Compound Growth and Decay

Understanding Compound Growth and Decay

  • Compound growth, also known as exponential growth, is when an amount is multiplied by a constant factor for every unit increase in the variable.
  • Compound decay, or exponential decay, is similar but the amount is divided by a constant factor for each unit increase in the variable.
  • It’s essential to note that in compound growth and decay, the change is a percentage of the current amount, not a fixed amount.
  • Compound growth can be used to calculate values in a variety of real-world scenarios like investments, populations, compound interest and more.
  • Compound decay is commonly used in depreciation of value, decay of radioactive substances, or any scenario where decrease occurs at a constantly changing rate.

Compound Growth and Decay Formulas

  • An essential formula for compound growth is A = P(1 + r/n)^(nt), where:
    • A is the final amount
    • P is the principal amount (initial amount)
    • r is the annual interest rate (in decimal form)
    • n is the number of times interest is compounded per year
    • t is the time the money is invested for (in years)
  • Compound decay can be calculated using A = P(1 - r/n)^(nt). It’s the same formula as compound growth but note the minus sign in the parenthesis.

Negative Exponents and Compound Decay

  • Negative exponents represent division by a base number’s positive exponent. So, a^-n equals 1/a^n.
  • Negative exponents are often seen in compound decay problems due to the “dividing by a constant factor” nature of the process.

Calculation Examples and Practice Problems

  • Calculate the final amount (A) when £1000 is deposited in a bank account that compounds interest at a rate of 5% per annum, yearly for 3 years.
    • Solution: A = 1000(1 + 5/100)^3 = £1157.63
  • A car’s value depreciates by 20% each year. If the initial value was £15,000, what would be its value after 5 years?
    • Solution: A = 15000(1 - 20/100)^5 = £4915.20

Final Notes

  • Ensure to convert any percentage given into a decimal form when inserting values into the formula.
  • Remember the difference between simple and compound growth or decay. Simple refers to a fixed amount of growth or decay, while compound refers to a changing growth or decay rate.
  • Work out many examples and practice questions to ensure a thorough understanding of how to handle problems relating to compound growth and decay.
  • Always bear in mind that any process that multiplies or divides by a constant factor at regular intervals is likely to involve compound growth or decay.