IGCSE Mathematics (0580) Study Notes: Percentages (Number Section)

Hello! Welcome to your guide on Percentages. This topic is one of the most practical and frequently tested areas in the IGCSE curriculum. Whether you are calculating a sale discount or understanding bank interest, percentages are essential. Mastery here builds confidence for many other topics!

Don’t worry if this seems tricky at first. We will break down every concept step-by-step, ensuring you understand the 'why' behind the 'how'.

1. The Fundamentals of Percentages

What Does 'Percent' Mean?

The word Percent literally means "per one hundred" (per centum in Latin). The symbol (\%) tells us we are dealing with a fraction out of 100.

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

Converting Between Forms

To use percentages effectively in calculations, you often need to convert them into decimals or fractions.

A. Percentage to Decimal:

To change a percentage to a decimal, simply divide by 100 (move the decimal point 2 places to the left).

  • \(40\% = 40 \div 100 = 0.40\)
  • \(3\% = 3 \div 100 = 0.03\)
  • \(150\% = 150 \div 100 = 1.5\) (Yes, percentages can be over 100!)

B. Percentage to Fraction:

Write the number over 100 and simplify the fraction to its lowest terms.

  • \(50\% = \frac{50}{100} = \frac{1}{2}\)
  • \(75\% = \frac{75}{100} = \frac{3}{4}\)

Memory Aid: The D/P Slide Trick

If you have a Decimal, slide the point 2 places right to get a Percentage. If you have a Percentage, slide the point 2 places left to get a Decimal.

Quick Review: Core Conversions

It's helpful to know these common conversions immediately:

  • \(\frac{1}{4} = 0.25 = 25\%\)
  • \(\frac{1}{2} = 0.5 = 50\%\)
  • \(\frac{3}{4} = 0.75 = 75\%\)
  • \(\frac{1}{10} = 0.1 = 10\%\)

2. Calculating a Given Percentage of a Quantity (C1.12.1)

This is the most common type of percentage calculation. The easiest and most efficient way to do this is using the decimal multiplier.

Step-by-Step Method (Using Decimals)

  1. Convert the percentage to a decimal.
  2. Multiply the quantity by this decimal.

Example: Calculate 18% of $500.

  • Step 1: Convert 18% to a decimal: \(18\% = 0.18\)
  • Step 2: Multiply: \(500 \times 0.18 = 90\)

Answer: 18% of $500 is $90.

Real-World Contexts: Discounts and Earnings

If you see a discount, you are calculating a percentage of the original price (the reduction amount).

Example: A shirt costs $45 and is on sale for 30% off (the discount).
Discount amount = \(45 \times 0.30 = 13.5\)
The saving is $13.50.

Did you know? Percentage calculations are crucial for understanding earnings, especially calculating tax deductions or commission (a percentage of sales earned).

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

Sometimes you need to know what percentage one number represents compared to a total or original number.

The core idea is to create a fraction, and then convert that fraction to a percentage.

The Formula

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

Analogy: Think about a test score. The 'Part' is the marks you got, and the 'Whole' is the total marks possible.

Example: In a class of 30 students, 6 students wear glasses. What percentage of the students wear glasses?

  • Part = 6
  • Whole = 30
  • Calculation: \(\frac{6}{30} \times 100\%\)
  • Simplifying the fraction first: \(\frac{1}{5} \times 100\% = 20\%\)

Answer: 20% of the students wear glasses.

4. Percentage Increase and Decrease (C1.12.3)

Calculating how a quantity changes over time (like inflation, price hikes, or depreciation) is done using percentage change.

Method A: Using the 'Change'

$$ \text{Percentage Change} = \frac{\text{Change (difference)}}{\text{Original Amount}} \times 100\% $$

We use the Original Amount in the denominator for standard percentage change questions.

Example: A house was bought for $200 000 and sold later for $250 000. What is the percentage profit (increase)?

  • Change (Profit) = \(250 000 - 200 000 = 50 000\)
  • Original Amount = \(200 000\)
  • Calculation: \(\frac{50 000}{200 000} \times 100\% = 0.25 \times 100\% = 25\%\)

Method B: Using the Multiplier (The Best Way for Complex Problems)

Using multipliers allows you to find the final amount in one step and is essential for repeated changes (like compound interest).

1. Percentage Increase:

If you increase a quantity by \(P\%\), the new amount is \(100\% + P\%\) of the original.

  • Multiplier Trick: Add the percentage increase to 100% and convert to a decimal.
  • To increase by 10%, the multiplier is \(1.10\) (from \(100\% + 10\% = 110\% \rightarrow 1.10\)).
  • New Value = Original Value \(\times\) Multiplier

Example: Increase $80 by 15%.

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

2. Percentage Decrease (Discount/Loss):

If you decrease a quantity by \(P\%\), the new amount is \(100\% - P\%\) of the original.

  • Multiplier Trick: Subtract the percentage decrease from 100% and convert to a decimal.
  • To decrease by 20%, the multiplier is \(0.80\) (from \(100\% - 20\% = 80\% \rightarrow 0.80\)).

Example: A phone valued at $450 loses 8% of its value (depreciation).

  • Multiplier: \(100\% - 8\% = 92\% = 0.92\)
  • Calculation: \(450 \times 0.92 = 414\)
  • Answer: The new value is $414. (This covers profit and loss as an amount or a percentage).
🔑 Key Takeaway: Multipliers

Using multipliers saves time and reduces errors:

  • Increase by 23%: Multiplier = \(1.23\)
  • Decrease by 7%: Multiplier = \(0.93\)
  • Increase by 150%: Multiplier = \(2.50\) (This links to calculations involving percentages over 100%).

5. Financial Mathematics: Simple and Compound Interest (C1.12.4)

These calculations apply percentage changes to money, often involving deposits or investments.

Important Note: The syllabus states that formulas for interest are not given in the examination papers. You must understand the method and be able to derive or apply the calculation structure.

A. Simple Interest

Simple Interest is calculated only on the original amount (the principal) you invested or borrowed. The interest earned is the same every period.

Formula Structure (Derive this!):
$$ \text{Simple Interest} = \text{Principal} \times \text{Rate} \times \text{Time} $$

Example: Calculate the simple interest earned on $1000 invested at 5% per year for 3 years.

  1. Calculate 5% of the principal: \(1000 \times 0.05 = 50\) (Interest earned per year).
  2. Multiply by the number of years: \(50 \times 3 = 150\)

Answer: The total simple interest is $150.

B. Compound Interest and Repeated Percentage Change

Compound Interest is interest calculated on the initial principal AND all accumulated interest from previous periods. The money grows faster because you earn "interest on interest."

This is an example of Repeated Percentage Change (Extended syllabus E1.12.4 includes this, but the Core syllabus C1.12.4 also expects understanding of compound interest).

Formula Structure (Using Multipliers):
$$ \text{Final Amount} = \text{Initial Amount} \times (\text{Multiplier})^{\text{Number of Periods}} $$

Example: $5000 is invested at 4% compound interest per year for 5 years. Find the final amount.

  • Initial Amount = 5000
  • Rate = 4% increase $\rightarrow$ Multiplier = 1.04
  • Time (Periods) = 5 years
  • Calculation: \(\text{Final Amount} = 5000 \times (1.04)^5\)
  • \(\text{Final Amount} = 5000 \times 1.21665... \approx 6083.26\)

Answer: The final amount is $6083.26 (Money answers should generally be given to 2 decimal places).

Tip for Depreciation (Loss of Value)

The compound interest formula structure also works for calculating depreciation (decrease in value).

If a car depreciates by 10% per year, the multiplier is 0.90. After 4 years: \(\text{Value} = \text{Original} \times (0.90)^4\).

6. Reverse Percentages (Extended Content Only - E1.12.5)

Reverse percentages involve finding the original amount before a percentage increase or decrease was applied.

Struggling students often try to find the percentage of the final amount and add/subtract it—this is the common mistake to avoid!

The Key Concept

We know that: Original Amount \(\times\) Multiplier = Final Amount.
To find the original amount, we must divide:

$$ \text{Original Amount} = \frac{\text{Final Amount}}{\text{Multiplier}} $$

Step-by-Step Method

Example: A dress is sold for $144 after a 20% discount. What was the original price?

  1. Identify the Multiplier: A 20% discount means the final price is \(100\% - 20\% = 80\%\) of the original. Multiplier = \(0.80\).
  2. Set up the Equation: We know Original \(\times 0.80 = 144\).
  3. Calculate the Original: Divide the final amount by the multiplier.
    \(\text{Original} = \frac{144}{0.80}\)
  4. Solve: \(144 \div 0.80 = 180\)

Answer: The original price was $180.

Real-World Analogy: Finding the price before tax (VAT/GST).

Example: A bill is $63.60, which includes 6% tax.

  • Final amount represents \(100\% + 6\% = 106\%\) of the original price.
  • Multiplier = 1.06.
  • Original Price = \(\frac{63.60}{1.06} = 60\)

The price before tax was $60.

Final Study Tip

The best way to master percentages is to always ask yourself: "Does my answer make sense?" If you calculate a 10% discount on a $100 item and get $110, you know immediately you made a mistake (the price should decrease!). Always estimate first!