Working alone, A can complete a task in 'a' days and B in 'b' days. They take turns in doing the task with each working 2 days at a time. If A starts they finish the task in exactly 10 days. If B starts, they take half a day more. How long does it take to complete the task if they both work together?

Options

A. `5(1)/(3)`days
B. `5(1)/(7)`days
C. `5(5)/(9)`days
D. `5(5)/(11)`days

Answer

To solve the problem step by step, we will first analyze the information given and then derive the necessary equations to find the solution.

### Step 1: Understand the Work Done by A and B
- Let the total work be denoted as W.
- A can complete the work in 'a' days, so A's work rate is \( \frac{W}{a} \) per day.
- B can complete the work in 'b' days, so B's work rate is \( \frac{W}{b} \) per day.

### Step 2: Work Done in 10 Days When A Starts
- A works for the first 2 days, then B works for the next 2 days, and this pattern continues.
- In 10 days, A works for 6 days (2 days at a time for 5 cycles) and B works for 4 days (2 days at a time for 4 cycles).

The work done by A in 6 days:
\[ \text{Work by A} = 6 \times \frac{W}{a} = \frac{6W}{a} \]

The work done by B in 4 days:
\[ \text{Work by B} = 4 \times \frac{W}{b} = \frac{4W}{b} \]

Since they complete the work in 10 days:
\[ \frac{6W}{a} + \frac{4W}{b} = W \]

Dividing the entire equation by W:
\[ \frac{6}{a} + \frac{4}{b} = 1 \quad \text{(Equation 1)} \]

### Step 3: Work Done in 10.5 Days When B Starts
- If B starts, they work for 10.5 days. B works for 6 days (2 days at a time for 5 cycles) and A works for 4.5 days (2 days at a time for 4 cycles and then half a day).

The work done by B in 6 days:
\[ \text{Work by B} = 6 \times \frac{W}{b} = \frac{6W}{b} \]

The work done by A in 4.5 days:
\[ \text{Work by A} = 4.5 \times \frac{W}{a} = \frac{4.5W}{a} \]

Since they complete the work in 10.5 days:
\[ \frac{6W}{b} + \frac{4.5W}{a} = W \]

Dividing the entire equation by W:
\[ \frac{6}{b} + \frac{4.5}{a} = 1 \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations
Now we have two equations:
1. \( \frac{6}{a} + \frac{4}{b} = 1 \)
2. \( \frac{6}{b} + \frac{4.5}{a} = 1 \)

From Equation 1, we can express \( \frac{4}{b} \):
\[ \frac{4}{b} = 1 - \frac{6}{a} \implies b = \frac{4a}{a - 6} \]

Substituting \( b \) in Equation 2:
\[ \frac{6(a - 6)}{4a} + \frac{4.5}{a} = 1 \]

Clearing the fractions and solving will yield the values of a and b.

### Step 5: Calculate the Time Taken When A and B Work Together
When A and B work together, their combined work rate is:
\[ \text{Combined Rate} = \frac{W}{a} + \frac{W}{b} = W \left( \frac{1}{a} + \frac{1}{b} \right) \]

The time taken to complete the work together is:
\[ T = \frac{W}{\left( \frac{W}{a} + \frac{W}{b} \right)} = \frac{1}{\left( \frac{1}{a} + \frac{1}{b} \right)} \]

Substituting the values of a and b will give the final answer.