A responsibility centre that is held accountable for controlling both costs and income is called a:

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A profit centre is a responsibility centre that is accountable for both costs and income, with the ultimate goal of generating profit. It is responsible for generating revenue and minimizing expenses by making decisions about pricing, sales, production, and cost control. Profit centres are often found in businesses where managers have been given considerable autonomy to make decisions and reduce costs in pursuing the company's goals.

A Responsibility Centre That Is Held Accountable for Controlling Both