5: Add the fractions
Now we can add the fractions:
\[ \text{Combined one day work} = \frac{3}{42} + \frac{2}{42} = \frac{5}{42} \]

### Step 6: Calculate the total time taken to complete the work
To find out how many days A and B will take to complete the work together, we take the reciprocal of their combined work rate:
\[ \text{Total days} = \frac{1}{\text{Combined one day work}} = \frac{1}{\frac{5}{42}} = \frac{42}{5} = 8.4 \text{ days} \]

Thus, A and B together will complete the work in **8.4 days**.