Fractions — Class 6 IMO

Types, Equivalent Fractions & Simplest Form

A proper fraction has numerator < denominator, an improper fraction has numerator ≥ denominator, and a mixed fraction is a whole number with a fraction.

Equivalent fractions show the same value (½ = 2/4). The simplest form is found by dividing the numerator and denominator by their HCF: 8/12 = 2/3.

Example 1: Write 8/12 in simplest form.

Divide both by 4: 8/12 = 2/3.

Example 2: Write 7/4 as a mixed fraction.

7 ÷ 4 = 1 remainder 3, so 1¾.

Quick recap

Proper < 1; improper ≥ 1; mixed = whole + fraction.
Simplest form: divide top and bottom by their HCF.

✓ Quick check

What is 6/9 in simplest form?

Divide both by 3: 6/9 = 2/3.

Which of these is an improper fraction?

In 5/2 the numerator is larger than the denominator.

Comparison, Addition & Subtraction

To compare or add/subtract unlike fractions, change them to the same denominator (the LCM) first, then work with the numerators.

So ⅓ + ¼ = 4/12 + 3/12 = 7/12, and ⅚ − ½ = 5/6 − 3/6 = ⅓.

Example 1: Add ⅓ + ¼.

Common denominator 12: 4/12 + 3/12 = 7/12.

Example 2: Subtract ⅚ − ½.

5/6 − 3/6 = 2/6 = ⅓.

Quick recap

Use a common denominator (the LCM) for unlike fractions.
Then add or subtract the numerators.

✓ Quick check

What is ⅖ + ⅗?

2/5 + 3/5 = 5/5 = 1.

What is ¾ − ¼?

3/4 − 1/4 = 2/4 = ½.

Multiplication, Division & Word Problems

The "Keep-Change-Flip" Rule

Dividing fractions is easy with this trick:

Keep the first fraction

Change ÷ to ×

Flip the second fraction (reciprocal)

Rule: (a)/(b) ÷ (c)/(d) = (a)/(b) × (d)/(c) = (a × d)/(b × c)

Examples:

⅔ ÷ ⅘ = ⅔ × 5/4 = 10/12 = (5)/(6)
⅚ ÷ ⅓ = ⅚ × 3/1 = 15/6 = 5/2 = 2½

Dividing Mixed Numbers:

Convert mixed numbers to improper fractions
Use Keep-Change-Flip
Simplify

KEEP-CHANGE-FLIP VISUAL:

    2   ÷   4    =    2   ×   5
    3       5         3       4
    
    KEEP    CHANGE    FLIP
    2/3      ×        5/4
    
    Multiply: 2×5=10, 3×4=12, 10/12 = 5/6


VISUAL REPRESENTATION: ½ ÷ ¼

    "How many ¼ are in ½?"
    
    ½ of a whole:    
    ┌─────────────────┐
    │░░░░░░░░░░░░░░░░░│
    └─────────────────┘
    
    ¼ of a whole:     
    ┌─────────┐
    │░░░░░░░░░│
    └─────────┘
    
    Count how many ¼ fit into ½:
    
    ┌─────────┬─────────┐
    │░░░░░░░░░│░░░░░░░░░│
    └─────────┴─────────┘
        ¼        ¼
    
    Answer: 2 (½ ÷ ¼ = 2)


DIVIDING MIXED NUMBERS:

    2⅓ ÷ 1½ = ?
    
    Step 1: Convert to improper fractions
    2⅓ = 7/3, 1½ = 3/2
    
    Step 2: Keep-Change-Flip
    7/3 ÷ 3/2 = 7/3 × 2/3
    
    Step 3: Multiply
    7×2=14, 3×3=9, 14/9 = 1⁵⁄₉

Example 1: Multiply ⅔ × ¾.

(2 × 3)/(3 × 4) = 6/12 = ½.

Example 2: Divide ¾ ÷ ½.

¾ × 2/1 = 6/4 = 1½.

Example 3:

Divide ⅔ ÷ ⅘

Keep: ⅔
Change: ×
Flip: ⅘ → 5/4
Multiply: ⅔ × 5/4 = (2×5)/(3×4) = 10/12 = (5)/(6)

Example 4:

Divide ⅚ ÷ ⅓

Keep: ⅚
Change: ×
Flip: ⅓ → 3/1
Multiply: ⅚ × 3/1 = (5×3)/(6×1) = 15/6 = 5/2 = 2½

Example 5:

Divide 2⅓ ÷ 1½

Convert: 2⅓ = (7)/(3), 1½ = (3)/(2)
Keep-Change-Flip: (7)/(3) × (2)/(3)
Multiply: (7×2)/(3×3) = 14/9 = 1⁵⁄₉

Example 6:

Divide ¾ ÷ 2

Write 2 as 2/1
Keep-Change-Flip: ¾ × ½ = (3)/(8)
Answer: (3)/(8)

Quick recap

Multiply: numerators × numerators, denominators × denominators.
Divide: multiply by the reciprocal of the second fraction.
Keep-Change-Flip: Keep 1st, Change ÷ to ×, Flip 2nd
Reciprocal = flipped fraction (a/b → b/a)
Convert mixed numbers before starting
Whole numbers = number over 1
Always simplify final answer

✓ Quick check

What is ⅖ × ⅚?

(2 × 5)/(5 × 6) = 10/30 = ⅓.

What is ½ ÷ ¼?

½ × 4/1 = 4/2 = 2.