A and B working together can complete a work in 10 days. If A works at half of his efficiency and B works at thrice of his efficiency, they can complete the work in 6 days. In how many days can A complete the work at his normal efficiency?

Sure, let's solve this step by step:

Step 1: Let's denote the work done by A in one day as 'a' and the work done by B in one day as 'b'. So, according to the first sentence, A and B together can complete the work in 10 days. This can be written as:

10(a + b) = 1 (work)

Step 2: According to the second sentence, if A works at half of his efficiency and B works at thrice of his efficiency, they can complete the work in 6 days. This can be written as:

6((1/2)a + 3b) = 1 (work)

Step 3: Now we have two equations, and we can solve them to find the values of 'a' and 'b'.

From the first equation, we get a + b = 1/10.

From the second equation, we get (1/2)a + 3b = 1/6.

Step 4: Multiply the second equation by 2, we get a + 6b = 1/3.

Step 5: Subtract the first equation from the new second equation, we get 5b = 1/3 - 1/10 = 1/15.

So, b = 1/75.

Step 6: Substitute b = 1/75 into the first equation, we get a = 1/10 - 1/75 = 2/30 = 1/15.

Step 7: Now we know that A can complete 1/15 of the work in one day at his normal efficiency. So, to complete the whole work, A will need 15 days.