Which expression represents continuously compounded interest growth of principal P at rate r over time t?

Which expression represents continuously compounded interest growth of principal P at rate r over time t?

• A. A = P (1 + r)^t
• B. A = P e^(r t)
• C. A = P (1 + r t)
• D. A = P / (1 + r)^t

Answer: (B)

Continuous compounding means the amount grows at every instant, so the rate of change of the amount is proportional to the current amount: dA/dt = rA. Solving this differential equation gives A(t) = P e^(rt), where P is the initial principal and t is time. The exponential factor e^(rt) captures growth that happens instantaneously and continuously, which is why this form is used for continuous compounding. This is different from discrete compounding, where the amount after t periods is A = P(1 + r)^t; that expression only exactly describes growth when interest is added at separate intervals and can approximate continuous growth only as compounding frequency becomes infinite. Simple interest, A = P(1 + rt), grows linearly rather than exponentially. The other option, P/(1 + r)^t, would decrease with time for positive r, which doesn’t represent growth.

Continuous compounding means the amount grows at every instant, so the rate of change of the amount is proportional to the current amount: dA/dt = rA. Solving this differential equation gives A(t) = P e^(rt), where P is the initial principal and t is time. The exponential factor e^(rt) captures growth that happens instantaneously and continuously, which is why this form is used for continuous compounding.

This is different from discrete compounding, where the amount after t periods is A = P(1 + r)^t; that expression only exactly describes growth when interest is added at separate intervals and can approximate continuous growth only as compounding frequency becomes infinite. Simple interest, A = P(1 + rt), grows linearly rather than exponentially. The other option, P/(1 + r)^t, would decrease with time for positive r, which doesn’t represent growth.