Introduction: Mastering the Parts of a Whole

Welcome to one of the most important chapters in your IGCSE Mathematics course! In this section, we are looking at fractions, decimals, ratios, and percentages. While they might look different, they are actually just different ways of describing the same thing: parts of a whole. Think of it like describing the same amount of water as "half a bottle" (fraction), "0.5 litres" (decimal), or "50%" (percentage).

In your exam, you will need to switch between these forms effortlessly and use them to solve real-world problems, such as sharing money or calculating a discount. Don't worry if these have been tricky before—we will break them down step-by-step!

1. Fractions: The Building Blocks

A fraction represents a part of a whole, written as \(\frac{\text{numerator}}{\text{denominator}}\).

The Four Operations with Fractions

Addition and Subtraction: To add or subtract fractions, you must have a common denominator.
Example: \(\frac{1}{4} + \frac{2}{3}\)
1. Find a common denominator (the LCM of 4 and 3 is 12).
2. Convert: \(\frac{1 \times 3}{4 \times 3} = \frac{3}{12}\) and \(\frac{2 \times 4}{3 \times 4} = \frac{8}{12}\).
3. Add: \(\frac{3+8}{12} = \frac{11}{12}\).

Multiplication: This is the easiest! Just multiply the top numbers and multiply the bottom numbers.
Example: \(\frac{2}{5} \times \frac{3}{4} = \frac{2 \times 3}{5 \times 4} = \frac{6}{20}\). Always simplify your answer: \(\frac{6}{20} = \frac{3}{10}\).

Division: Use the "Keep, Change, Flip" rule.
1. Keep the first fraction.
2. Change the \(\div\) to \(\times\).
3. Flip the second fraction upside down.
Example: \(\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}\).

Quick Review: Always check if your final answer can be simplified by dividing the top and bottom by the same number!

2. Decimals and Rounding

Decimals are another way to show parts of a whole using a place-value system based on 10.

Operations with Decimals

When adding or subtracting, the most important rule is to line up the decimal points. If the numbers have different lengths, you can add "placeholder" zeros at the end.
Example: \(12.5 + 3.07 \rightarrow 12.50 + 3.07 = 15.57\).

Rounding: Accuracy is Key

The syllabus requires you to round to a specific number of Decimal Places (d.p.) or Significant Figures (s.f.).

Decimal Places: Count the digits after the decimal point.
Example: Round \(5.6782\) to \(2\) d.p. \(\rightarrow\) Look at the 3rd digit (\(8\)). Since it is \(5\) or more, round up to \(5.68\).

Significant Figures: Start counting from the first non-zero digit.
Example: Round \(0.004562\) to \(2\) s.f. \(\rightarrow\) The first significant figure is \(4\). The second is \(5\). Look at the next digit (\(6\)). Round up to \(0.0046\).

3. Interchange: Switching Forms

You need to be able to "swap" between these forms depending on what the question asks. Here is a quick guide:

  • Fraction to Decimal: Divide the top by the bottom. \(\frac{3}{4} = 3 \div 4 = 0.75\).
  • Decimal to Percentage: Multiply by \(100\). \(0.85 \times 100 = 85\%\).
  • Percentage to Fraction: Put the number over \(100\) and simplify. \(40\% = \frac{40}{100} = \frac{2}{5}\).
  • Fraction to Percentage: Change the fraction to a decimal, then multiply by \(100\).

Key Takeaway: If a question involves different types (e.g., a fraction and a percentage), convert them all to the same form (usually decimals) first to make comparing them easier!

4. Ratio and Proportion

Ratios compare two or more quantities. In this course, you will see ratios with up to three parts, such as \(a : b : c\).

Simplifying Ratios

Divide all parts by the same number.
Example: \(10 : 15 : 25\) can be simplified by dividing by \(5\) to get \(2 : 3 : 5\).

Sharing in a Ratio

To share a total amount into a ratio, follow these steps:
1. Add the total number of parts.
2. Divide the total amount by the number of parts to find the value of "one share".
3. Multiply that value by each number in the ratio.
Example: Share \(\$60\) in the ratio \(1 : 2 : 3\).
Total parts: \(1 + 2 + 3 = 6\).
Value of one share: \(\$60 \div 6 = \$10\).
Shares: \(1 \times 10 = \$10\), \(2 \times 10 = \$20\), \(3 \times 10 = \$30\).

Proportion

Direct proportion means as one thing increases, the other increases at the same rate.
Example: If \(5\) pens cost \(\$10\), how much do \(8\) pens cost?
First, find the cost of one (the unitary method): \(10 \div 5 = \$2\).
Then, multiply for the new amount: \(8 \times 2 = \$16\).

5. Percentages

Percentages are parts of \(100\).

Percentage Change

To find the percentage increase or decrease, use this formula:
\(\text{Percentage Change} = \frac{\text{Change}}{\text{Original Amount}} \times 100\)

Common Mistake: Students often divide by the new amount. Always divide by the original starting value!

Reverse Percentages

This is used when you know the value after a percentage change and need to find the original value.
Example: A coat is on sale for \(\$72\) after a \(10\%\) discount. What was the original price?
1. If there was a \(10\%\) discount, \(\$72\) represents \(90\%\) of the original price (\(100\% - 10\% = 90\%\)).
2. Find \(1\%\): \(72 \div 90 = 0.8\).
3. Find \(100\%\) (the original): \(0.8 \times 100 = \$80\).

Summary: Tips for Success

1. Show your working: Even if you use a calculator, write down the steps like the common denominator or the ratio parts.
2. Read the rounding instructions: Check if the question asks for \(3\) significant figures or \(2\) decimal places.
3. Sanity Check: If you are calculating a sale price, make sure it is lower than the original! If you are sharing money, make sure the individual parts add up to the original total.

Note: For related topics like Standard Form or Upper and Lower Bounds, please refer to the specific chapters within the "Number" section.