Identify that will lead to dissolution of partnership

The correct option is: When there is change in existing profit-sharing ratio among partners

To understand why this option is correct, it is essential to distinguish between the dissolution of a partnership and the dissolution of a firm under partnership law:
1. Dissolution of Partnership: This refers to the termination of the original agreement between the partners and the reconstitution of the firm. While the relationship between the partners changes, the business of the firm continues under a new agreement. A change in the profit-sharing ratio among existing partners is a classic example of reconstitution (dissolution of partnership), as it terminates the old partnership agreement and brings a new one into effect without closing down the business.
2. Dissolution of Firm: This involves the complete breakdown of the relation between all the partners, leading to the winding up of the business, realization of assets, and settlement of liabilities.

Let's analyze the other options:
- When the business of the firm becomes illegal: This makes it unlawful to carry on the business, leading to compulsory dissolution of the firm.
- When a partner becomes insane: This is a ground on which the court may order the dissolution of the firm.
- When a partner persistently commits breach of partnership agreement: This is also a ground for the court to decree the dissolution of the firm.

Therefore, a change in the existing profit-sharing ratio among partners leads to the dissolution of the partnership (reconstitution of the firm) rather than the dissolution of the firm itself.