Decimals — Class 6 IMO

Place Value & Fraction–Decimal Conversion

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... ✗

Example 1: Write 0.75 as a fraction in simplest form.

0.75 = 75/100 = ¾.

Example 2: In 3.246, which place is the digit 6 in?

The thousandths place.

Example 3:

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

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

Example 4:

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

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

Example 5:

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

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

Example 6:

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

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

Quick recap

Decimal places: tenths, hundredths, thousandths.
Decimal to fraction: digits over 10/100/1000, then simplify.
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

Write ⅖ as a decimal.

⅖ = 4/10 = 0.4.

0.125 is the same as which fraction?

0.125 = 125/1000 = ⅛.

Comparing, Adding & Subtracting Decimals

To compare decimals, line up the points and compare place by place, adding zeros if needed. To add or subtract, line up the decimal points and work column by column.

So 2.45 + 1.55 = 4.00, and 5.6 − 2.35 = 3.25.

Example 1: Add 2.45 + 1.55.

2.45 + 1.55 = 4.00.

Example 2: Subtract 5.6 − 2.35.

5.60 − 2.35 = 3.25.

Quick recap

Compare and operate by lining up the decimal points.
Add zeros to give the same number of places if needed.

✓ Quick check

Which is greater, 0.7 or 0.68?

0.7 = 0.70 > 0.68.

What is 3.2 + 0.85?

3.20 + 0.85 = 4.05.

Multiplying & Dividing Decimals

To multiply a decimal by a whole number, multiply as usual and place the decimal point so there are the same number of decimal places as in the question. To divide, share equally and keep the point lined up.

So 1.2 × 3 = 3.6, and 4.5 ÷ 5 = 0.9.

Example 1: Multiply 1.2 × 3.

12 × 3 = 36, with one decimal place: 3.6.

Example 2: Divide 4.5 ÷ 5.

4.5 ÷ 5 = 0.9.

Quick recap

Multiply as whole numbers, then place the decimal point.
Divide and keep the decimal point lined up.

✓ Quick check

What is 0.6 × 4?

6 × 4 = 24, with one decimal place: 2.4.

What is 2.4 ÷ 6?

2.4 ÷ 6 = 0.4.