**Solution:**

Let's assume that the total work to be completed is represented by W.

**Step 1: Calculate the work done by A and B in one day**

We need to find out how much work A and B can complete in one day.

Given that A can complete the work in 18 days, we can say that A can complete 1/18th of the work in one day.

Similarly, B can complete 1/40th of the work in one day as B can complete the work in 40 days.

**Step 2: Calculate the work done in alternate days**

In this scenario, A and B are working on alternate days. So, let's calculate the work done in one cycle of two days.

On the first day, A will work and complete 1/18th of the work.

On the second day, B will work and complete 1/40th of the work.

Therefore, in two days, the total work completed is (1/18 + 1/40) of the total work.

**Step 3: Calculate the total cycles required to complete the work**

To find out the total number of cycles required to complete the work, we divide the total work (W) by the work done in one cycle (1/18 + 1/40).

Total Cycles = W / (1/18 + 1/40)

**Step 4: Calculate the total days required to complete the work**

As we know that one cycle consists of two days, we can multiply the total number of cycles by 2 to get the total number of days required to complete the work.

Total Days = Total Cycles * 2

**Step 5: Simplify the equation and find the solution**

Let's simplify the equation to find the solution.

Total Cycles = W / (1/18 + 1/40)

By finding the least common multiple of 18 and 40, we can rewrite the equation as follows:

Total Cycles = W / (40/360 + 18/360)

Total Cycles = W / (58/360)

Simplifying further, we get:

Total Cycles = (360 * W) / 58

Total Days = Total Cycles * 2

Total Days = (720 * W) / 58

Therefore, the work will be completed in (720 * W) / 58 days.