Grade 8 · Common Fractions

Operations with Common Fractions

Interactive lesson · Addition · Subtraction · Multiplication · Division

Section 1 of 5
➕ Addition of Common Fractions
Adding fractions with like and unlike denominators, including mixed numbers

Key Rule

You can only add fractions that have the same denominator (like fractions).
If the denominators are different, first find the Lowest Common Denominator (LCD) — the smallest number both denominators divide into evenly.

Case 1 — Same Denominator

Add the numerators, keep the denominator the same. Simplify if possible.

Example
1
\(\dfrac{2}{9} + \dfrac{4}{9}\) — same denominator, add numerators
2
\(= \dfrac{2+4}{9} = \dfrac{6}{9}\)
3
Simplify (÷3): \(\dfrac{2}{3}\)

Case 2 — Different Denominators

Find the LCD, convert each fraction to an equivalent fraction with that denominator, then add.

Example  –  \(\dfrac{3}{4} + \dfrac{2}{3}\)
1
LCD of 4 and 3 = 12
2
\(\dfrac{3}{4} = \dfrac{3 \times 3}{4 \times 3} = \dfrac{9}{12}\)
3
\(\dfrac{2}{3} = \dfrac{2 \times 4}{3 \times 4} = \dfrac{8}{12}\)
4
\(\dfrac{9}{12} + \dfrac{8}{12} = \dfrac{17}{12}\)
5
Convert improper fraction: \(1\dfrac{5}{12}\)

Case 3 — Mixed Numbers

Add the whole-number parts and the fraction parts separately. If the fraction part is improper, carry over to the whole number.

Example  –  \(2\dfrac{1}{3} + 1\dfrac{3}{4}\)
1
Whole numbers: \(2 + 1 = 3\)
2
Fractions: \(\dfrac{1}{3} + \dfrac{3}{4}\), LCD = 12
3
\(\dfrac{4}{12} + \dfrac{9}{12} = \dfrac{13}{12} = 1\dfrac{1}{12}\)
4
\(3 + 1\dfrac{1}{12}\) = \(4\dfrac{1}{12}\)
⚠️ Common mistake: NEVER add denominators!  \(\dfrac{1}{2} + \dfrac{1}{3} \neq \dfrac{2}{5}\)
Check Your Understanding
Calculate: \(\dfrac{1}{4} + \dfrac{2}{4}\)
Calculate: \(\dfrac{1}{2} + \dfrac{1}{3}\)
Calculate: \(1\dfrac{1}{2} + 2\dfrac{3}{4}\). Type your answer as a mixed number
LCD = 4. \(\dfrac{2}{4} + \dfrac{3}{4} = \dfrac{5}{4} = 1\dfrac{1}{4}\). Total: \(1+2+1\dfrac{1}{4} = 4\dfrac{1}{4}\)
➖ Subtraction of Common Fractions
Subtracting fractions with like and unlike denominators, including borrowing with mixed numbers

Key Rule

Same rule as addition — denominators must be the same before you subtract. Find the LCD, convert, then subtract numerators.

Case 1 — Same Denominator

Example  –  \(\dfrac{7}{8} - \dfrac{3}{8}\)
1
Same denominator — subtract numerators: \(\dfrac{7-3}{8} = \dfrac{4}{8}\)
2
Simplify (÷4): \(\dfrac{1}{2}\)

Case 2 — Different Denominators

Example  –  \(\dfrac{5}{6} - \dfrac{1}{4}\)
1
LCD of 6 and 4 = 12
2
\(\dfrac{5}{6} = \dfrac{10}{12}\)    \(\dfrac{1}{4} = \dfrac{3}{12}\)
3
\(\dfrac{10}{12} - \dfrac{3}{12} = \) \(\dfrac{7}{12}\)

Case 3 — Mixed Numbers with Borrowing

When the fraction part of the first mixed number is smaller than the fraction part being subtracted, you need to borrow 1 from the whole number.

Example  –  \(3\dfrac{1}{4} - 1\dfrac{3}{4}\)
1
Fractions: \(\dfrac{1}{4} - \dfrac{3}{4}\) — can't do this yet (1 < 3)
2
Borrow 1 from the 3:   \(3\dfrac{1}{4} = 2\dfrac{4}{4} + \dfrac{1}{4} = 2\dfrac{5}{4}\)
3
\(2\dfrac{5}{4} - 1\dfrac{3}{4} = (2-1)\dfrac{5-3}{4} = \) \(1\dfrac{2}{4} = 1\dfrac{1}{2}\)
Another example  –  \(5\dfrac{1}{3} - 2\dfrac{5}{6}\)
1
LCD = 6:   \(\dfrac{1}{3} = \dfrac{2}{6}\). Need \(\dfrac{2}{6} - \dfrac{5}{6}\) — borrow!
2
\(5\dfrac{2}{6} = 4\dfrac{8}{6}\)
3
\(4\dfrac{8}{6} - 2\dfrac{5}{6} = \) \(2\dfrac{3}{6} = 2\dfrac{1}{2}\)
💡 Tip — borrowing:
When you borrow 1 whole, convert it to the fraction form using the LCD.

e.g. borrow 1 with LCD 6: add \(\dfrac{6}{6}\) to the fraction part.
Check Your Understanding
Calculate: \(\dfrac{5}{8} - \dfrac{1}{8}\)
Calculate: \(\dfrac{3}{4} - \dfrac{1}{3}\)
Calculate: \(4\dfrac{1}{5} - 1\dfrac{3}{5}\). Type your answer as a mixed number
\(\dfrac{1}{5} < \dfrac{3}{5}\) → borrow: \(4\dfrac{1}{5} = 3\dfrac{6}{5}\). Then \(3\dfrac{6}{5} - 1\dfrac{3}{5} = 2\dfrac{3}{5}\)
✖️ Multiplication of Common Fractions
Multiply numerators together and denominators together — no common denominator needed!

Key Rule

Multiply straight across: numerator × numerator, denominator × denominator.
\[\dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{a \times c}{b \times d}\] No LCD needed! Always simplify your answer.

Case 1 — Fraction × Fraction

Example  –  \(\dfrac{2}{3} \times \dfrac{3}{4}\)
1
Multiply numerators: \(2 \times 3 = 6\)
2
Multiply denominators: \(3 \times 4 = 12\)
3
\(\dfrac{6}{12}\) — simplify (÷6): \(\dfrac{1}{2}\)

Shortcut — Cancel Before Multiplying

Simplify diagonally before multiplying to keep numbers small.

Example  –  \(\dfrac{4}{9} \times \dfrac{3}{8}\)
1
Cancel: 4 and 8 share factor 4 → \(\dfrac{\cancel{4}^1}{9} \times \dfrac{3}{\cancel{8}^2}\)
2
Cancel: 3 and 9 share factor 3 → \(\dfrac{1}{\cancel{9}^3} \times \dfrac{\cancel{3}^1}{2}\)
3
\(\dfrac{1 \times 1}{3 \times 2} = \) \(\dfrac{1}{6}\)

Case 2 — Mixed Numbers

Convert mixed numbers to improper fractions first, then multiply.

To convert: \(a\dfrac{b}{c} = \dfrac{(a \times c) + b}{c}\)    e.g. \(2\dfrac{3}{4} = \dfrac{11}{4}\)
Example  –  \(1\dfrac{1}{2} \times 2\dfrac{2}{3}\)
1
Convert: \(1\dfrac{1}{2} = \dfrac{3}{2}\)    \(2\dfrac{2}{3} = \dfrac{8}{3}\)
2
Multiply: \(\dfrac{3}{2} \times \dfrac{8}{3} = \dfrac{24}{6}\)
3
Simplify: \(4\)

Case 3 — Fraction of a Quantity ("of" = ×)

Example  –  \(\dfrac{3}{4}\) of 24
1
"of" means multiply: \(\dfrac{3}{4} \times \dfrac{24}{1}\)
2
Cancel: 4 into 24 = 6 → \(\dfrac{3 \times 6}{1} = \) \(18\)
Check Your Understanding
Calculate: \(\dfrac{2}{5} \times \dfrac{5}{6}\)
Find \(\dfrac{2}{3}\) of 30.
Calculate: \(1\dfrac{1}{3} \times 2\dfrac{1}{4}\). Type your answer as a whole number or mixed number
\(\dfrac{4}{3} \times \dfrac{9}{4} = \dfrac{36}{12} = 3\)
➗ Division of Common Fractions
Dividing by a fraction is the same as multiplying by its reciprocal

Key Rule — Keep, Change, Flip (KCF)

To divide by a fraction: Keep the first fraction, Change ÷ to ×, Flip the second fraction (use its reciprocal).

\[\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c}\]
What is a reciprocal? Flip the numerator and denominator.
Reciprocal of \(\dfrac{3}{4}\) is \(\dfrac{4}{3}\)  |  Reciprocal of 5 is \(\dfrac{1}{5}\)  |  Reciprocal of \(2\dfrac{1}{3}\) = \(\dfrac{3}{7}\) (convert first!)

Case 1 — Fraction ÷ Fraction

Example  –  \(\dfrac{3}{4} \div \dfrac{1}{2}\)
1
Keep \(\dfrac{3}{4}\), flip \(\dfrac{1}{2}\) → \(\dfrac{2}{1}\)
2
\(\dfrac{3}{4} \times \dfrac{2}{1} = \dfrac{6}{4}\)
3
Simplify: \(\dfrac{3}{2} = 1\dfrac{1}{2}\)
Example  –  \(\dfrac{2}{5} \div \dfrac{4}{15}\)
1
\(\dfrac{2}{5} \times \dfrac{15}{4}\)
2
Cancel: \(\dfrac{\cancel{2}^1}{\cancel{5}^1} \times \dfrac{\cancel{15}^3}{\cancel{4}^2}\)
3
\(\dfrac{3}{2} = 1\dfrac{1}{2}\)

Case 2 — Mixed Numbers

Convert all mixed numbers to improper fractions first, then apply KCF.

Example  –  \(2\dfrac{1}{2} \div 1\dfrac{1}{4}\)
1
Convert: \(2\dfrac{1}{2} = \dfrac{5}{2}\)    \(1\dfrac{1}{4} = \dfrac{5}{4}\)
2
KCF: \(\dfrac{5}{2} \times \dfrac{4}{5}\)
3
Cancel: \(\dfrac{\cancel{5}^1}{\cancel{2}^1} \times \dfrac{\cancel{4}^2}{\cancel{5}^1} = \dfrac{2}{1} = \) \(2\)

Case 3 — Dividing by a Whole Number

Example  –  \(\dfrac{3}{4} \div 3\)
1
Write 3 as \(\dfrac{3}{1}\), then flip → \(\dfrac{1}{3}\)
2
\(\dfrac{3}{4} \times \dfrac{1}{3} = \dfrac{3}{12}\) = \(\dfrac{1}{4}\)
⚠️ Common mistake: Students sometimes flip the FIRST fraction instead of the second. Only flip the divisor (the number after ÷).
Check Your Understanding
Calculate: \(\dfrac{1}{2} \div \dfrac{1}{4}\)
Calculate: \(\dfrac{5}{6} \div \dfrac{5}{3}\)
Calculate: \(3\dfrac{1}{2} \div 1\dfrac{3}{4}\). Type your answer as a whole number or mixed number
\(\dfrac{7}{2} \div \dfrac{7}{4} = \dfrac{7}{2} \times \dfrac{4}{7} = \dfrac{28}{14} = 2\)
🎯 Practice Quiz
10 questions covering all four operations — check your understanding!