Understanding the problem:
We are given a set of 10 positive integers and we need to find the new average when each number in the set is increased by 3. We are also given that the average of the original set is A.

Approach:
To find the new average, we need to understand the relationship between the sum of the original set, the sum of the new set, and the number of elements in each set.

Step 1: Finding the sum of the original set:
To find the sum of the original set, we need to multiply the average (A) by the number of elements (10). This can be represented as:
Sum of original set = A * 10

Step 2: Finding the sum of the new set:
To find the sum of the new set, we need to consider that each number in the original set is increased by 3. So, we need to add 3 to each number and then find the sum. This can be represented as:
Sum of new set = (Number 1 + 3) + (Number 2 + 3) + ... + (Number 10 + 3)

Step 3: Finding the new average:
To find the new average, we need to divide the sum of the new set by the number of elements in the set (which remains the same). This can be represented as:
New average = (Sum of new set) / 10

Step 4: Simplifying the equation:
To simplify the equation, we can expand the sum of the new set and then simplify it further. Let's consider the expanded form:
Sum of new set = Number 1 + 3 + Number 2 + 3 + ... + Number 10 + 3

Step 5: Simplifying further:
By rearranging the terms, we can write:
Sum of new set = (Number 1 + Number 2 + ... + Number 10) + (3 + 3 + ... + 3)
= Sum of original set + 3 * 10
= (A * 10) + 30

Step 6: Finding the new average:
Substituting the value of the sum of the new set in the equation for the new average, we get:
New average = ((A * 10) + 30) / 10
= A + 3

Conclusion:
The new average, when each number in the set is increased by 3, is A + 3.