Lesson 8 ~30 min Unit 1 · Fractions +90 XP

Adding and Subtracting Fractions

Same denominator? Just add the tops. Different denominators? Find the LCD first. Mixed numbers? Convert, compute, simplify.

Today's hook: Can you add $rac{1}{2}$ and $rac{1}{3}$? Not directly — they speak different languages. Find a common language (denominator) first!

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Think First

warm-up

Before you read on — quickly: $rac{1}{2} + rac{1}{4} = ?$ Can you picture it? And what about $rac{1}{2} + rac{1}{3}$? Why is that harder?

Record your answer in your workbook.

The Big Idea

+5 XP

Same denominator? Just add or subtract the numerators. Different denominators? Find the lowest common denominator (LCD), convert both fractions, then add or subtract. Mixed numbers? Convert to improper fractions first (or handle whole and fraction parts separately).

Think of fractions as slices of pizza. You can only add slices that are the same size. $rac{1}{2} + rac{1}{4}$ doesn't work directly because halves and quarters are different sizes. Convert $rac{1}{2}$ to $rac{2}{4}$, then add: $rac{2}{4} + rac{1}{4} = rac{3}{4}$. The key rule: denominators must match before you add or subtract.

$\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}$

Same size slices

Denominators must match before adding or subtracting.

Find the LCD

Lowest Common Denominator = LCM of the denominators.

Simplify at the end

Always write your final answer in lowest terms.

What You'll Master

objectives

Know

Fractions need common denominators to add or subtract
LCD is the LCM of the denominators
Mixed numbers can be converted to improper fractions

Understand

Why different denominators mean different-sized pieces
Why converting to equivalent fractions preserves value
When to use improper vs mixed number strategies

Can Do

Add and subtract fractions with the same denominator
Find the LCD and convert fractions
Add and subtract mixed numbers

Words You Need

vocabulary

Common denominatorA shared denominator that both fractions can be converted to.

LCDLowest Common Denominator. The smallest number both denominators divide into.

LCMLowest Common Multiple. The smallest number that is a multiple of both numbers.

ConvertChange a fraction to an equivalent fraction with a different denominator.

Mixed number strategyAdd/subtract whole numbers and fractions separately, or convert to improper first.

SimplifyReduce the final answer to lowest terms by dividing by the HCF.

Spot the Trap

heads-up

Wrong: $\frac{1}{2} + \frac{1}{3} = \frac{2}{5}$. No! You cannot add denominators. Different sizes need conversion first.

Right: LCD of 2 and 3 is 6. $\frac{1}{2} = \frac{3}{6}$, $\frac{1}{3} = \frac{2}{6}$. Then $\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$.

Wrong: $2\frac{1}{4} + 1\frac{3}{4} = 3\frac{4}{8}$. Don\'t add denominators! The denominator stays 4.

Right: $2\frac{1}{4} + 1\frac{3}{4} = (2+1) + (\frac{1}{4}+\frac{3}{4}) = 3 + \frac{4}{4} = 3 + 1 = 4$.

Adding Fractions

+5 XP

Same denominator: add the numerators, keep the denominator. Different denominators: find LCD, convert both, then add. Always simplify your final answer.

Calculate $\frac{2}{5} + \frac{1}{3}$. LCD of 5 and 3 = 15. Convert: $\frac{2}{5} = \frac{2 \times 3}{5 \times 3} = \frac{6}{15}$. And $\frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}$. Now add: $\frac{6}{15} + \frac{5}{15} = \frac{11}{15}$. Check: HCF(11, 15) = 1, so fully simplified.

$rac{a}{b} + rac{c}{d} = rac{ad + bc}{bd}$

Find LCD first

Always convert to common denominators before adding.

Multiply strategically

Multiply each fraction by the other\'s denominator/LCD.

Check at the end

Always simplify your final answer to lowest terms.

Subtracting Fractions

+5 XP

Subtracting fractions follows the exact same process as adding. Find the LCD, convert both fractions, subtract the numerators, and simplify.

Calculate $\frac{5}{6} - \frac{1}{4}$. LCD of 6 and 4 = 12. Convert: $\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$. And $\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$. Now subtract: $\frac{10}{12} - \frac{3}{12} = \frac{7}{12}$. Check: HCF(7, 12) = 1, fully simplified.

$rac{a}{b} - rac{c}{d} = rac{ad - bc}{bd}$

Same process as adding

LCD, convert, subtract numerators, simplify.

Watch for borrowing

Mixed numbers may need borrowing (like 3 - 1 1/2).

Negative result?

If top < bottom, the answer is negative. That\'s OK!

Adding and Subtracting Mixed Numbers

+5 XP

Method 1: Convert to improper fractions, add/subtract, convert back. Method 2: Handle wholes and fractions separately. If the fraction part goes negative, borrow 1 from the whole number.

Calculate $3\frac{1}{4} - 1\frac{3}{4}$. Method 2: Wholes: 3 - 1 = 2. Fractions: $\frac{1}{4} - \frac{3}{4} = -\frac{2}{4}$. So we have $2 - \frac{2}{4}$. Borrow 1: $1 + \frac{4}{4} - \frac{2}{4} = 1\frac{2}{4} = 1\frac{1}{2}$. Method 1 check: $\frac{13}{4} - \frac{7}{4} = \frac{6}{4} = 1\frac{2}{4} = 1\frac{1}{2}$.

$arac{b}{c} - drac{e}{c} = (a-d) + rac{b-e}{c}$

Two methods

Improper fractions OR separate whole and fraction parts.

Borrow when needed

If fraction part goes negative, borrow 1 from the whole.

Check with method 2

Use both methods to verify your answer is correct.

Watch Me Solve It · 3 examples

step-by-step

Example 1: Adding with different denominators

Calculate $\frac{3}{8} + \frac{5}{12}$

Find the LCD of 8 and 12. Multiples of 8: 8, 16, 24, 32... Multiples of 12: 12, 24, 36... LCD = 24.

Convert: $\frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24}$ and $\frac{5}{12} = \frac{5 \times 2}{12 \times 2} = \frac{10}{24}$.

Add: $\frac{9}{24} + \frac{10}{24} = \frac{19}{24}$. Check: HCF(19, 24) = 1, fully simplified.

$\frac{3}{8} + \frac{5}{12} = \frac{19}{24}$

Example 2: Subtracting with borrowing

Calculate $4\frac{1}{6} - 2\frac{5}{6}$

Same denominator (6), so subtract whole and fraction parts separately. Wholes: 4 - 2 = 2. Fractions: $\frac{1}{6} - \frac{5}{6} = -\frac{4}{6}$. We have $2 - \frac{4}{6}$.

Borrow 1 from the 2: $1 + \frac{6}{6} - \frac{4}{6} = 1\frac{2}{6}$. Simplify: $\frac{2}{6} = \frac{1}{3}$.

Check with improper fractions: $\frac{25}{6} - \frac{17}{6} = \frac{8}{6} = 1\frac{2}{6} = 1\frac{1}{3}$.

$4\frac{1}{6} - 2\frac{5}{6} = 1\frac{1}{3}$

Example 3: Adding mixed numbers with different denominators

Calculate $2\frac{1}{3} + 1\frac{3}{4}$

Method: convert to improper fractions. $2\frac{1}{3} = \frac{7}{3}$ and $1\frac{3}{4} = \frac{7}{4}$.

LCD of 3 and 4 = 12. Convert: $\frac{7}{3} = \frac{28}{12}$ and $\frac{7}{4} = \frac{21}{12}$.

Add: $\frac{28}{12} + \frac{21}{12} = \frac{49}{12}$. Convert back: $\frac{49}{12} = 4\frac{1}{12}$.

$2\frac{1}{3} + 1\frac{3}{4} = 4\frac{1}{12}$

Common Pitfalls

avoid these

Mistake: Adding denominators. $\frac{1}{2} + \frac{1}{3} \ne \frac{2}{5}$. Denominators never add!

Fix: Find LCD, convert, then add numerators only. $\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}$.

Mistake: Forgetting to simplify the final answer. $\frac{6}{8}$ should be $\frac{3}{4}$.

Fix: Always check: are numerator and denominator both even? If yes, divide by 2.

Mistake: Borrowing incorrectly with mixed numbers. $3\frac{1}{4} - 1\frac{3}{4} \ne 2\frac{-2}{4}$.

Fix: Borrow 1 from the whole number. $3\frac{1}{4} = 2\frac{5}{4}$, then $2\frac{5}{4} - 1\frac{3}{4} = 1\frac{2}{4} = 1\frac{1}{2}$.

Copy Into Your Books

essential notes

Same denominator: $\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}$ (just add tops)

Different denominators: find LCD, convert both, then add/subtract.

Mixed numbers: convert to improper OR handle whole and fraction parts separately.

Always simplify your final answer to lowest terms.

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Add & Subtract

4 problems

Four drill problems to build your fraction addition and subtraction fluency. Work each, then reveal the answer.

1 $\frac{3}{7} + \frac{2}{7}$

Same denominator, so add tops: 3 + 2 = 5. Keep denominator 7.5/7
2 $\frac{5}{6} - \frac{1}{4}$

LCD(6,4) = 12. 5/6 = 10/12, 1/4 = 3/12. 10/12 - 3/12 = 7/12.7/12
3 $2\frac{1}{5} + 1\frac{3}{5}$

Wholes: 2 + 1 = 3. Fractions: 1/5 + 3/5 = 4/5. So 3 4/5.3 4/5
4 $4\frac{1}{3} - 2\frac{2}{3}$

Wholes: 4 - 2 = 2. Fractions: 1/3 - 2/3 = -1/3. Borrow 1: 1 + 3/3 - 1/3 = 1 2/3.1 2/3

Complete in your workbook.

Quick Check · 5 questions

$\frac{3}{8} + \frac{4}{8} = ?$

+10 XP

$\frac{3}{4} - \frac{1}{6} = ?$

+10 XP

$1\frac{1}{2} + 2\frac{1}{2} = ?$

+10 XP

$2\frac{3}{4} - 1\frac{1}{4} = ?$

+10 XP

$\frac{2}{3} + \frac{4}{5} = ?$

+10 XP

Show Your Working · 3 questions

Show Your Working

11 marks total

ApplyMedium4 MARKS

Q6. (a) Calculate $\frac{7}{10} + \frac{2}{5}$. Show the LCD and conversion steps. (b) Calculate $\frac{5}{6} - \frac{3}{8}$. Show all working.

Answer in your workbook.

ApplyMedium4 MARKS

Q7. A recipe needs $2\frac{1}{4}$ cups of flour and $1\frac{2}{3}$ cups of sugar. (a) What is the total amount? (b) How much more flour than sugar is needed? Show all working.

Answer in your workbook.

ReasonHard3 MARKS

Q8. A student calculates $\frac{2}{3} + \frac{1}{6} = \frac{3}{9} = \frac{1}{3}$. Identify TWO mistakes they made, explain the correct method, and give the right answer.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — Same denominator: 3/8 + 4/8 = 7/8.

2. C — LCD(4,6) = 12. 3/4 = 9/12, 1/6 = 2/12. 9/12 - 2/12 = 7/12.

3. A — 1 + 2 = 3, 1/2 + 1/2 = 1. So 3 + 1 = 4.

4. B — 11/4 - 5/4 = 6/4 = 1 1/2.

5. D — LCD(3,5) = 15. 2/3 = 10/15, 4/5 = 12/15. 10/15 + 12/15 = 22/15 = 1 7/15.

Show Your Working Model Answers

Q6 (4 marks): (a) LCD(10,5) = 10 [0.5]. 7/10 stays, 2/5 = 4/10 [0.5]. 7/10 + 4/10 = 11/10 [1]. (b) LCD(6,8) = 24 [0.5]. 5/6 = 20/24, 3/8 = 9/24 [0.5]. 20/24 - 9/24 = 11/24 [1].

Q7 (4 marks): (a) 2 1/4 + 1 2/3 = 9/4 + 5/3 [0.5]. LCD = 12: 27/12 + 20/12 = 47/12 = 3 11/12 cups [1.5]. (b) 9/4 - 5/3 [0.5]. 27/12 - 20/12 = 7/12 cups more flour [1.5].

Q8 (3 marks): Mistake 1: added denominators (3 + 6 = 9) [1]. Mistake 2: did not find LCD first [0.5]. Correct: LCD(3,6) = 6, so 2/3 = 4/6, then 4/6 + 1/6 = 5/6 [1.5].

Stretch Challenge · +25 XP, +10 coins

The Egyptian Fraction Puzzle

Ancient Egyptians only used fractions with numerator 1 (like $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$). They wrote other fractions as sums of these "unit fractions." Can you write $\frac{5}{6}$ as a sum of two different unit fractions? What about $\frac{7}{12}$? Find at least one solution for each.

Reveal solution

$\frac{5}{6} = \frac{1}{2} + \frac{1}{3}$ (since $\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$). $\frac{7}{12} = \frac{1}{3} + \frac{1}{4}$ (since $\frac{4}{12} + \frac{3}{12} = \frac{7}{12}$). There are often multiple solutions!

Quick Review

Same den

Add/subtract tops only

LCD

Find before converting

Convert

Multiply top and bottom equally

Mixed numbers

Convert to improper OR separate

Borrow

When fraction part goes negative

Simplify

Always final step

Interactive: Fraction Steps

Drag the steps into the correct order to solve fraction addition and subtraction problems. Remember: find the LCD first!

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