Two technicians, A and B, can each complete a maintenance task alone in 14 days. They begin the task together, but B does not work on every third day. How many days will they need to finish the task?

The correct option is 8 days.

To find the total number of days needed to complete the maintenance task, we can analyze the daily work rates of technician A and technician B step-by-step.

Step 1: Calculate individual daily work rates
Since each technician can complete the task alone in 14 days, their individual work rate per day is:

$\text{Rate of Technician A}=\frac{1}{14}\text{of the task per day}$

$\text{Rate of Technician B}=\frac{1}{14}\text{of the task per day}$

Step 2: Determine the work done in a 3-day cycle
Technician B does not work on every third day (Day 3, Day 6, etc.). This forms a repeating 3-day cycle:

• Day 1: Both A and B work.
$\text{Work done on Day 1}=\frac{1}{14}+\frac{1}{14}=\frac{2}{14}$

• Day 2: Both A and B work.
$\text{Work done on Day 2}=\frac{1}{14}+\frac{1}{14}=\frac{2}{14}$

• Day 3: Technician B takes the day off, so only technician A works.
$\text{Work done on Day 3}=\frac{1}{14}$

Total work completed in one 3-day cycle is:

$\text{Work in 3 days}=\frac{2}{14}+\frac{2}{14}+\frac{1}{14}=\frac{5}{14}$

Step 3: Calculate work completed over multiple cycles
In 2 full 3-day cycles (6 days total):

$\text{Work done in 6 days}=2\times \frac{5}{14}=\frac{10}{14}$

The remaining portion of the task after 6 days is:

$\text{Remaining work}=1-\frac{10}{14}=\frac{4}{14}$

Step 4: Calculate the remaining days
On Day 7, both technicians work together:

$\text{Work done on Day 7}=\frac{2}{14}$

$\text{Remaining work after Day 7}=\frac{4}{14}-\frac{2}{14}=\frac{2}{14}$

On Day 8, both technicians work together again:

$\text{Work done on Day 8}=\frac{2}{14}$

$\text{Remaining work after Day 8}=\frac{2}{14}-\frac{2}{14}=0$

Therefore, the task is completely finished at the end of the 8th day.

Thus, Technicians A and B will need 8 days to finish the task.