🔢 Chapter 1: Percentages (Number Section)

Hello future mathematician! This chapter is all about percentages. You might think percentages only appear in math class, but they are everywhere: calculating sale prices, checking your bank interest, or understanding statistics in the news. Mastery of percentages is crucial not just for your IGCSE exam (0607), but for real life!

We’ll break down every type of percentage problem step-by-step, making sure you feel confident with both calculator and non-calculator methods.
Let's dive in!

1. The Fundamental Concept of Percentages

What is a Percentage?

The word "percent" literally means "per hundred" or "out of one hundred". It’s simply a way of representing a fraction or a proportion where the denominator is always 100.

Example: 25% means 25 out of 100, or \(\frac{25}{100}\).

1.1 Converting Between Forms (C1.4.2 / E1.4.2)

Before you calculate anything, you must be comfortable converting percentages into decimals, as this is the quickest method for calculations.

Percentage ↔ Decimal Conversion: The Easy Rule
  • Percentage to Decimal: Divide by 100 (move the decimal point 2 places to the left).
    Example: \(45\% \rightarrow 45 \div 100 = 0.45\)
  • Decimal to Percentage: Multiply by 100 (move the decimal point 2 places to the right).
    Example: \(0.7 \rightarrow 0.7 \times 100 = 70\%\)
Converting to Fractions: Simplest Form (C1.4.2 / E1.4.2)

To convert a percentage to a fraction, put the percentage value over 100 and simplify:

Example: Convert \(30\%\) to a fraction.
\(\frac{30}{100} = \frac{3}{10}\)

⚠️ Common Mistake Alert!

Students often forget that percentages can be greater than 100%!
Example: \(150\%\) of a quantity is 1.5 times the original quantity.
\(150\% \rightarrow 150 \div 100 = 1.5\) (or \(\frac{3}{2}\))

Key Takeaway: Decimals are your best friend for quick percentage calculations. To convert to a decimal, just divide by 100!

2. Calculating Percentages (C1.12.1 / E1.12.1)

Calculating a given percentage of a quantity is the most basic skill. This often involves calculating things like a deposit, a discount, or earnings.

Step-by-Step Method

To find X% of a quantity Y:

  1. Convert the percentage X% into a decimal (or fraction).
  2. Multiply the decimal (or fraction) by the quantity Y.

Example: Calculate \(18\%\) of $450.

  • Step 1: Convert \(18\%\) to a decimal: \(18 \div 100 = 0.18\)
  • Step 2: Multiply: \(0.18 \times \$450 = \$81\)

"Did you know?" Mental Math Trick:

You can use simple percentages to find trickier ones:

  • To find \(10\%\), divide by 10. (e.g., \(10\%\) of 90 is 9).
  • To find \(1\%\), divide by 100. (e.g., \(1\%\) of 500 is 5).
  • To find \(15\%\), calculate \(10\%\) and add half of that value (\(5\%\)).

Key Takeaway: "Percentage of" usually means "Decimal times the quantity."

3. Expressing One Quantity as a Percentage of Another (C1.12.2 / E1.12.2)

This skill is used when you want to know what proportion one value represents compared to a total value, such as calculating your test score or market share.

The Test Score Analogy

Imagine you scored 38 marks out of a maximum of 50 on a test. You want to know your score as a percentage.

You need to find the fraction first, and then multiply by 100%.

\[ \text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100\% \]

Calculation:
\[ \text{Score Percentage} = \frac{38}{50} \times 100\% = 76\% \]

Example: A company made a profit of $5,000 last year on total revenue of $80,000. What percentage of the revenue was profit?

\[ \text{Profit Percentage} = \frac{\$5,000}{\$80,000} \times 100\% = 6.25\% \]

Key Takeaway: Always set up the calculation as a fraction first: (the part you are interested in) / (the original total).

4. Percentage Increase and Decrease (C1.12.3 / E1.12.3)

Calculating percentage change can be done in two ways, but the Multiplier Method is far more efficient, especially for complex problems.

Method A: Using the Change Amount (The Long Way)

  1. Calculate the actual increase or decrease amount.
  2. Divide the change amount by the original amount.
  3. Multiply by 100 to get the percentage.

\[ \text{Percentage Change} = \frac{\text{Change Amount}}{\text{Original Amount}} \times 100\% \]

Method B: Using Multipliers (The Efficient Way)

The multiplier is a single decimal number you multiply the original quantity by to get the final quantity immediately.

How to find the Multiplier (M):
  • For a Percentage Increase (e.g., VAT, Profit):
    \(M = 1 + \text{decimal change}\)
    Example: A \(10\%\) increase. \(M = 1 + 0.10 = 1.10\)
  • For a Percentage Decrease (e.g., Discount, Depreciation):
    \(M = 1 - \text{decimal change}\)
    Example: A \(25\%\) discount. \(M = 1 - 0.25 = 0.75\)

The Calculation:
\[ \text{New Amount} = \text{Original Amount} \times \text{Multiplier} \]

Example: A bicycle costing $320 is discounted by \(15\%\). Find the new price.

  • The decrease is \(15\%\), so the new amount is \(100\% - 15\% = 85\%\) of the original price.
  • Multiplier \(M = 0.85\).
  • New Price \( = 320 \times 0.85 = \$272\).

Key Takeaway: Use multipliers! They save time and are essential for interest and reverse percentage calculations.

5. Simple and Compound Interest (C1.12.4 / E1.12.4)

Interest calculations are crucial for understanding deposits, investments, and loans. Remember: Formulas for Simple and Compound Interest are NOT given in the syllabus. You must understand the method.

5.1 Simple Interest

Simple Interest means that the interest is calculated only on the original principal amount (P) each year. The interest earned is the same every year.

Simple Interest Calculation

Interest (I) is calculated using:
\[ I = \frac{PRT}{100} \]

Where P = Principal (starting amount), R = Rate (percentage per year), T = Time (in years).

Example: $500 is invested at \(4\%\) simple interest for 3 years.

1. Calculate the interest earned in one year:
\(500 \times 0.04 = \$20\) per year.

2. Calculate total interest for 3 years:
\(3 \times \$20 = \$60\)

3. Calculate the total amount (Principal + Interest):
\(\$500 + \$60 = \$560\)

5.2 Compound Interest (Repeated Percentage Change)

Compound Interest means that the interest is calculated on the principal plus any interest already earned. This is an example of repeated percentage change.

Analogy: Imagine a snowball rolling down a hill. It gets bigger each time, so it picks up even more snow on the next roll.

Compound Interest Calculation (Using the Multiplier)

Total Amount (A) after T years:
\[ A = P \times (M)^T \]

Where P = Principal, M = Multiplier (1 + rate as a decimal), T = Number of time periods.

Example: $500 is invested at \(4\%\) compound interest for 3 years.

  • Principal (P) = $500
  • Rate = \(4\%\), so Multiplier (M) = 1.04
  • Time (T) = 3 years

\[ A = 500 \times (1.04)^3 \]

\[ A = 500 \times 1.124864 \approx \$562.43 \]

Note: The compound interest earned ($62.43) is slightly higher than the simple interest ($60), demonstrating the power of compounding!

🌟 Quick Review: Multipliers and Interest

  • Simple Interest: Interest is calculated ONCE on the original P. \(I = PRT/100\).
  • Compound Interest: Interest is applied REPEATEDLY using the multiplier. \(A = P \times M^T\).
  • This formula also works for depreciation (a decrease), where M would be less than 1. (e.g., \(M = 0.9\) for \(10\%\) depreciation).

6. Reverse Percentages (Extended Content E1.12.5)

Don't worry if this seems tricky at first! Reverse percentages involve finding the original amount when you only know the final amount after a percentage change has occurred.

This is extremely common in price increase/decrease problems (e.g., finding the pre-sale price). The key is always to determine what percentage the final amount represents.

The Reverse Percentage Method

\[ \text{Original Amount} = \frac{\text{Final Amount}}{\text{Multiplier}} \]

Step 1: Identify the percentage change and determine the final percentage (relative to the original \(100\%\)).

Step 2: Convert this final percentage into the multiplier (M).

Step 3: Divide the final amount by the multiplier.

Example 1: Finding the Original Price (Decrease)
A shirt is sold for $48 after a \(20\%\) discount. What was the original price?

  1. Final Percentage: \(100\% - 20\% = 80\%\).
  2. Multiplier (M): \(0.80\).
  3. Original Price \( = \frac{\$48}{0.80} = \$60\).

Example 2: Finding the Pre-Tax Price (Increase)
A receipt shows that a laptop cost $952, including \(19\%\) VAT (tax). Find the cost before VAT.

  1. Final Percentage: \(100\% + 19\% = 119\%\).
  2. Multiplier (M): \(1.19\).
  3. Original Cost \( = \frac{\$952}{1.19} = \$800\).
⛔ Common Reverse Percentage Error

Do NOT calculate \(20\%\) of the final price and add it back!

In Example 1, \(20\%\) of $48 is $9.60. If you add this back ($48 + $9.60 = $57.60), you get the wrong original price because the \(20\%\) discount was applied to the *original* $60, not the discounted $48.

Always use the multiplier method to work backwards!

Key Takeaway: Reverse percentages use division by the multiplier to undo the percentage change.

Final Chapter Summary

  • To find X% of Y, use the decimal multiplier: \(Y \times (X/100)\).
  • To express A as a percentage of B: \(\frac{A}{B} \times 100\%\).
  • For percentage change problems, the Multiplier Method is the fastest and most reliable approach.
  • Simple Interest adds the same amount yearly: \(A = P + I\).
  • Compound Interest uses repeated multiplication: \(A = P \times M^T\).
  • Reverse Percentages (Extended): Divide the final amount by the multiplier: \(\text{Original} = \frac{\text{Final}}{M}\).