Cambridge IGCSE Mathematics (0580) Study Notes

Chapter 1: Fractions, Decimals, and Percentages (The Foundation of Number)

Hello mathematicians! This chapter is absolutely fundamental to your entire IGCSE Maths course. Fractions, decimals, and percentages (FDP) are just three different ways of representing the same thing: parts of a whole. Mastering the links between them will make complex problems much easier. Don't worry if this seems tricky at first—we’ll break down all the conversions step-by-step!

I. Defining the Three Forms (C1.4/E1.4)

1. Fractions (Parts of a Whole)

A fraction represents how many parts we have out of the total number of parts. It has two main components:

  • Numerator: The top number (how many parts you have).
  • Denominator: The bottom number (the total number of parts in the whole).

We need to understand three main types of fractions:

  • Proper Fractions:

    Here, the numerator is smaller than the denominator. The value is always less than 1.
    Example: If you eat 3 slices of a pizza cut into 8 slices, you ate \( \frac{3}{8} \).

  • Improper Fractions:

    Here, the numerator is greater than or equal to the denominator. The value is 1 or more.
    Example: If you have 1 full pizza (8 slices) and 3 extra slices from a second pizza, you have \( \frac{11}{8} \) slices in total.

  • Mixed Numbers:

    These combine a whole number and a proper fraction.
    Example: \( 1 \frac{3}{8} \) (This is the same as \( \frac{11}{8} \)).

Conversion Tip: Mixed to Improper (and back!)

To convert \( 2 \frac{1}{3} \) to an improper fraction:
1. Multiply the whole number by the denominator: \( 2 \times 3 = 6 \).
2. Add the numerator: \( 6 + 1 = 7 \).
3. Put the result over the original denominator: \( \frac{7}{3} \).

2. Decimals (Place Value)

Decimals represent parts of a whole using powers of ten (tenths, hundredths, thousandths, etc.).

Example: \( 0.75 \) means \( 7 \) tenths and \( 5 \) hundredths, or simply \( \frac{75}{100} \).

3. Percentages (Out of 100)

The word "percent" literally means "per one hundred" (per centum in Latin). A percentage is a fraction where the denominator is fixed at 100.

Example: \( 62\% \) is the same as \( \frac{62}{100} \).

Did you know? Using percentages is very common in everyday life—think about calculating sales tax, discounts, or finding profit margins!

Key Takeaway for Section I: FDP are interchangeable representations of value. Understand what each form means before attempting conversions.

II. Essential Skills: Converting Between Forms (C1.4/E1.4)

The IGCSE syllabus requires you to be fluent in converting between all three forms.

A. Converting Fractions to Decimals and Percentages

The most reliable way to convert a fraction to a decimal is to remember that the fraction bar means division.

1. Fraction to Decimal (F \(\rightarrow\) D)

Method: Divide the numerator by the denominator.

Example 1: Convert \( \frac{3}{4} \).
\( 3 \div 4 = 0.75 \)

Example 2: Convert \( \frac{5}{8} \).
\( 5 \div 8 = 0.625 \)

2. Fraction to Percentage (F \(\rightarrow\) P)

Method: Convert to a decimal first, then multiply by 100.

Example: Convert \( \frac{3}{5} \).
Step 1: F \(\rightarrow\) D: \( 3 \div 5 = 0.6 \)
Step 2: D \(\rightarrow\) P: \( 0.6 \times 100 = 60 \)
Answer: \( 60\% \)

B. Converting Decimals to Fractions and Percentages
1. Decimal to Fraction (D \(\rightarrow\) F)

Method: Use place value. Write the decimal digits over the correct power of 10, then simplify the fraction.

Memory Aid:

  • 0.X $\rightarrow$ over 10
  • 0.XX $\rightarrow$ over 100
  • 0.XXX $\rightarrow$ over 1000

Example: Convert \( 0.45 \).
Step 1: \( 0.45 \) is 45 hundredths: \( \frac{45}{100} \)
Step 2: Simplify (divide top and bottom by 5): \( \frac{9}{20} \)

2. Decimal to Percentage (D \(\rightarrow\) P)

Method: Multiply by 100 (move the decimal point two places to the right).

Example: Convert \( 0.082 \).
\( 0.082 \times 100 = 8.2 \)
Answer: \( 8.2\% \)

C. Converting Percentages to Decimals and Fractions
1. Percentage to Decimal (P \(\rightarrow\) D)

Method: Divide by 100 (move the decimal point two places to the left).

Example: Convert \( 135\% \).
\( 135 \div 100 = 1.35 \)
(Note: A percentage over 100% simply means the value is greater than 1 whole.)

2. Percentage to Fraction (P \(\rightarrow\) F)

Method: Write the percentage over 100, then simplify.

Example: Convert \( 70\% \).
Step 1: Write as a fraction: \( \frac{70}{100} \)
Step 2: Simplify (divide top and bottom by 10): \( \frac{7}{10} \)

D. Simplifying Fractions (A crucial requirement)

When asked for a fraction answer, it must always be given in its simplest form unless otherwise stated. This means finding the largest number that divides evenly into both the numerator and the denominator (the HCF).

Example: Simplify \( \frac{18}{24} \).
The largest number that divides into both 18 and 24 is 6.
\( 18 \div 6 = 3 \)
\( 24 \div 6 = 4 \)
Simplified answer: \( \frac{3}{4} \)

QUICK REVIEW: Conversions Cheat Sheet

\(\mathbf{F \rightarrow D}\): Divide (Numerator \(\div\) Denominator)
\(\mathbf{D \rightarrow P}\): Multiply by 100
\(\mathbf{P \rightarrow D}\): Divide by 100
\(\mathbf{P \rightarrow F}\): Put over 100, then simplify

III. Ordering Quantities (C1.5/E1.5)

Sometimes you are asked to order a mixture of fractions, decimals, and percentages. To do this, you must convert them all into one uniform format, usually decimals.

Example: Order the following quantities from smallest to largest: \( \frac{2}{5} \), \( 0.45 \), \( 42\% \).

Step 1: Convert all to decimals:
1. \( \frac{2}{5} \): \( 2 \div 5 = 0.4 \)
2. \( 0.45 \): (Already a decimal)
3. \( 42\% \): \( 42 \div 100 = 0.42 \)

Step 2: Order the decimals:
\( 0.4, \quad 0.42, \quad 0.45 \)

Step 3: Write the final answer using the original forms:
Smallest to largest: \( \frac{2}{5}, \quad 42\%, \quad 0.45 \)

Remember the symbols for ordering:

  • \( \mathbf{=} \): Equal to
  • \( \mathbf{\ne} \): Not equal to
  • \( \mathbf{>} \): Greater than
  • \( \mathbf{<} \): Less than
  • \( \mathbf{\ge} \): Greater than or equal to
  • \( \mathbf{\le} \): Less than or equal to

Common Mistake to Avoid: When comparing fractions and decimals, students often get confused by the number of digits. Always line up the decimal places and compare place values.
For example: \( 0.40 \) is smaller than \( 0.45 \), but \( 0.45 \) is smaller than \( 0.5 \) (which is \( 0.50 \)).

Key Takeaway for Section II & III: Conversion is the key. To compare or order different number types, pick one format (usually decimal) and convert everything to it. Always simplify fractions unless told otherwise.

IV. Calculating with Fractions and Decimals (C1.6/E1.6)

While this section focuses on conversions, you must be prepared to use the four operations (+, -, \(\times\), \(\div\)) with all these number types, including negative numbers and mixed numbers.

Working with Decimals and Integers

Standard addition, subtraction, multiplication, and division rules apply. If you are doing calculations without a calculator (Paper 1/2), make sure you line up the decimal points for addition and subtraction!

Example: \( 3.5 + 1.25 = 4.75 \)

Working with Fractions

When dealing with fractions, especially mixed numbers, convert them to improper fractions before performing multiplication or division.

Addition/Subtraction: You MUST find a Common Denominator (LCM) before adding or subtracting.
Example: \( \frac{1}{3} + \frac{1}{4} \). The LCM of 3 and 4 is 12.
\( \frac{1 \times 4}{3 \times 4} + \frac{1 \times 3}{4 \times 3} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12} \)

Multiplication: Multiply the numerators together and the denominators together.
\( \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} \)

Division: "Keep, Change, Flip." Keep the first fraction, change division to multiplication, and flip (take the reciprocal of) the second fraction.
\( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \)

Remember BODMAS/BIDMAS!
Always ensure you follow the correct order of operations when dealing with mixed calculations (especially those involving brackets).

Key Takeaway for Section IV: Be systematic. Convert mixed numbers to improper fractions for multiplication and division, and find common denominators for addition and subtraction.