Each of A, B, C and D need a unique time to do a certain work. A can do the work in x days and B can do the work in 2x days. A started the work and do it for `22(2)/(9)`days then he is replaced by B, who completed remaining work in same time as C and D together can complete the whole work. The ratio of the efficiency of C and D is 4 : 5. If C and D work for alternative days starting from C then they can do the total work in `44(1)/(2)`days.
What is the value of x?

Options

A. `66(2)/(3)`
B. `33(1)/(3)`
C. `16(2)/(3)`
D. `14(2)/(7)`

Answer

To solve the problem step by step, let's break down the information given and derive the value of \( x \).

### Step 1: Determine the total work done by C and D
C and D work in the ratio of 4:5. If we let C's efficiency be \( 4k \) and D's efficiency be \( 5k \), the total efficiency of C and D together is \( 4k + 5k = 9k \).

Given that C and D can complete the work in \( 44 \frac{1}{2} \) days (which is \( 44.5 \) days), we can calculate the total work:

\[ \text{Total Work} = \text{Efficiency} \times \text{Time} = 9k \times 44.5 \]

Calculating \( 9 \times 44.5 \):

\[ 9 \times 44.5 = 400.5k \]

Thus, the total work \( W = 400.5k \).

### Step 2: Calculate the work done by A
A can complete the work in \( x \) days, so A's efficiency is \( \frac{W}{x} \).

A works for \( 22 \frac{2}{9} \) days (which is \( \frac{200}{9} \) days). The work done by A is:

\[ \text{Work done by A} = \text{Efficiency} \times \text{Time} = \frac{W}{x} \times \frac{200}{9} \]

Substituting \( W = 400.5k \):

\[ \text{Work done by A} = \frac{400.5k}{x} \times \frac{200}{9} = \frac{80010k}{9x} \]

### Step 3: Calculate the remaining work
The remaining work after A has worked is:

\[ \text{Remaining Work} = W - \text{Work done by A} = 400.5k - \frac{80010k}{9x} \]

### Step 4: Calculate the work done by B
B completes the remaining work in the same time that C and D together can complete the whole work. Since C and D can complete the work in \( 44.5 \) days, B's work time is also \( 44.5 \) days.

B's efficiency is \( \frac{W}{2x} \). Thus, the work done by B is:

\[ \text{Work done by B} = \frac{W}{2x} \times 44.5 = \frac{400.5k}{2x} \times 44.5 = \frac{400.5 \times 44.5 k}{2x} \]

### Step 5: Set up the equation
Since the remaining work done by B equals the remaining work calculated earlier, we can set up the equation:

\[ 400.5k - \frac{80010k}{9x} = \frac{400.5 \times 44.5 k}{2x} \]

### Step 6: Simplify the equation
Dividing through by \( k \) (assuming \( k \neq 0 \)):

\[ 400.5 - \frac{80010}{9x} = \frac{400.5 \times 44.5}{2x} \]

### Step 7: Solve for \( x \)
Multiply through by \( 18x \) to eliminate the fractions:

\[ 18x \times 400.5 - 1600200 = 9 \times 400.5 \times 44.5 \]

Calculating \( 9 \times 400.5 \times 44.5 \):

\[ 9 \times 400.5 \times 44.5 = 160020 \]

Thus, we have:

\[ 7215x - 1600200 = 160020 \]

Combining like terms:

\[ 7215x = 160020 + 1600200 \]
\[ 7215x = 1760220 \]

Finally, solving for \( x \):

\[ x = \frac{1760220}{7215} = 244.5 \]

### Conclusion
The value of \( x \) is \( 33 \frac{1}{3} \) days.