Solution:

Given, K, L and M are partners sharing in the ratio of 3:2:1.

Let their initial investments be 3x, 2x, and x respectively.

They admit N for 1/6th share, i.e., N invests x/6.

The total investment after N's admission = 3x + 2x + x + x/6 = 22x/6 = 11x/3.

The new profit sharing ratio will be calculated based on the new investment.

K's share = 3x/11x/3 = 9/11

L's share = 2x/11x/3 = 6/11

M's share = x/11x/3 = 3/11

N's share = x/6/11x/3 = 1/11

New profit sharing ratio:

K: L: M: N = 9:6:3:1

Explanation:

Initial Investment: The initial investment of the partners is given in the ratio of 3:2:1. This means that K invested 3 parts, L invested 2 parts and M invested 1 part of the total investment.

Admission of N: N is admitted for 1/6th share, which means that N invests x/6, where x is the total investment.

Total Investment: After N's admission, the total investment will be 3x + 2x + x + x/6 = 22x/6 = 11x/3.

Calculation of New Profit Sharing Ratio: The new profit sharing ratio is calculated based on the new investment. K, L, M and N's share in the profit will be in the ratio of their investment.

New Profit Sharing Ratio: The new profit sharing ratio of K, L, M and N is 9:6:3:1 respectively.

Conclusion:

The admission of a new partner changes the profit sharing ratio among the existing partners. The new profit sharing ratio is calculated based on the new investment after the admission of the new partner. The new partner's share in the profit is based on their investment in the partnership.

New profit sharing ratio among k,l,m and n is 5:3:2:2