Final answer:
True, the composition of partner's equity does not change when a new partner directly purchases an existing partner's interest, as the overall equity of the business remains the same.
Step-by-step explanation:
The statement "The admission of a partner does not change the composition of partner's equity if the new partner purchases the old partner's interest by paying the old partner directly" is generally true. When a new partner buys out an existing partner's interest, the equity of the business as a whole does not change, because the transaction is between individuals outside of the business accounts. For example, if the value of a partner's equity in the business is $250,000, and a new partner pays this amount to the old partner, the total equity remains unchanged, although ownership has transferred.