Compound Growth and Decay

Understanding Compound Growth and Decay

  • Compound growth and decay is a logical progression of the principles of ratios and proportions. It is used in various subjects, including maths, physics, and economics, to describe a process that increases or decreases exponentially.

  • Compound growth, also called exponential growth, means a quantity increases by a fixed percentage rate from one period to the next. Real-world examples can include population growth or interest on a savings account.

  • Compound decay, also known as exponential decay, refers to a quantity that decreases by a fixed percentage rate. For instance, depreciation of a car’s value or the radioactive decay of elements.

Calculating Compound Growth

  • The formula for calculating compound growth is: A = P(1 + r/n)^(nt) Where:
    • A is the amount of money accumulated after n years, including interest.
    • P is the principal amount (the initial amount of money).
    • r is the annual interest rate (decimal).
    • n is the number of times that interest is compounded per year.
    • t is the time the money is invested or borrowed for, in years.
  • Remember to convert the percentage rate into a decimal in your calculations. For example, a growth rate of 5% is equal to 0.05 in decimal form.

Calculating Compound Decay

  • The formula for calculating compound decay, especially useful for depreciation, is: A = P(1 - r)^(t) Where:
    • A is the final amount remaining after the decay.
    • P is the principal amount (the initial quantity).
    • r is the rate of decay per period (decimal form).
    • t is the time period over which the decay is taking place.

Applying Compound Growth and Decay Formulae

  • Take care to correctly identify the initial amount (P), the growth or decay rate (r), and the time period (t) in the problem scenario and then substitute these values into the appropriate formula.

  • Always review your answers to make sure they’re reasonable in the context of the problem. For instance, you should not end up with a negative value when dealing with quantities like population or money.

Graphical Representation of Compound Growth and Decay

  • Compound growth can be represented graphically as an upward-curving line that becomes steeper as time progresses. The steeper the curve, the higher the growth rate.

  • Compound decay is represented by a downward-curving line that becomes less steep as time progresses. The steeper the initial decline, the higher the rate of decay.