Fractions

1. What is a Fraction?

A fraction represents a part of a whole. It is written as a/b, where:

• a is the numerator (the part we have).
• b is the denominator (the total number of equal parts).

Example: In 3/5, the whole is divided into 5 equal parts, and we have 3 of them.

2. Types of Fractions

Proper Fractions

Numerator < denominator. Value < 1.

• Examples: 2/3, 5/8, 7/12.

Improper Fractions

Numerator >= denominator. Value >= 1.

• Examples: 7/4, 11/8, 9/9.

Mixed Fractions

A whole number and a proper fraction combined.

• Example: 2 1/3 means 2 + 1/3.

Conversion: Improper to Mixed

11/4 = Divide 11 by 4: quotient = 2, remainder = 3.
Mixed fraction = 2 3/4.

Conversion: Mixed to Improper

3 2/5 = (3 x 5 + 2) / 5 = (15 + 2) / 5 = 17/5.

Common Mistake: Writing 3 2/5 as (3 x 2 + 5)/5. Always multiply the whole number by the denominator first.

Exam Focus (2 marks): 'Convert 17/3 to a mixed fraction.'

17/3 = 5 remainder 2, so the answer is 5 2/3.

3. Equivalent Fractions

Equivalent fractions represent the same value. Multiply OR divide numerator and denominator by the same number (not zero).

1/2 = 2/4 = 3/6 = 4/8 = 5/10

Worked Example: Find the missing number: 3/7 = ?/28.

To get 28 from 7, multiply by 4. So numerator = 3 x 4 = 12. Answer: 12/28.

Worked Example: Reduce 18/24 to its simplest form.

Divide numerator and denominator by 6: 18/24 = 3/4. In simplest form = 3/4.

Checking Equivalence: Cross-Multiplication

a/b = c/d if and only if a x d = b x c.

Example: Are 4/7 and 12/21 equivalent?
4 x 21 = 84, 7 x 12 = 84. Yes, they are equivalent!

4. Comparing Fractions

Like Fractions (same denominator)

Compare numerators directly: 5/8 > 3/8 because 5 > 3.

Unlike Fractions (different denominators)

Method 1: Find LCM of denominators, convert to like fractions, compare.
Method 2: Cross-multiply.

Worked Example: Compare 3/5 and 5/8.

Cross-multiplication: 3 x 8 = 24, 5 x 5 = 25.
Since 24 < 25, 3/5 < 5/8.

Common Mistake: Comparing numerators and denominators separately, e.g., thinking 3/5 > 2/3 because 5 > 3. Always use cross-multiplication or LCM.

5. Addition and Subtraction of Fractions

Adding Like Fractions

Add the numerators, keep the denominator the same.

3/7 + 2/7 = (3 + 2)/7 = 5/7.

Adding Unlike Fractions

Find LCM, convert, then add.

Worked Example: 2/3 + 3/4.

LCM of 3 and 4 = 12.
2/3 = 8/12, 3/4 = 9/12.
8/12 + 9/12 = 17/12 = 1 5/12.

Adding Mixed Fractions

Method 1: Convert to improper, then add.
Method 2: Add whole parts separately, add fractions separately.

Worked Example: 2 1/4 + 3 2/5.

Method 2: Whole: 2 + 3 = 5. Fractions: 1/4 + 2/5 = (5 + 8)/20 = 13/20.
Answer: 5 13/20.

Common Mistake: Forgetting to add the whole number parts. Always handle wholes and fractions separately.

6. Comparison Table: Fraction Types

7. Subtraction of Fractions

Worked Example: 5/6 - 1/4.

LCM of 6 and 4 = 12.
5/6 = 10/12, 1/4 = 3/12.
10/12 - 3/12 = 7/12.

8. Self-Test

1. Convert 37/5 to a mixed fraction.
2. Convert 6 2/9 to an improper fraction.
3. Find the equivalent: 4/9 = ?/54.
4. Reduce 36/48 to simplest form.
5. Compare using >, <, or =: 5/6 and 7/9.
6. Add: 2 1/2 + 3 2/3.
7. Subtract: 9/4 - 3/8.
8. Are 6/15 and 8/20 equivalent? Justify.

9. Answers to Self-Test

1. 37/5 = 7 2/5.
2. 6 2/9 = (54 + 2)/9 = 56/9.
3. 4/9 = 24/54 (multiply by 6).
4. 36/48 = 3/4 (divide by 12).
5. 5 x 9 = 45, 6 x 7 = 42. 45 > 42, so 5/6 > 7/9.
6. 5/2 + 11/3 = (15 + 22)/6 = 37/6 = 6 1/6.
7. 9/4 - 3/8 = 18/8 - 3/8 = 15/8 = 1 7/8.
8. 6 x 20 = 120, 15 x 8 = 120. They are equivalent.