Fraction Arithmetic: Add, Subtract, Multiply, and Divide Step by Step

Fractions appear constantly in programming (ratios, probabilities, measurements) and everyday life. This guide walks through every operation with step-by-step explanations so the rule makes sense rather than just being memorised.

What is a fraction?

A fraction a/b means a parts out of b equal parts. The top number is the numerator, the bottom is the denominator:

  3   ← numerator
 ─── 
  4   ← denominator

A proper fraction has a numerator smaller than the denominator (e.g. 3/4). An improper fraction has a numerator ≥ denominator (e.g. 7/4). A mixed number combines an integer and a proper fraction (1¾).

Simplifying fractions (reducing to lowest terms)

Divide both numerator and denominator by their Greatest Common Divisor (GCD):

12/18 → GCD(12, 18) = 6 → 12÷6 / 18÷6 = 2/3

The Euclidean algorithm finds GCD efficiently:

GCD(12, 18):
  18 mod 12 = 6
  12 mod 6  = 0
  → GCD = 6

Addition and subtraction

Same denominator: add or subtract numerators directly:

3/8 + 2/8 = (3+2)/8 = 5/8

Different denominators: find the Least Common Denominator (LCD), convert each fraction, then add:

1/4 + 1/6
  LCD(4, 6) = 12
  1/4 = 3/12
  1/6 = 2/12
  3/12 + 2/12 = 5/12

The LCD is the Least Common Multiple (LCM) of the denominators. You can always use the product as a common denominator (then simplify afterwards):

LCM(4, 6) = (4 × 6) / GCD(4, 6) = 24 / 2 = 12

Subtraction follows the same rules:

5/6 − 1/4
  LCD = 12
  5/6 = 10/12
  1/4 = 3/12
  10/12 − 3/12 = 7/12

Multiplication

Multiply numerators together and denominators together, then simplify:

2/3 × 3/5 = (2×3) / (3×5) = 6/15 = 2/5

Cross-cancellation shortcut: simplify before multiplying:

2/3 × 3/5
  GCD(2,5)=1, GCD(3,3)=3
  Cancel the 3s: 2/1 × 1/5 = 2/5

For mixed numbers, convert to improper fractions first:

1½ × 2⅓
  = 3/2 × 7/3
  = 21/6
  = 7/2
  = 3½

Division

Dividing by a fraction = multiplying by its reciprocal:

3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1⅞

The reciprocal of a/b is b/a. This rule comes from the definition of division: dividing by X is multiplying by 1/X, and the reciprocal of a fraction is just flipping numerator and denominator.

Comparing fractions

To compare, convert to a common denominator:

Which is larger: 3/5 or 5/8?
  LCD = 40
  3/5 = 24/40
  5/8 = 25/40
  25/40 > 24/40 → 5/8 > 3/5

Alternatively, cross-multiply: 3 × 8 = 24, 5 × 5 = 25. Since 25 > 24, 5/8 > 3/5.

Common mistakes

Mistake	Wrong	Correct
Adding denominators	1/4 + 1/4 = 2/8	1/4 + 1/4 = 2/4 = 1/2
Forgetting to find LCD	1/3 + 1/4 = 2/7	4/12 + 3/12 = 7/12
Dividing instead of using reciprocal	1/2 ÷ 1/3 = 1/6	1/2 × 3/1 = 3/2
Not simplifying	leaving 6/9	2/3

Quick reference formulas

Addition:       a/b + c/d = (ad + bc) / bd   (then simplify)
Subtraction:    a/b - c/d = (ad - bc) / bd
Multiplication: a/b × c/d = ac / bd
Division:       a/b ÷ c/d = ad / bc

Use the Fraction Calculator to compute any operation with step-by-step working. For GCD and LCM calculations, see GCD Calculator and LCM Calculator.