Learning goals
- Simplify a fraction and build equivalent fractions with a common denominator
- Add, subtract, multiply, and divide fractions with reliable step-by-step methods
- Convert between improper fractions and mixed numbers when needed
What a fraction means
A fraction a/b means a equal parts of a whole that is split into b equal parts. The top number is the numerator; the bottom is the denominator. The denominator cannot be zero.
Two fractions are equivalent when they represent the same amount — for example 1/2 = 2/4 = 3/6. You create equivalents by multiplying or dividing the numerator and denominator by the same nonzero number.
Before comparing or adding unlike fractions, rewrite them with a common denominator. Multiplying and dividing do not require a common denominator — those operations follow different rules.
Add and subtract (common denominator)
a/b ± c/d = (a·d ± c·b) / (b·d)
Rewrite both fractions over a common denominator (here b·d works, though the least common denominator is often smaller), then add or subtract the numerators and keep the denominator.
Multiply and divide
(a/b) × (c/d) = (a·c)/(b·d); (a/b) ÷ (c/d) = (a/b) × (d/c)
Multiply straight across. To divide, multiply by the reciprocal of the second fraction (flip numerator and denominator). Cancel common factors before multiplying when you can.
Example 1 — Add unlike fractions
Given
Compute 2/3 + 1/4.
Steps
- Common denominator: 3 × 4 = 12.
- 2/3 = 8/12 and 1/4 = 3/12.
- Add numerators: 8/12 + 3/12 = 11/12.
Answer
2/3 + 1/4 = 11/12.
Tip: 11 and 12 share no common factor greater than 1, so 11/12 is already simplified.
Example 2 — Multiply and simplify
Given
Compute (3/4) × (8/9).
Steps
- Cancel a factor of 4: 3/4 × 8/9 → 3/1 × 2/9.
- Cancel a factor of 3: 1/1 × 2/3.
- Multiply: 2/3.
Answer
(3/4) × (8/9) = 2/3.
Tip: Canceling before multiplying keeps numbers small and reduces mistakes.
Example 3 — Divide by a fraction
Given
Compute (5/6) ÷ (2/3).
Steps
- Rewrite as multiplication by the reciprocal: (5/6) × (3/2).
- Cancel a common factor of 3 from 6 and 3: the product becomes (5/2) × (1/2).
- Multiply: (5 × 1) / (2 × 2) = 5/4.
Answer
(5/6) ÷ (2/3) = 5/4 (or 1¼ as a mixed number).
Tip: “Dividing by 2/3” asks how many groups of 2/3 fit into 5/6 — an answer above 1 is expected.
Adding numerators and denominators separately
Common mistake
2/3 + 1/4 → (2+1)/(3+4) = 3/7.
Better approach
Only numerators combine after a shared denominator: 8/12 + 3/12 = 11/12. Never add denominators when adding fractions.
Forgetting to flip when dividing
Common mistake
(5/6) ÷ (2/3) → (5÷2)/(6÷3) = 2.5/2 without using the reciprocal rule carefully.
Better approach
Keep the first fraction, flip the second, multiply: (5/6) × (3/2) = 5/4.
Check your understanding
1.Simplify 18/24.
Answer: Divide by 6 → 3/4.
2.Compute 1/2 + 1/5.
Answer: 5/10 + 2/10 = 7/10.
3.Compute (2/5) × (15/8).
Answer: After canceling: 3/4.
4.Compute (3/4) ÷ (1/2).
Answer: (3/4) × (2/1) = 3/2.
Key takeaways
- Equivalent fractions scale numerator and denominator by the same factor.
- Add/subtract only after a common denominator; then combine numerators.
- Multiply across; divide by multiplying by the reciprocal.
- Simplify (cancel common factors) whenever you can to keep work clean.