A can lay railway track between two given stations in 22 days and B can do the same job in 11 days. With the help of C, they did the job in 5 days only. Then, C alone can do the job in

The correct option is 110/7.

Step 1: Determine individual work rates per day.
Let the total work of laying the railway track be represented as 1 unit.

A can complete the work in 22 days, so A's work done in 1 day is:

$\text{A's 1 day work}=\frac{1}{22}$

B can complete the work in 11 days, so B's work done in 1 day is:

$\text{B's 1 day work}=\frac{1}{11}$

Step 2: Determine combined work rate.
With the help of C, A, B, and C together completed the work in 5 days. Therefore, the combined 1 day work of (A + B + C) is:

$\text{(A + B + C)'s 1 day work}=\frac{1}{5}$

Step 3: Calculate C's work rate per day.
C's 1 day work can be found by subtracting A's and B's 1 day work from the combined 1 day work:

$\text{C's 1 day work}=\frac{1}{5}-\left(\frac{1}{22}+\frac{1}{11}\right)$

First, simplify the sum of A's and B's work:

$\frac{1}{22}+\frac{1}{11}=\frac{1}{22}+\frac{2}{22}=\frac{3}{22}$

Now, subtract this from the combined rate:

$\text{C's 1 day work}=\frac{1}{5}-\frac{3}{22}$

Find a common denominator for 5 and 22, which is 110:

$\text{C's 1 day work}=\frac{22}{110}-\frac{15}{110}=\frac{7}{110}$

Step 4: Calculate the total time C takes alone.
Since C completes $\frac{7}{110}$ of the work in 1 day, the total number of days C takes to complete the entire job alone is the reciprocal of C's 1 day work:

$\text{Days taken by C alone}=\frac{1}{7/110}=\frac{110}{7}\text{days}$