Introduction to Exponential Growth and Decay
Have you ever heard of a video "going viral" or a population "exploding" in size? In Mathematics, we describe these rapid changes using Exponential Growth. On the flip side, when things vanish quickly—like the value of a new car or the amount of medicine in your bloodstream—we call it Exponential Decay.
This topic is part of the Extended syllabus (Section 1.17). It is closely linked to Compound Interest (Section 1.13), but it applies to much more than just money! Don't worry if it looks intimidating at first; once you master the "Multiplier Method," these problems become much easier to solve.
What Does "Exponential" Actually Mean?
In simple growth (like simple interest), you add the same amount every time. In exponential growth, the amount added gets bigger every time because it is calculated based on the new total, not the starting one. It's like a snowball rolling down a hill—the bigger it gets, the more snow it picks up!
Key Differences
Exponential Growth: The quantity increases by a fixed percentage over a specific period. (e.g., a population growing by \(3\%\) every year).
Exponential Decay: The quantity decreases by a fixed percentage over a specific period. (e.g., a radioactive substance losing \(5\%\) of its mass every hour).
The Universal Formula
To solve these problems, we use one main formula. You don't need to learn fancy constants like \(e\) for this syllabus; you just need to understand how percentages work.
The formula for the value after \(n\) periods is:
\( \text{Final Value} = \text{Initial Amount} \times (1 \pm \frac{r}{100})^n \)
Breaking Down the Formula:
1. Initial Amount: The starting value (sometimes called the Principal).
2. \(r\): The percentage rate of growth or decay.
3. \(n\): The number of time periods (years, hours, days, etc.).
4. The Sign (\(\pm\)): Use \(+\) for growth and \(-\) for decay.
The "Multiplier" Shortcut
Most successful students prefer using the multiplier method. Instead of writing out the fraction every time, you can convert the percentage change into a single decimal.
For Growth (Increase): Add the percentage to \(100\%\) and convert to a decimal.
Example: A \(6\%\) increase means you now have \(106\%\) of the original. Multiplier = \(1.06\).
\( \text{Final Value} = \text{Initial Amount} \times 1.06^n \)
For Decay (Decrease): Subtract the percentage from \(100\%\) and convert to a decimal.
Example: A \(4\%\) decrease means you have \(96\%\) left. Multiplier = \(0.96\).
\( \text{Final Value} = \text{Initial Amount} \times 0.96^n \)
Step-by-Step Examples
Example 1: Exponential Growth (Population)
The population of a town is \(5000\). It is expected to grow by \(2\%\) every year. Calculate the population after \(3\) years.
Step 1: Identify the values.
Initial Amount = \(5000\)
Rate (\(r\)) = \(2\%\)
Time (\(n\)) = \(3\)
Step 2: Find the multiplier.
Growth of \(2\%\) means \(100\% + 2\% = 102\%\).
Decimal = \(1.02\)
Step 3: Plug into the formula.
\(5000 \times 1.02^3\)
Step 4: Calculate.
\(5000 \times 1.061208 = 5306.04\)
Since we are talking about people, we round to the nearest whole number: \(5306\) people.
Example 2: Exponential Decay (Depreciation)
A laptop costs \(\$800\). Its value decreases (depreciates) by \(15\%\) each year. Work out its value after \(4\) years.
Step 1: Identify the values.
Initial Amount = \(\$800\)
Rate (\(r\)) = \(15\%\)
Time (\(n\)) = \(4\)
Step 2: Find the multiplier.
Decay of \(15\%\) means \(100\% - 15\% = 85\%\).
Decimal = \(0.85\)
Step 3: Plug into the formula.
\(800 \times 0.85^4\)
Step 4: Calculate.
\(800 \times 0.52200625 = 417.605\)
Rounding to \(2\) decimal places for money: \(\$417.61\).
Common Mistakes to Avoid
1. Mixing up Simple and Exponential: If the question says "compounds," "per year," or "exponentially," use the powers/indices. If it says "simple interest," don't use powers!
2. Rounding too early: Keep the full number in your calculator until the very end. If you round the multiplier, your final answer will be slightly off.
3. Time units: Ensure the rate (\(r\)) and the time (\(n\)) use the same units. If the rate is "per month," then \(n\) must be the number of months.
Quick Review Checklist
• Growth: Use \( (1 + \frac{r}{100}) \) or a multiplier \( > 1 \).
• Decay: Use \( (1 - \frac{r}{100}) \) or a multiplier \( < 1 \).
• Calculation: Always apply the power (\(n\)) to the multiplier before multiplying by the initial amount.
• Context: Check if your answer makes sense. (Is it bigger for growth? Smaller for decay?)
• Accuracy: Give non-exact answers to \(3\) significant figures unless the question is about money (use \(2\) decimal places).
Summary Key Takeaway
Exponential growth and decay happen when a value changes by a percentage of its current state. Use the formula \( \text{Final} = \text{Initial} \times (\text{Multiplier})^n \) to solve these efficiently on your Paper 4 calculator exam.