Operations on Fractions

By the end of this lesson you will be able to:

1. Add and subtract fractions and mixed numbers using the LCM method
2. Multiply fractions by working across numerators and denominators
3. Divide fractions by multiplying by the reciprocal
4. Apply BODMAS to expressions with fractions, treating "of" as multiply
5. Solve real-life word problems that require one or more fraction operations

Why Can't We Just Add the Numbers?

• Tailor: m cloth m lining. Total?
• Try: , — answer ?
• No: is almost a metre — total must be more.

The denominator names the unit. You cannot add different units directly.

What Is in Twelfths?

We need a shared unit. Recall equivalent fractions:

• → multiply top and bottom by →
• → multiply top and bottom by →

Both fractions now speak the same language: twelfths.

Why 12? — Finding the LCM

• Multiples of 4: 4, 8, 12, … · Multiples of 3: 3, 6, 9, 12, …

Steps: (1) Find LCM · (2) Convert both fractions · (3) Operate on numerators · (4) Simplify

Worked: — The Tailor's Cloth

The tailor needs m of cloth altogether.

Worked: Subtraction — Cloth Remaining

The tailor cuts m from a m piece. How much is left?

• LCM(6, 4) = 12
• ,
• m remaining

Same four steps — subtraction is just adding a negative amount.

Worked: Mixed Numbers —

Mixed numbers must be converted to improper fractions first.

• LCM(3, 4) = 12
• ,

Solo Check-In:

Work this out before the next slide.

1. LCM(4, 6) = ?
2. Convert both fractions
3. Subtract the numerators
4. Simplify if needed

Write every step — don't guess.

Guided Practice:

LCM(5, 3) = 15

Mixed Numbers: The Conversion Rule

Before any operation, convert mixed numbers to improper fractions.

Applies to all four operations: +, −, ×, ÷

Why Does Work Differently?

• How many cells are doubly shaded?
• Total cells:
• Doubly shaded:
• Answer:

Multiply: Numerator × Numerator, Denominator × Denominator

No common denominator needed — multiply straight across.

Cancel common factors first to keep numbers small.

Worked Example:

Step 1: Convert mixed numbers first.

Step 2: Cancel before multiplying.

The answer is a whole number — that happens sometimes with fractions!

Solo Check-In:

Try this before entering the division section.

• Can you cancel any factors before multiplying?
• What is the simplified answer?

Write your working step by step.

The answer is — did you cancel the 3s?

Is Dividing the Same as Multiplying?

— same as

• ÷ 3 = × · ÷ = × · ÷ = ×
• Always flip the divisor — the fraction on the right of ÷

Divide: Multiply by the Reciprocal

Keep the first fraction. Flip the second fraction. Multiply.

KFC: Keep — Flip — Calculate

Worked Example:

1. Keep the dividend:
2. Flip the divisor:
3. Calculate:
4. Sense check: dividing by a number less than 1 gives a bigger result — ✓

Probe: Find the Division Error

A student wrote:

What went wrong? Fix it.

Hint: Which fraction should have been flipped?

Worked Example:

Step 1: Convert mixed numbers.

Step 2: Keep, Flip, Calculate.

Step 3: Cancel, then multiply.

Independent Practice:

Work this independently.

• Convert any mixed numbers? (none here)
• Keep, Flip, Calculate
• Cancel if possible
• Simplify the final answer

Answer:

BODMAS with Fractions: Same Order, New Content

Special rule: "of" means multiply.

Brackets Can Change Your Answer

Same numbers. Different order. Different answer.

Word Problem: Sack of Maize

A farmer sells of a full sack in the morning, then of what is left in the afternoon. What fraction remains?

Morning: remains.

Afternoon: sold.

Remaining:

Word Problem: Jerrycan of Water

A jerrycan holds 5 litres. It is full. Someone pours out of the water present. How much water remains?

Water present: litres.

Poured out: litres.

Remaining: litre.

Worked: of

Step 1: Brackets first.

Step 2: "of" means multiply.

Answer:

Worked:

Unscaffolded: Apply the Full BODMAS Chain

Work this out completely with no hints.

Show every step in your exercise book.

Check each operation follows BODMAS order.

Key Takeaways: Four Operations on Fractions

✓ Add/subtract: LCM → convert → operate → simplify
✓ Multiply: straight across; cancel first; no LCM
✓ Divide: KFC — Keep, Flip the divisor, Calculate
✓ BODMAS: same order; "of" means multiply

Never: add denominators · skip mixed-number conversion · flip the dividend · ignore BODMAS