Fractions
Parts of a Fraction
A fraction represents a part of a whole.
Numerator — the top number (how many parts we have) Denominator — the bottom number (how many equal parts the whole is divided into)
In ¾: numerator = 3, denominator = 4 → we have 3 out of 4 equal parts.
Fractions can also represent division: 3/4 means 3 ÷ 4.
Types of Fractions
| Type | Definition | Example |
|---|---|---|
| Proper fraction | Numerator < denominator | 3/4 |
| Improper fraction | Numerator > denominator | 7/4 |
| Mixed number | Whole number + proper fraction | 1¾ |
| Unit fraction | Numerator = 1 | 1/5 |
Equivalent Fractions
Equivalent fractions look different but represent the same value.
½ = 2/4 = 3/6 = 4/8
To create an equivalent fraction, multiply or divide both the numerator and denominator by the same number.
2/3 × (2/2) = 4/6 ← equivalent to 2/3
Simplifying Fractions
Divide both numerator and denominator by their Greatest Common Factor (GCF) until no common factor remains.
18/24 → GCF of 18 and 24 is 6 → 18÷6 / 24÷6 = 3/4
A fraction is in its simplest form (or lowest terms) when the numerator and denominator share no common factor other than 1.
Adding and Subtracting Fractions
Same denominator: add or subtract the numerators; keep the denominator.
3/7 + 2/7 = 5/7
Different denominators: find the Lowest Common Denominator (LCD), convert to equivalent fractions, then add or subtract.
½ + ¼ → LCD = 4 → 2/4 + 1/4 = 3/4
⅔ - ½ → LCD = 6 → 4/6 - 3/6 = 1/6
Step-by-step for 3/4 + 2/5:
- LCD of 4 and 5 is 20
- 3/4 = 15/20 and 2/5 = 8/20
- 15/20 + 8/20 = 23/20 = 1 and 3/20
Multiplying Fractions
Multiply numerators together, then denominators together. Simplify the result.
¾ × ⅔ = (3 × 2) / (4 × 3) = 6/12 = ½
Tip: Cancel common factors before multiplying to keep numbers small: 3/4 × 2/3 → cancel 3s and cancel 2 and 4: = 1/2
Dividing Fractions
Use Keep, Change, Flip (also called "multiply by the reciprocal"):
- Keep the first fraction unchanged
- Change ÷ to ×
- Flip the second fraction (swap numerator and denominator)
¾ ÷ ½ = ¾ × 2/1 = 6/4 = 3/2 = 1½
2/5 ÷ 4/3 = 2/5 × 3/4 = 6/20 = 3/10
Mixed Numbers and Improper Fractions
A mixed number has a whole number and a fraction: 2¾ An improper fraction has a numerator larger than the denominator: 11/4
Convert mixed → improper: multiply whole number by denominator, add numerator, keep denominator. 2¾ = (2 × 4 + 3) / 4 = 11/4
Convert improper → mixed: divide numerator by denominator, write quotient as whole number and remainder as numerator. 11/4 = 2 remainder 3 = 2¾
Comparing Fractions
To compare fractions, give them the same denominator, then compare numerators.
Which is bigger: 3/5 or 5/8?
- LCD = 40
- 3/5 = 24/40 and 5/8 = 25/40
- 25 > 24, so 5/8 is bigger
Quick rule: for fractions with the same numerator, the one with the smaller denominator is larger (1/3 > 1/5).
Fractions of a Quantity
To find a fraction of a quantity, divide by the denominator then multiply by the numerator.
3/4 of 28: 28 ÷ 4 = 7, then 7 × 3 = 21
2/5 of 60: 60 ÷ 5 = 12, then 12 × 2 = 24
Common Mistakes
- Adding denominators: ½ + ¼ ≠ 2/6. Never add denominators — find the LCD instead.
- Forgetting to simplify: always reduce the final answer to its simplest form.
- Flipping the wrong fraction when dividing: flip the second fraction (the divisor), not the first.
- Mixing up improper fraction conversion: remember to add the numerator after multiplying, not multiply the whole number by the fraction.
Tips and Tricks
- To quickly compare two fractions a/b and c/d, cross-multiply: compare a × d with b × c. The larger product belongs to the larger fraction.
- When multiplying mixed numbers, convert to improper fractions first.
- "Of" in a word problem means multiply: "3/4 of 20" = 3/4 × 20 = 15.
- To add three or more fractions, find the LCD for all denominators at once.