In the decay formula y = a(1 - r)^t, what does r represent?

• A. Initial amount
• B. Time
• C. Final amount
• D. Rate of decay

Answer: (D)

The key idea is what the parameter r means in an exponential decay model. In y = a(1 - r)^t, a is the starting amount and t is the number of time periods. The factor (1 − r) is the amount that remains after each period, so r is the rate of decay expressed as a decimal. It tells you how much of the starting amount is lost each period; for example, if r = 0.2, you keep 0.8 of the amount every period, so the value multiplies by 0.8 each time. After t periods you get a(0.8)^t, and so on. If r were 0, nothing decays; if r is large, more is lost each period. The final amount y is whatever remains after t periods, and is not r itself nor the initial amount a, nor the time t.