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:

1. LCD of 4 and 5 is 20
2. 3/4 = 15/20 and 2/5 = 8/20
3. 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"):

1. Keep the first fraction unchanged
2. Change ÷ to ×
3. 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.