Introduction to Growth and Decay

In this chapter, we explore how quantities change over time. Whether it is the money in a savings account growing or the value of a new car dropping, these changes often happen at a percentage rate. We will learn how to calculate these changes efficiently using multipliers and look at how mathematicians use repetitive steps, called iteration, to solve complex equations.

1. Understanding Multipliers

To master growth and decay, you must be comfortable with percentage multipliers. A multiplier is a single number (usually a decimal) that you multiply by to change a quantity.

  • For Growth (Increase): Add the percentage to \(100\%\) and convert to a decimal. Example: A \(5\%\) increase is \(100\% + 5\% = 105\%\), which is a multiplier of \(1.05\).
  • For Decay (Decrease): Subtract the percentage from \(100\%\) and convert to a decimal. Example: A \(12\%\) decrease is \(100\% - 12\% = 88\%\), which is a multiplier of \(0.88\).

Quick Tip: If the multiplier is greater than 1, the value is growing. If it is less than 1, the value is decaying (shrinking).

2. Compound Growth and Interest

In the "Percentages and Financial Maths" chapter, you may have seen simple interest, where the amount added stays the same every year. However, in compound interest, you earn interest on your interest!

The Edexcel formula sheet provides this formula for the total amount accrued:

\(\text{Total accrued} = P(1 + \frac{r}{100})^n\)

Where:

  • \(P\) is the Principal (the starting amount).
  • \(r\) is the percentage rate of interest.
  • \(n\) is the number of time periods (usually years).
  • \((1 + \frac{r}{100})\) is simply your multiplier.

Example: Savings Account

Question: You invest \(\$2000\) at a compound interest rate of \(3\%\) per year. How much is in the account after \(5\) years?

Step 1: Find the multiplier. \(100\% + 3\% = 103\%\), which is \(1.03\).
Step 2: Apply the formula. \(2000 \times 1.03^5\).
Step 3: Calculate. \(2000 \times 1.1592... = \$2318.55\) (rounded to 2 decimal places for money).

3. Compound Decay (Depreciation)

Growth isn't always about money going up. Depreciation is when the value of something (like a car or a computer) decreases over time by a certain percentage.

The method is the same, but your multiplier will be less than 1.

Example: Car Value

Question: A car is bought for \(\$15,000\). It depreciates by \(20\%\) each year. What is its value after \(3\) years?

Step 1: Find the multiplier. \(100\% - 20\% = 80\%\), which is \(0.8\).
Step 2: Set up the calculation. \(15000 \times 0.8^3\).
Step 3: Calculate. \(15000 \times 0.512 = \$7680\).

Common Mistake to Avoid: Don't just multiply the percentage by the number of years (e.g., \(20\% \times 3 = 60\%\)). This would be simple interest, not compound decay. In compound decay, the amount lost gets smaller each year because the car is worth less than it was the year before!

4. Iterative Processes (Higher Tier Only)

Iteration means repeating a process over and over again. In maths, we use an iterative formula to find an approximate solution to an equation that might be too hard to solve normally.

An iterative formula looks like this: \(x_{n+1} = f(x_n)\).
This just means: "To find the next value (\(x_{n+1}\)), plug your current value (\(x_n\)) into the formula."

How to perform iteration on a calculator:

  1. Type your starting value (usually called \(x_0\)) into your calculator and press [=].
  2. Type the iterative formula, but instead of \(x_n\), use the [ANS] button.
  3. Press [=] to get \(x_1\).
  4. Press [=] again to get \(x_2\), again for \(x_3\), and so on.

Example: Solve \(x^3 - 5x + 1 = 0\) using the iteration \(x_{n+1} = \frac{x_n^3 + 1}{5}\) starting with \(x_0 = 0\).

  • Press \(0\) then [=].
  • Type \(\frac{ANS^3 + 1}{5}\) and press [=].
  • \(x_1 = 0.2\)
  • \(x_2 = 0.2016\)
  • \(x_3 = 0.20164... \)

Notice how the numbers start to settle (converge) around a specific value. This value is the approximate solution to the equation.

Key Takeaways
  • Multipliers: Growth = \(1 + decimal\); Decay = \(1 - decimal\).
  • Compound Interest: Use the formula \(P \times Multiplier^n\).
  • Iteration (H): Use the [ANS] button on your calculator to repeat the formula and find where the sequence "settles."
  • Units: Always round money to \(2\) decimal places unless told otherwise.

Don't worry if iteration feels strange at first! It is just a "number loop" that gets you closer and closer to the right answer. Practice using the [ANS] button, and it will become one of your favorite calculator tricks.