🍰 Fractions

Cambridge Lower Secondary · Grade 8 · Number

Adding & Subtracting Mixed Numbers
Convert → Improper Fractions → LCD → Simplify
e.g. 2¾ + 1⅔ → 11/4 + 5/3 → LCD 12 → 45/12 = 3¾
Multiplying Mixed Numbers
Convert to improper fractions, then multiply top × top ÷ bottom × bottom
2½ × 1⅓ = 5/2 × 4/3 = 20/6 = 3⅓
Dividing Mixed Numbers — KCF
Keep · Change · Flip
3½ ÷ 1¾ = 7/2 ÷ 7/4 = 7/2 × 4/7 = 2

Watch fractions build up! 🍭

What you'll learn:

  • Adding & subtracting fractions with different denominators
  • Adding & subtracting mixed numbers (convert → improper → LCD)
  • Multiplying mixed numbers (convert to improper fractions first)
  • Dividing mixed numbers using Keep · Change · Flip (KCF)
  • Multi-step fraction problems and fractions of fractions
  • Solving equations that contain fractions
  • Comparing and ordering fractions
  • Fractions in real contexts: recipes, sharing, scaling
Created by Mr Josh

📖 Learn: Fractions — Stage 8

Part 1: Adding & Subtracting Fractions (Different Denominators)

To add or subtract fractions, the denominators must be the same. Find the Lowest Common Denominator (LCD).

ab + cd = ad + bcbd (then simplify)
Step 1: Find the LCD of the denominators.
Step 2: Convert each fraction to an equivalent fraction with the LCD.
Step 3: Add or subtract the numerators. Keep the denominator.
Step 4: Simplify (cancel) the result if possible.
Example: 3⁄4 + 5⁄6 → LCD = 12 → 9⁄12 + 10⁄12 = 19⁄12 = 17⁄12
💡 LCD tip: list multiples of each denominator until you find the first match. For 4 and 6: multiples of 4 → 4, 8, 12; multiples of 6 → 6, 12. LCD = 12.
Example: 7⁄8 − 2⁄5 → LCD = 40 → 35⁄40 − 16⁄40 = 19⁄40

Part 2: Adding & Subtracting Mixed Numbers

A mixed number like 2¾ has a whole part (2) and a fraction part (¾). The best method for Stage 8:

Step 1: Convert each mixed number to an improper fraction.
Rule: ab⁄c = (a×c + b)⁄c
Step 2: Find the LCD and add/subtract as normal.
Step 3: Convert the answer back to a mixed number if needed. Simplify.
Example: 2¾ + 12⁄3 → 11⁄4 + 5⁄3 → LCD = 12 → 33⁄12 + 20⁄12 = 53⁄12 = 45⁄12
💡 Always convert to improper fractions before adding — it avoids mistakes with carrying over wholes.
Example (subtract): 31⁄3 − 15⁄6 → 10⁄3 − 11⁄6 → 20⁄6 − 11⁄6 = 9⁄6 = 3⁄2 = 11⁄2
⚠️ When subtracting, if the fraction part of the second number is larger, convert to improper fractions first to avoid errors!

Part 3: Multiplying Mixed Numbers

Always convert to improper fractions first, then multiply numerators together and denominators together.

ab × cd = a × cb × d
Example: 2½ × 1⅓ → 5⁄2 × 4⁄3 = 20⁄6 = 10⁄3 = 3⅓
Example: 3¾ × 2⅖ → 15⁄4 × 12⁄5 = 180⁄20 = 9
✂️ Cross-cancel before multiplying to keep numbers small. E.g. 15⁄4 × 12⁄5: cancel 15 and 5 → 3, cancel 12 and 4 → 3 → 3×3⁄1×1 = 9
Example: 1⅘ × 2⅔ → 9⁄5 × 8⁄3 = 72⁄15 = 24⁄5 = 44⁄5

Part 4: Dividing Mixed Numbers — Keep · Change · Flip (KCF)

Dividing by a fraction is the same as multiplying by its reciprocal (flip it upside down).

ab ÷ cd = ab × dc

Keep the first fraction · Change ÷ to × · Flip the second fraction

Example: 3½ ÷ 1¾ → 7⁄2 ÷ 7⁄4 → KEEP 7⁄2 · CHANGE to × · FLIP → 7⁄2 × 4⁄7 = 28⁄14 = 2
Example: 2⅔ ÷ 1⅓ → 8⁄3 ÷ 4⁄3 → 8⁄3 × 3⁄4 = 24⁄12 = 2
✅ Always convert to improper fractions BEFORE applying KCF. Never try to "flip" a mixed number.
Example: 4½ ÷ 2¼ → 9⁄2 ÷ 9⁄4 → 9⁄2 × 4⁄9 = 36⁄18 = 2

Part 5: Multi-Step Problems, Equations & Comparing

Fractions of fractions: "find ¾ of 2⅔" means multiply: ¾ × 8⁄3 = 24⁄12 = 2

Multi-step: A recipe needs 2¼ cups of flour. To make 1½ times the recipe, how much flour?
→ 9⁄4 × 3⁄2 = 27⁄8 = 3⅜ cups

Solving equations with fractions: multiply both sides by the denominator.

Solve: x⁄3 + 1⁄2 = 2 → multiply everything by 6: 2x + 3 = 12 → 2x = 9 → x = 4.5 = 4½
Solve: 2x⁄5 = 1⅖ → 2x⁄5 = 7⁄5 → 2x = 7 → x = 3½

Comparing fractions: convert to a common denominator or decimals.

Which is greater: 7⁄9 or 5⁄6? → LCD = 18 → 14⁄18 vs 15⁄18 → 5⁄6 is greater.
📐 Order fractions by finding LCD, then comparing numerators. Or convert to decimals: 7÷9 ≈ 0.778, 5÷6 ≈ 0.833
Created by Mr Josh

💡 Worked Examples

Example 1: Adding Fractions with Different Denominators

Calculate 5⁄6 + 3⁄8. Give your answer as a mixed number in its simplest form.

Step 1: Find the LCD of 6 and 8.
Multiples of 6: 6, 12, 18, 24 · Multiples of 8: 8, 16, 24 → LCD = 24
Step 2: Convert: 5⁄6 = 20⁄24    3⁄8 = 9⁄24
Step 3: Add: 20⁄24 + 9⁄24 = 29⁄24
Step 4: Convert to mixed number: 29⁄24 = 15⁄24
✅ 29 ÷ 24 = 1 remainder 5 → 15⁄24. Check: GCD(5,24)=1 so already simplified.

Example 2: Subtracting Mixed Numbers

Calculate 41⁄4 − 15⁄6. Give your answer as a mixed number in its simplest form.

Step 1: Convert to improper fractions:
4¼ = (4×4+1)⁄4 = 17⁄4    15⁄6 = (1×6+5)⁄6 = 11⁄6
Step 2: LCD of 4 and 6 = 12.
17⁄4 = 51⁄12    11⁄6 = 22⁄12
Step 3: Subtract: 51⁄12 − 22⁄12 = 29⁄12
Step 4: Convert: 29⁄12 = 25⁄12
📌 If you tried to subtract without converting: ¼ − 5⁄6 is negative — that's why converting to improper first is safer!

Example 3: Multiplying Mixed Numbers

Calculate 2⅗ × 1⅞. Simplify fully.

Step 1: Convert: 2⅗ = 13⁄5    1⅞ = 15⁄8
Step 2: Cross-cancel: GCD(15,5)=5 → 13⁄1 × 3⁄8
Step 3: Multiply: 13×3⁄1×8 = 39⁄8
Step 4: Convert: 39⁄8 = 4⅞
✂️ Cross-cancel: 15÷5=3 and 5÷5=1. This simplifies before multiplying.

Example 4: Dividing Mixed Numbers (KCF)

Calculate 5¼ ÷ 1⅖. Simplify fully.

Step 1: Convert: 5¼ = 21⁄4    1⅖ = 7⁄5
Step 2 (KCF): KEEP 21⁄4 · CHANGE ÷ to × · FLIP → × 5⁄7
Step 3: Cross-cancel: GCD(21,7)=7 → 3⁄4 × 5⁄1 = 15⁄4
Step 4: Convert: 15⁄4 = 3¾
✅ KCF in 3 words: Keep the first · Change the sign · Flip the second.
Created by Mr Josh

🎨 Fraction Bar Visualizer

Enter two mixed numbers, choose an operation, and see the fraction bars animate step by step!

First number (e.g. 1 2/3) ⁄
Operation
Second number (e.g. 2 1/4) ⁄
Created by Mr Josh

✏️ Exercise 1: Add & Subtract Proper Fractions

Type your answer as a fraction (e.g. 7/12) or mixed number (e.g. 1 5/12)

Created by Mr Josh

✏️ Exercise 2: Add & Subtract Mixed Numbers

Type your answer as a fraction (e.g. 19/12) or mixed number (e.g. 1 7/12)

Created by Mr Josh

✏️ Exercise 3: Multiply Mixed Numbers

Type your answer as a fraction or mixed number in simplest form.

Created by Mr Josh

✏️ Exercise 4: Divide Mixed Numbers (KCF)

Remember: Keep · Change · Flip. Type your answer as a fraction or mixed number.

Created by Mr Josh

🔍 Exercise 5: Multi-Step Fraction Problems

These questions involve more than one step. Work carefully!

Created by Mr Josh

📝 Practice: 20 Questions

Mixed questions from all topics. Type answers as fractions or mixed numbers.

Created by Mr Josh

🏆 Challenge: 8 Hard Problems

Equations with fractions, multi-step real-world problems. Aim for full marks!

Created by Mr Josh