Q.Define a partnership and distinguish between a simple partnership and a compound partnership.

A partnership is a jointly-run business whose profit is shared in an agreed ratio; the simple/compound distinction turns on whether capitals are invested for equal or unequal times.

In a simple partnership every partner's capital stays in for the same period, so time cancels and only the capitals matter; in a compound partnership the times differ, so each partner's capital must be multiplied by its own time.

Partnership. A partnership is a form of business in which two or more persons, called partners, agree to run a business together and to share its profits and losses in an agreed proportion known as the profit-sharing ratio. The business itself is called the firm.

Simple partnership. In a simple partnership, every partner's capital remains invested in the business for the same length of time (for instance, the whole year). Because the time is common to all partners, it has no effect on the comparison between them and cancels out. Profit is therefore divided directly in the ratio of the capitals: if the capitals are $C_1, C_2, \dots$, the profit ratio is $C_1 : C_2 : \dots$.

Compound partnership. In a compound partnership, the partners' capitals remain invested for different lengths of time. Time can no longer be ignored, so each partner's capital is multiplied by the time for which it was invested, and profit is divided in the ratio of these products: for capitals $C_1, C_2, \dots$ invested for times $t_1, t_2, \dots$, the profit ratio is $C_1 t_1 : C_2 t_2 : \dots$.

The link between them. A simple partnership is really the special case of a compound partnership in which all the times are equal: $C_1 t : C_2 t : \dots = C_1 : C_2 : \dots$, so the common time cancels and the ratio reduces to the ratio of capitals. This is why the compound rule (capital × time) can be regarded as the single master rule of the whole topic.

Students sometimes describe the distinction as being about the amount of capital rather than the time — but both simple and compound partnerships can involve unequal capitals; what separates them is whether the periods of investment are equal (simple) or different (compound).