Fatima and Zahira can do a piece of work in 12 days and 15 days respectively. If they work for alternate day and Fatima starts the work first, then in how many days the work will be completed?

Options

A. `12(1)/(5)`
B. `13(1)/(4)`
C. `13(1)/(5)`
D. 15

Answer

To solve the problem step by step, we need to determine how many days Fatima and Zahira will take to complete the work when they work alternately, starting with Fatima.

### Step 1: Determine the total work
We will calculate the total work based on the least common multiple (LCM) of the days taken by Fatima and Zahira to complete the work.

- Fatima can complete the work in 12 days.
- Zahira can complete the work in 15 days.

**LCM of 12 and 15:**
- The prime factorization of 12 is \(2^2 \times 3\).
- The prime factorization of 15 is \(3 \times 5\).
- The LCM is \(2^2 \times 3 \times 5 = 60\).

So, the total work is 60 units.

### Step 2: Calculate the work done by each person in one day
Now, we will find out how much work each person can do in one day.

- Work done by Fatima in one day:
\[ \text{Fatima's work per day} = \frac{60 \text{ units}}{12 \text{ days}} = 5 \text{ units/day} \]

- Work done by Zahira in one day:
\[ \text{Zahira's work per day} = \frac{60 \text{ units}}{15 \text{ days}} = 4 \text{ units/day} \]

### Step 3: Calculate the total work done in two days
Since they work on alternate days, we will calculate the total work done in a 2-day cycle (1 day by Fatima and 1 day by Zahira).

- Work done in 2 days:
\[ \text{Total work in 2 days} = \text{Fatima's work} + \text{Zahira's work} = 5 + 4 = 9 \text{ units} \]

### Step 4: Determine how many complete 2-day cycles are needed
Now, we need to find out how many complete 2-day cycles are required to get close to 60 units of work.

- Total work done in \(n\) cycles of 2 days:
\[ \text{Total work in } n \text{ cycles} = 9n \text{ units} \]

To find \(n\):
\[ 9n \leq 60 \implies n \leq \frac{60}{9} \approx 6.67 \]
So, the maximum complete cycles \(n = 6\).

### Step 5: Calculate the total work done after 12 days
After 6 complete cycles (12 days):
\[ \text{Total work done} = 9 \times 6 = 54 \text{ units} \]

### Step 6: Calculate the remaining work
Now, we need to find out how much work is left after 12 days:
\[ \text{Remaining work} = 60 - 54 = 6 \text{ units} \]

### Step 7: Determine who works on the 13th day
On the 13th day, it will be Fatima's turn to work again. She can do 5 units of work in one day.

- After Fatima works on the 13th day:
\[ \text{Work done on 13th day} = 5 \text{ units} \]
\[ \text{Remaining work after 13th day} = 6 - 5 = 1 \text{ unit} \]

### Step 8: Determine how much time Zahira will take to finish the remaining work
On the 14th day, it will be Zahira's turn to work. She can do 4 units of work in one day.

To finish the remaining 1 unit of work, Zahira will take:
\[ \text{Time taken by Zahira} = \frac{1 \text{ unit}}{4 \text{ units/day}} = \frac{1}{4} \text{ days} \]

### Step 9: Calculate the total time taken
Total time taken to complete the work:
\[ \text{Total days} = 13 + \frac{1}{4} = 13 \frac{1}{4} \text{ days} \]

### Final Answer
The work will be completed in \(13 \frac{1}{4}\) days.

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