Welcome to Discrete Growth and Decay!
Hi there! In this chapter, we are going to explore how things change over time. Whether it's your savings account getting bigger or a new car losing its value, we use Discrete Growth and Decay to calculate exactly what's happening.
Don't worry if this seems a bit "maths-heavy" at first. We’ll break it down into simple steps, using real-life examples like money and shopping to make sense of it all!
1. The Secret Weapon: Multipliers
Before we dive in, we need to master one quick skill: the multiplier. This is a single number that does the work of a percentage increase or decrease in one simple multiplication.
How to find a multiplier:
- For Growth (Increase): Start with \(100\%\), add the percentage, then divide by \(100\).
Example: A \(5\%\) increase. \(100\% + 5\% = 105\%\). The multiplier is \(1.05\). - For Decay (Decrease): Start with \(100\%\), subtract the percentage, then divide by \(100\).
Example: A \(12\%\) decrease. \(100\% - 12\% = 88\%\). The multiplier is \(0.88\).
Quick Review Box:
\(10\%\) increase \(\rightarrow\) Multiplier is \(1.10\)
\(10\%\) decrease \(\rightarrow\) Multiplier is \(0.90\)
2. Simple Interest
Simple Interest is the "straightforward" version of growth. You calculate the interest based only on the original amount you started with. It stays the same every single year.
Analogy: Imagine you have a library fine of \(\$1\) per day. It doesn't matter how long you've had the book; the fine only grows by that same \(\$1\) every day. It's predictable!
Step-by-Step: How to calculate Simple Interest
- Find the percentage of the original amount.
- Multiply that amount by the number of years (or time intervals).
- Add this to the original amount to find the total.
Example: You invest \(\$200\) at \(3\%\) simple interest for \(4\) years.
\(3\%\) of \(\$200 = 0.03 \times 200 = \$6\) interest per year.
In \(4\) years: \(4 \times \$6 = \$24\).
Total amount: \(\$200 + \$24 = \$224\).
Key Takeaway: Simple interest is like a set of stairs—you go up by the exact same amount every step.
3. Compound Interest (Growth)
Compound Interest is much more exciting (and profitable!). This is "interest on interest." At the end of the first year, the interest is added to your account. In the second year, you earn interest on your original money and on the interest from the first year.
Analogy: The Snowball Effect. As a snowball rolls down a hill, it picks up more snow. Because it's bigger, it picks up even more snow on the next turn. It grows faster and faster!
The Step-by-Step Method (Foundation)
If the percentages change each year, just multiply step-by-step.
Example: A house worth \(\$200,000\) grows by \(10\%\) in year one and \(5\%\) in year two.
Year 1: \(\$200,000 \times 1.10 = \$220,000\)
Year 2: \(\$220,000 \times 1.05 = \$231,000\)
The Formula Method (Higher Tier)
When the percentage is the same every year, we use powers to save time:
\( \text{Total Amount} = \text{Initial Amount} \times (\text{Multiplier})^n \)
...where \(n\) is the number of years.
Key Takeaway: Compound interest grows exponentially. The longer you leave it, the faster it earns!
4. Depreciation (Decay)
Depreciation is just growth in reverse. It's when something loses value over time. Most "things" we buy (like cars or phones) depreciate.
Syllabus Example: The Depreciating Car
A car is worth \(\$15,000\) new. It depreciates by \(30\%\) in the first year, \(20\%\) in the second, and \(15\%\) in the third.
Let's solve this step-by-step using multipliers:
- Year 1 multiplier (\(30\%\) loss): \(0.70\)
- Year 2 multiplier (\(20\%\) loss): \(0.80\)
- Year 3 multiplier (\(15\%\) loss): \(0.85\)
Calculation: \(\$15,000 \times 0.70 \times 0.80 \times 0.85 = \$7,140\).
Wow! The car lost over half its value in just three years.
Key Takeaway: For decay, your multiplier will always be less than 1 (because you are keeping less than \(100\%\) of the value).
5. Formula Summary (Higher Tier Focus)
If you are aiming for the Higher Tier, you should be comfortable writing these as a general formula. For an initial amount \(P\) and a rate \(r\) (as a decimal):
Exponential Growth: \( A = P(1 + r)^n \)
Exponential Decay: \( A = P(1 - r)^n \)
Did you know?
Albert Einstein reportedly called compound interest "the eighth wonder of the world." He said, "He who understands it, earns it; he who doesn't, pays it!"
Common Mistakes to Avoid
- Confusing Simple and Compound: Always read the question carefully. If it says "Simple," don't use powers!
- Incorrect Multipliers for Decay: If a value drops by \(20\%\), the multiplier is \(0.8\), NOT \(0.2\). Remember: you are calculating what is left, not what was lost.
- Order of Operations: When using the formula \( P \times R^n \), always do the power (\(R^n\)) before multiplying by \(P\).
Final Quick Review
Simple Interest: Fixed amount added every time. Based on the start value.
Compound Interest: Growing amount added. Multiplier is greater than \(1\).
Depreciation: Value dropping over time. Multiplier is less than \(1\).
Multipliers: The easiest way to do percentage changes in one go!
Don't worry if this feels like a lot to take in right now. The best way to get better is to try a few practice questions. You've got this!