Fractions and Decimals

What you'll learn

• Classify fractions as proper, improper, or mixed
• Find equivalent fractions and simplify fractions
• Add, subtract, multiply, and divide fractions
• Perform operations on decimals
• Convert between fractions and decimals

Key concepts

Proper, Improper, and Mixed Fractions

Converting between improper and mixed:

Improper → Mixed: Divide numerator by denominator.

• 17/5 → 17 ÷ 5 = 3 remainder 2 → 3⅖

Mixed → Improper: (Whole × Denominator + Numerator) / Denominator

• 4⅔ → (4 × 3 + 2) / 3 = 14/3

Equivalent Fractions

Equivalent fractions represent the same value.

To find equivalent fractions: Multiply (or divide) both numerator and denominator by the same non-zero number.

Simplifying (reducing) fractions: Divide both terms by their HCF.

Worked Example: Simplify 36/48. HCF(36, 48) = 12 36/48 = (36÷12)/(48÷12) = 3/4

Comparing fractions:

• Same denominator → compare numerators: 3/7 < 5/7
• Different denominators → find LCM, convert, then compare.

Worked Example: Compare 3/4 and 5/6. LCM(4, 6) = 12 3/4 = 9/12, 5/6 = 10/12 9/12 < 10/12, so 3/4 < 5/6

Operations on Fractions

Addition and Subtraction

Same denominator: Add/subtract numerators, keep denominator.

• 3/8 + 1/8 = 4/8 = 1/2

Different denominators: Convert to same denominator using LCM.

Worked Example: 2/3 + 3/4 LCM(3, 4) = 12 2/3 = 8/12, 3/4 = 9/12 8/12 + 9/12 = 17/12 = 1⁵⁄₁₂

Worked Example (subtraction): 5/6 − 1/4 LCM(6, 4) = 12 5/6 = 10/12, 1/4 = 3/12 10/12 − 3/12 = 7/12

Mixed number addition: Add whole parts and fraction parts separately, then combine.

• 2⅓ + 1½ = (2+1) + (1/3 + 1/2) = 3 + (2/6 + 3/6) = 3 + 5/6 = 3⅚

Multiplication

Fraction × Fraction: Multiply numerators together and denominators together.

Formula: (a/b) × (c/d) = ac/bd

Worked Example: 3/5 × 4/7 = 12/35

Mixed number multiplication: Convert to improper fraction first.

• 2½ × 1⅓ = 5/2 × 4/3 = 20/6 = 10/3 = 3⅓

Fraction × Whole number:

• 3/4 × 8 = 3 × 8 / 4 = 24/4 = 6

Division

Dividing fractions: Multiply by the reciprocal of the divisor.

Formula: (a/b) ÷ (c/d) = (a/b) × (d/c) = ad/bc

Worked Example: How many pieces of 2/5 m can be cut from 4 m of rope? 4 ÷ (2/5) = 4 × 5/2 = 20/2 = 10 pieces

Decimal Operations

Addition and Subtraction

Rule: Align decimal points, then add or subtract as with whole numbers.

  23.450        47.300
+  8.375      -  9.625
--------      --------
  31.825        37.675

Worked Example: 12.6 + 3.45 + 0.007 Write as 12.600 + 3.450 + 0.007 = 16.057

Multiplication

Decimal × Whole number: Multiply, then place decimal point.

• 3.14 × 6 = 18.84

Decimal × Decimal: Multiply as whole numbers, then count total decimal places.

Worked Example: 0.25 × 0.4 25 × 4 = 100; decimal places = 2+1 = 3 Answer: 0.100 = 0.1

Division

Decimal ÷ Whole number: Divide normally, carry the decimal point.

• 8.4 ÷ 4 = 2.1

Decimal ÷ Decimal: Convert divisor to whole number by multiplying both by power of 10.

• 6.4 ÷ 0.8 = 64 ÷ 8 = 8

Worked Example: 3.75 ÷ 0.05 Multiply both by 100: 375 ÷ 5 = 75

Converting Fractions to Decimals

Method 1 — Division: Divide numerator by denominator.

• 3/8 → 3 ÷ 8 = 0.375
• 2/3 → 2 ÷ 3 = 0.666… = 0.6̄

Method 2 — Equivalent fraction with power-of-10 denominator:

• 3/4 = 75/100 = 0.75
• 7/25 = 28/100 = 0.28

Converting decimals to fractions:

• 0.6 = 6/10 = 3/5
• 0.125 = 125/1000 = 1/8
• 1.75 = 175/100 = 7/4 = 1¾

Summary table:

Quick check

1. Convert 4⅗ to an improper fraction, then to a decimal.
2. Calculate: 5/6 + 3/4 − 1/3. Express as a mixed number.
3. Find: 2¼ × 1⅗
4. A tank holds 18.6 litres. If 6 equal portions are taken out, how much is each portion?
5. Arrange in ascending order: 3/4, 0.7, 5/8, 0.65

Open the Practice tab for graded questions on Fractions and Decimals.