Explanation:

Given, A, B and C are partners sharing profits in the ratio of 1/2, 1/3 and 1/6 respectively. B retires and his share is taken by A and C in the ratio of 5:3.

Step 1: Finding the share ratio of A, B and C before retirement
- Let the total profit be x
- A's share = 1/2x
- B's share = 1/3x
- C's share = 1/6x
- Total share ratio = 1/2 : 1/3 : 1/6 = 3 : 2 : 1

Step 2: Finding the share of B after retirement
- B's share is taken by A and C in the ratio of 5:3
- Let the share of B be y
- A's share = 5/8y (5/8 = 5/8 + 3/8)
- C's share = 3/8y (3/8 = 3/8 + 5/8)

Step 3: Finding the new share ratio of A, B and C after retirement
- A's new share = A's previous share + B's share taken by A
= 1/2x + 5/8y
- C's new share = C's previous share + B's share taken by C
= 1/6x + 3/8y
- Total share ratio = (1/2x + 5/8y) : 0 : (1/6x + 3/8y)
= (4x + 5y) : 0 : (2x + 3y)
= 4x + 5y : 0 : 2x + 3y

Step 4: Equating the share ratios before and after retirement
- The share ratios before and after retirement should be equal
- 3 : 2 : 1 = 4x + 5y : 0 : 2x + 3y

Step 5: Solving for x and y
- Equating the corresponding ratios, we get:
4x + 5y = 3k
0 = 2k
2x + 3y = k
where k is a constant
- Solving these equations, we get:
x = k/11
y = 2k/11

Step 6: Finding the new share ratio of A and C after retirement
- Substituting the values of x and y, we get:
A's new share = (4x + 5y)/(4x + 2y + 5y) = 20/33
C's new share = (2x + 3y)/(4x + 2y + 5y) = 13/33

Step 7: Checking the answer
- Verifying the answer, we get:
A's share ratio = 1/2 : 5/8 = 4 : 5
C's share ratio = 1/6 : 3/8 = 1 : 4
- The new share ratio of A and C after retirement is 20 : 13, which is in the ratio of 4 : 5, as expected.