A and B do a work individually on alternate days. If A start the work, whole work is completed in 17 days. But when B start the work, work is completed in `17(2)/(3)` days. In how many days, A and B together can do the work?

Options

A. `8(3)/(7)`
B. `8(1)/(5)`
C. `8(2)/(5)`
D. `8(3)/(4)`

Answer

To solve the problem, we need to determine how many days A and B together can complete the work. Let's break it down step by step.

### Step 1: Determine the work done by A and B individually.

Let the total work be represented as 1 unit.

1. **When A starts the work:**
- A works on the 1st day, B on the 2nd day, A on the 3rd day, and so on.
- A completes the work in 17 days.
- In these 17 days, A works for 9 days (1st, 3rd, 5th, ..., 17th) and B works for 8 days (2nd, 4th, 6th, ..., 16th).

Let A's work per day be \( a \) and B's work per day be \( b \).

From the information given:
\[ 9a + 8b = 1 \quad \text{(Equation 1)} \]

2. **When B starts the work:**
- B works on the 1st day, A on the 2nd day, B on the 3rd day, and so on.
- The work is completed in \( 17 \frac{2}{3} \) days, which is \( \frac{53}{3} \) days.
- In these \( \frac{53}{3} \) days, B works for 9 days (1st, 3rd, ..., 17th) and A works for 8 days.

Thus:
\[ 9b + 8a = 1 \quad \text{(Equation 2)} \]

### Step 2: Solve the equations.

We have the two equations:
1. \( 9a + 8b = 1 \)
2. \( 9b + 8a = 1 \)

We can solve these equations simultaneously.

From Equation 1:
\[ 9a + 8b = 1 \quad \text{(Multiply by 8)} \]
\[ 72a + 64b = 8 \quad \text{(Equation 3)} \]

From Equation 2:
\[ 9b + 8a = 1 \quad \text{(Multiply by 9)} \]
\[ 72b + 64a = 9 \quad \text{(Equation 4)} \]

Now we can rearrange Equation 3 and Equation 4:
- From Equation 3: \( 72a + 64b = 8 \)
- From Equation 4: \( 64a + 72b = 9 \)

### Step 3: Eliminate one variable.

Multiply Equation 3 by 9 and Equation 4 by 8:
\[ 648a + 576b = 72 \quad \text{(Equation 5)} \]
\[ 512a + 576b = 72 \quad \text{(Equation 6)} \]

Now, subtract Equation 6 from Equation 5:
\[ (648a - 512a) + (576b - 576b) = 72 - 72 \]
\[ 136a = 0 \implies a = 0 \]

This indicates we made an error in the elimination process. Let's go back and solve the equations directly.

### Step 4: Solve for a and b directly.

From Equation 1:
\[ 9a + 8b = 1 \implies 8b = 1 - 9a \implies b = \frac{1 - 9a}{8} \]

Substituting \( b \) in Equation 2:
\[ 9\left(\frac{1 - 9a}{8}\right) + 8a = 1 \]
\[ \frac{9 - 81a}{8} + 8a = 1 \]
Multiply through by 8:
\[ 9 - 81a + 64a = 8 \]
\[ -17a = -1 \implies a = \frac{1}{17} \]

Now substitute \( a \) back to find \( b \):
\[ b = \frac{1 - 9(\frac{1}{17})}{8} = \frac{1 - \frac{9}{17}}{8} = \frac{\frac{8}{17}}{8} = \frac{1}{17} \]

### Step 5: Calculate combined work.

Now we have:
- \( a = \frac{1}{17} \)
- \( b = \frac{1}{17} \)

The combined work done by A and B in one day:
\[ a + b = \frac{1}{17} + \frac{1}{17} = \frac{2}{17} \]

### Step 6: Calculate the total time taken by A and B together.

To find the time taken by A and B together to complete the work:
\[ \text{Time} = \frac{1 \text{ unit of work}}{\frac{2}{17}} = \frac{17}{2} = 8.5 \text{ days} \]

### Final Answer

Thus, A and B together can complete the work in **8.5 days**.