A and B together can do a job in 15 days and A alone could do the same job in 20 days how many days would B take to do half the job if he worked alone

Options

A. 60
B. 30
C. 45
D. 40

Answer

To solve the problem step by step, we will determine how many days B would take to complete half the job alone.

### Step 1: Determine the work done by A and B together
A and B together can complete the job in 15 days. This means that in one day, they complete:
\[ \text{Work done by A and B together in one day} = \frac{1}{15} \]

### Step 2: Determine the work done by A alone
A alone can complete the job in 20 days. This means that in one day, A completes:
\[ \text{Work done by A in one day} = \frac{1}{20} \]

### Step 3: Determine the work done by B alone
To find out how much work B can do in one day, we can use the information from Steps 1 and 2. The work done by B alone can be calculated as follows:
\[ \text{Work done by B in one day} = \text{Work done by A and B together} - \text{Work done by A} \]
Substituting the values we found:
\[ \text{Work done by B in one day} = \frac{1}{15} - \frac{1}{20} \]

### Step 4: Find a common denominator and calculate B's work
The least common multiple of 15 and 20 is 60. We can express both fractions with a denominator of 60:
\[ \frac{1}{15} = \frac{4}{60}, \quad \frac{1}{20} = \frac{3}{60} \]
Now, substituting these into the equation:
\[ \text{Work done by B in one day} = \frac{4}{60} - \frac{3}{60} = \frac{1}{60} \]

### Step 5: Determine how long B takes to complete the whole job
If B can do \(\frac{1}{60}\) of the job in one day, then to complete the entire job, B would take:
\[ \text{Days taken by B to complete the whole job} = \frac{1}{\frac{1}{60}} = 60 \text{ days} \]

### Step 6: Determine how long B takes to complete half the job
To find out how long B would take to complete half the job:
\[ \text{Days taken by B to complete half the job} = \frac{60}{2} = 30 \text{ days} \]

### Final Answer
B would take **30 days** to complete half the job if he worked alone.

---