Introduction to Percentages, Growth, and Decay

Percentages are one of the most useful parts of maths because we see them every day—from "20% off" sales in shops to the interest rates on bank accounts. In this chapter, we will learn how to calculate percentage changes, work backward to find original values, and understand how things grow or shrink over time using growth and decay models.

Don't worry if percentages have felt confusing before. We are going to break everything down into simple steps, focusing on using multipliers, which is the "secret weapon" for making these problems much easier.

1. Percentage Basics: "Parts per Hundred"

The word "percentage" literally means "out of 100." So, \(45\%\) is just another way of writing the fraction \(\frac{45}{100}\) or the decimal \(0.45\).

Using Multipliers

To find a percentage of an amount, the quickest way is to use a decimal multiplier. You find this by dividing the percentage by \(100\).

  • To find \(7\%\), the multiplier is \(0.07\)
  • To find \(15\%\), the multiplier is \(0.15\)
  • To find \(120\%\), the multiplier is \(1.2\)

Step-by-step Example: Calculate \(18\%\) of \(\$600\).
1. Convert \(18\%\) to a decimal: \(18 \div 100 = 0.18\).
2. Multiply the amount by this decimal: \(600 \times 0.18 = 108\).
3. The answer is \(\$108\).

2. Percentage Increase and Decrease

When something grows or shrinks by a percentage, we can do it in one single step using a multiplier.

Percentage Increase

Think of the original amount as \(100\%\). If we increase it by \(5\%\), we now have \(105\%\) of the original.

The Rule: Add the percentage to \(100\%\), then convert to a decimal multiplier.

Example: Increase \(\$80\) by \(15\%\).
\(100\% + 15\% = 115\%\). Multiplier = \(1.15\).
Calculation: \(80 \times 1.15 = 92\). Answer: \(\$92\).

Percentage Decrease

If we decrease an amount by \(20\%\), we are left with \(80\%\) of the original (\(100\% - 20\% = 80\%\)).

The Rule: Subtract the percentage from \(100\%\), then convert to a decimal multiplier.

Example: A \(\$200\) coat is in a \(30\%\) off sale.
\(100\% - 30\% = 70\%\). Multiplier = \(0.70\).
Calculation: \(200 \times 0.7 = 140\). Answer: \(\$140\).

Quick Review: To increase, the multiplier is greater than \(1\). To decrease, the multiplier is less than \(1\).

3. Expressing One Quantity as a Percentage of Another

Sometimes you need to find out what percentage a score or a value represents. For example, if you get \(42\) out of \(60\) in a test.

The Method: Write the numbers as a fraction, then multiply by \(100\).
\(\frac{\text{Part}}{\text{Whole}} \times 100\)

Example: What is \(12\) as a percentage of \(40\)?
\(\frac{12}{40} = 0.3\).
\(0.3 \times 100 = 30\%\).

4. Reverse Percentages (Original Value Problems)

This is a common exam topic! This is when you are told the new value after a change, and you need to find the original value.

Common Mistake: Never try to calculate the percentage of the new price and add/subtract it. It won't work because the percentage was originally calculated on the old price.

The Trick: Find the multiplier and divide by it.

Example: A car's value decreased by \(20\%\) and is now worth \(\$8,000\). What was its original price?
1. A \(20\%\) decrease means a multiplier of \(0.80\).
2. Let the original price be \(x\). So, \(x \times 0.8 = 8000\).
3. To find \(x\), divide: \(8000 \div 0.8 = 10000\).
4. The original price was \(\$10,000\).

5. Growth and Decay: Simple vs. Compound Interest

In real-world finance, we often deal with interest over several years.

Simple Interest

Interest is calculated only on the original amount every year. The amount of interest stays the same each year.

Example: \(\$500\) invested at \(3\%\) simple interest for \(4\) years.
\(3\%\) of \(\$500 = 15\).
Over \(4\) years: \(15 \times 4 = 60\).
Total: \(500 + 60 = \$560\).

Compound Interest (Growth)

Interest is calculated on the new balance each year. This means you earn "interest on your interest." For your exams, you are given a formula on the formula sheet!

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

  • \(P\) = The starting amount (Principal)
  • \(r\) = The percentage interest rate
  • \(n\) = The number of time periods (usually years)

Example: \(\$2,000\) is invested at \(4\%\) compound interest for \(3\) years.
Multiplier = \(1.04\).
Calculation: \(2000 \times (1.04)^3\).
Using a calculator: \(2000 \times 1.124864 = \$2,249.73\).

Compound Decay (Depreciation)

This is when a value falls by a percentage every year (like a car losing value). We use the same idea, but the multiplier will be less than \(1\).

Example: A machine worth \(\$5,000\) depreciates by \(10\%\) each year. What is it worth after \(2\) years?
Multiplier = \(0.90\).
Calculation: \(5000 \times (0.90)^2 = 5000 \times 0.81 = \$4,050\).

6. Higher Tier Only: Iterative Processes

For Higher tier students, growth and decay can be seen as an iterative process. This simply means a process that repeats over and over. Each "output" becomes the "input" for the next step.

If a population \(P\) grows by \(2\%\) every year, we can write a formula for the next year (\(P_{n+1}\)) based on the current year (\(P_n\)):
\(P_{n+1} = 1.02 \times P_n\)

By repeating this calculation, we can model long-term growth trends.

Key Takeaways Summary

  • Multipliers are the most efficient way to solve percentage problems.
  • Increase: \(Original \times (1 + \text{decimal percentage})\).
  • Decrease: \(Original \times (1 - \text{decimal percentage})\).
  • Reverse Percentages: Always divide by the multiplier to find the original value.
  • Compound Interest: Use the formula \(P(1 + \frac{r}{100})^n\) for growth over time.

Did you know? Banks use compound interest, which is why debt can grow so quickly if not paid off, but it's also why saving money early in life is a great idea!