Fractions, Decimals & Percents (Mastery) • Topic 3 of 3

Converting Fractions to Decimals

Two Types of Decimals:

Type	Description	Example
Terminating Decimal	Ends/stops after finite digits	1/4 = 0.25
Repeating Decimal	Digits repeat forever	1/3 = 0.333...

Converting Fraction to Decimal:

Method: Divide numerator by denominator

(1)/(4) = 1 ÷ 4 = 0.25 (terminating)
(1)/(3) = 1 ÷ 3 = 0.333... (repeating)
(5)/(8) = 5 ÷ 8 = 0.625 (terminating)
(2)/(3) = 2 ÷ 3 = 0.666... (repeating)

When does a fraction terminate?

A fraction in simplest form terminates if the denominator's prime factors are only 2 and/or 5!

(1)/(4) = 1/2² → terminates (0.25)
1/8 = 1/2³ → terminates (0.125)
3/20 = 3/(2²×5) → terminates (0.15)
(1)/(3) = (1)/(3) → repeats (0.333...)
1/6 = 1/(2×3) → repeats (0.1666...)

LONG DIVISION - FRACTION TO DECIMAL:

    1 ÷ 4 = 0.25 (terminating)
    
        0.25
    4 ) 1.00
        0
        ──
        10
         8
        ──
         20
         20
        ──
          0


    1 ÷ 3 = 0.333... (repeating)
    
        0.333...
    3 ) 1.000...
        0
        ──
        10
         9
        ──
         10
          9
         ──
         10


REPEATING DECIMAL NOTATION:

    1/3 = 0.333... = 0.̅3
    
    2/3 = 0.666... = 0.̅6
    
    1/6 = 0.1666... = 0.1̅6
    (the bar goes over only the repeating part)
    
    1/7 = 0.142857142857... = 0.̅142857


DENOMINATOR RULE - VISUAL:

    Terminating if denominator = 2ᵃ × 5ᵇ:
    
    1/2 = 0.5 ✓         1/3 = 0.333... ✗
    1/4 = 0.25 ✓        1/6 = 0.1666... ✗
    1/5 = 0.2 ✓         1/7 = 0.142857... ✗
    1/8 = 0.125 ✓       1/9 = 0.111... ✗
    1/10 = 0.1 ✓        1/11 = 0.0909... ✗

Solved Examples

Worked Example

Convert (3)/(8) to a decimal.

Solution

Divide: 3 ÷ 8 = 0.375
Answer: 0.375 (terminating)

Worked Example

Convert (5)/(6) to a decimal.

Solution

Divide: 5 ÷ 6 = 0.83333...
3 repeats: 0.8̅3 or 0.8333...
Answer: 0.8̅3 (repeating)

Worked Example

Without dividing, determine if 7/40 terminates or repeats.

Solution

Simplify: 7/40 is already simplified
Denominator 40 = 2³ × 5
Only prime factors are 2 and 5 → terminates
Answer: Terminates

Worked Example

Convert 1/7 to a decimal (to 6 decimal places).

Solution

1 ÷ 7 = 0.142857142857...
The pattern "142857" repeats
Answer: 0.̅142857

Key Points

Fraction → Decimal: divide numerator by denominator
Terminating decimal = ends (denominator has only 2 and 5 as factors)
Repeating decimal = digits repeat forever
Use bar notation (̅) to show repeating digits
Some fractions have very long repeating patterns

Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.To convert a fraction to a decimal, divide the numerator by the:

Explanation: Denominator.

Q2.$\tfrac12=$

Explanation: $0.5$.

Q3.$\tfrac34=$

Explanation: $0.75$.

Q4.$\tfrac14=$

Explanation: $0.25$.

Practice more MCQs → 📝 Assignment · 35 marks