Mastering the language of parts before algebra demands it.
Section 01
What Is a Fraction?
A fraction represents a part of a whole. It expresses a relationship between two integers: the number of parts you have, and the total number of equal parts the whole is divided into.
numeratordenominator
=
parts you havetotal equal parts
Key Terms
Term
Meaning
Example in 3/7
Numerator
Top — parts counted
3
Denominator
Bottom — total equal parts
7
Fraction bar
Means division (÷)
3 ÷ 7
Fraction Types
Type
Definition
Example
Proper fraction
Numerator < Denominator
3/5
Improper fraction
Numerator ≥ Denominator
7/4
Mixed number
Integer + proper fraction
1¾
Unit fraction
Numerator = 1
1/8
Think of a fraction as a slice of pizza: the denominator is how many equal slices the pizza was cut into; the numerator is how many slices you’re holding.
Section 02
Equivalent Fractions & Simplifying
Two fractions are equivalent when they represent the same portion of a whole. You create equivalent fractions by multiplying or dividing both numerator and denominator by the same non-zero number.
12
=
24
=
510
=
50100
Simplifying (Reducing) a Fraction
Divide both numerator and denominator by their Greatest Common Factor (GCF) until no common factor remains.
Worked Example — Simplify 18/24
1
List factors of 18: 1, 2, 3, 6, 9, 18. List factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
2
GCF = 6
3
18 ÷ 6 = 3 | 24 ÷ 6 = 4
4
Result: 3/4 — fully simplified (GCF of 3 and 4 is 1)
Common Mistake
Only divide by a common factor. Students often divide the numerator and denominator by different numbers — this changes the fraction’s value and is wrong. The rule: whatever you do to the top, do to the bottom.
Section 03
Multiplying & Dividing Fractions
Multiplication
Multiply straight across: numerator × numerator, denominator × denominator. Simplify before or after.
ab
×
cd
=
a × cb × d
Worked Example — Multiply 3/5 × 4/7
1
Multiply numerators: 3 × 4 = 12
2
Multiply denominators: 5 × 7 = 35
3
Check if simplifiable: GCF(12, 35) = 1. Answer: 12/35
Division — Keep, Change, Flip (KCF)
To divide by a fraction, multiply by its reciprocal (flip the second fraction).
ab
÷
cd
=
ab
×
dc
=
adbc
Worked Example — Divide 2/3 ÷ 4/9
1
Keep the first fraction: 2/3
2
Change ÷ to ×: 2/3 ×
3
Flip the second fraction: 2/3 × 9/4
4
Multiply: 18/12 = 3/2 → 1½
Section 04
Adding & Subtracting Fractions
You can only add or subtract fractions when they share a common denominator. Once denominators match, operate on numerators only — the denominator stays.
Same Denominator
ac
+
bc
=
a+bc
Just add the tops. Keep the bottom.
Different Denominators
Find the Lowest Common Denominator (LCD), convert each fraction, then add/subtract numerators.
Worked Example — Add 2/3 + 3/4
1
LCD of 3 and 4: multiples of 4 are 4, 8, 12 ✓. LCD = 12
2
Convert: 2/3 = 8/12 | 3/4 = 9/12
3
Add numerators: 8 + 9 = 17
4
Result: 17/12 = 1 5/12 (improper → mixed)
Worked Example — Subtract 5/6 − 1/4
1
LCD of 6 and 4 = 12
2
Convert: 5/6 = 10/12 | 1/4 = 3/12
3
Subtract: 10 − 3 = 7
4
Answer: 7/12
Section 05
Mixed Numbers & Improper Fractions
Mixed → Improper
2¾
1
Multiply integer × denominator: 2 × 4 = 8
2
Add numerator: 8 + 3 = 11
3
Result: 11/4
Improper → Mixed
17/5
1
Divide: 17 ÷ 5 = 3 remainder 2
2
Integer part = 3, remainder = 2
3
Result: 3 2/5
Section 06
Fractions in Algebra
In algebra, variables appear in numerators and denominators. The same fraction rules apply — the letters simply travel along with the numbers.
Algebraic Fraction Examples
Operation
Example
Result
Simplify
6x / 4x
3/2 (x ≠ 0)
Multiply
(2/3) × (x/5)
2x/15
Add (same denom.)
x/7 + 3/7
(x+3)/7
Add (diff. denom.)
x/2 + x/3
3x/6 + 2x/6 = 5x/6
Critical Rule
When a variable appears in the denominator, you must note it cannot equal zero. 1/x is undefined when x = 0. This restriction becomes essential in later lessons.