Grade 8 · CAPS Mathematics

Common & Decimal Fractions

Interactive lesson — work through each section and check your answers

Section 1 of 5

Equivalent Forms Day 1

Common fractions · Decimal fractions · Percentages — three ways to say the same thing

Key Idea: Common fractions, decimal fractions and percentages are simply different notations for the same number. A decimal fraction has a power of ten in the denominator.

\(\dfrac{2}{10} = 0{,}2\) ← one decimal place = tenths | \(\dfrac{26}{100} = 0{,}26\) ← two decimal places = hundredths

Place Value Chart

Thousands	Hundreds	Tens	Units	,	Tenths	Hundredths	Thousandths
—	—	—	0	,	2	6	8
The number above is 0,268 — read as: nought comma two six eight

How to Convert a Common Fraction → Decimal

1Identify the denominator. Ask: what do I multiply it by to get a power of 10 (10, 100, 1000…)?

2Multiply both numerator and denominator by that number (equivalent fraction rule).

3Write using decimal notation.

Example: Convert \(\dfrac{3}{25}\) to a decimal.

Denominator 25 × 4 = 100, so multiply top and bottom by 4:

\dfrac{3}{25} = \dfrac{3 \times 4}{25 \times 4} = \dfrac{12}{100} = 0{,}12

Example: Convert \(2\dfrac{3}{5}\) to a decimal.

2\dfrac{3}{5} = 2 + \left(\dfrac{3}{5} \times \dfrac{2}{2}\right) = 2 + \dfrac{6}{10} = 2 + 0{,}6 = 2{,}6

The Three-Way Table

Common Fraction	Percentage	Decimal Fraction
\(\dfrac{1}{4}\)	25%	0,25
\(\dfrac{1}{2}\)	50%	0,5
\(\dfrac{3}{4}\)	75%	0,75
\(\dfrac{3}{10}\)	30%	0,3
\(\dfrac{7}{20}\)	35%	0,35

Check Your Understanding

1Convert \(\dfrac{4}{5}\) to a decimal fraction.

Solution: Denominator 5 × 20 = 100.
\(\dfrac{4}{5} = \dfrac{4 \times 20}{5 \times 20} = \dfrac{80}{100} = 0{,}80 = \mathbf{0{,}8}\)

2Write \(\dfrac{125}{1\,000}\) as a decimal fraction.

Solution: 3 decimal places (thousandths).
\(\dfrac{125}{1\,000} = \mathbf{0{,}125}\)

3Which percentage is equivalent to \(0{,}35\)?

Solution: \(0{,}35 = \dfrac{35}{100} = 35\%\)

4Convert \(3\dfrac{2}{4}\) to a decimal. (Type your answer as a decimal)

Solution: \(3\dfrac{2}{4} = 3\dfrac{1}{2}\)
\(\dfrac{1}{2} = \dfrac{5}{10} = 0{,}5\)
\(\therefore 3\dfrac{2}{4} = \mathbf{3{,}5}\)

Ordering & Comparing Day 2

Compare decimal fractions to at least three decimal places

Rule: Compare decimal fractions digit by digit, starting from the tenths, then hundredths, then thousandths.

Trailing zeros do not change the value:
\(0{,}1 = 0{,}10 = 0{,}100\) — all the same number.

Strategy: Make Decimal Places Equal First

Example: Which is bigger — 3,5 or 3,05?

Write both with the same decimal places: 3,50 and 3,05
Compare tenths: 5 > 0 → 3,5 > 3,05

Ordering on a number line: A number line between 7 and 8 has 10 intervals; each small division = 0,1 (one tenth). A half-division = 0,05.

Symbols to know

Symbol	Meaning	Example
<	less than	\(0{,}45 < 0{,}451\)
>	greater than	\(0{,}9 > 0{,}87\)
=	equal to	\(0{,}10 = 0{,}1\)

Check Your Understanding

1Which symbol completes: \(0{,}62 \;\square\; 0{,}26\)?

Solution: Tenths: 6 > 2, so \(0{,}62 \mathbf{>} 0{,}26\)

2Arrange in ascending order (smallest → biggest): 0,925 | 0,012 | 0,125 | 0,87

Solution: Compare tenths: 0 (0,012) < 1 (0,125) < 8 (0,87) < 9 (0,925)
∴ 0,012 ; 0,125 ; 0,87 ; 0,925

3Is \(0{,}1 = 0{,}10\)?

Solution: Yes — trailing zeros don't change the value. \(0{,}10 = \dfrac{10}{100} = \dfrac{1}{10} = 0{,}1\)

4Which is the smallest? 0,004 | 0,04 | 0,4

Solution: \(0{,}004 = \dfrac{4}{1000}\), \(0{,}04 = \dfrac{4}{100}\), \(0{,}4 = \dfrac{4}{10}\)
Smaller denominator = larger piece, but here numerator is same — smaller denominator means larger fraction.
∴ smallest is 0,004

Rounding Off Day 3

Round to nearest whole number, 1 or 2 decimal places

The Golden Rule: Always look at the digit one position to the right of where you are rounding.

• If that digit is 5 or more → round up
• If that digit is less than 5 → round down (keep the same)

Rounding to	Look at	Example	Result
Nearest whole number	Tenths digit	5,7	6
Nearest whole number	Tenths digit	5,3	5
1 decimal place	Hundredths digit	5,78	5,8
1 decimal place	Hundredths digit	5,43	5,4
2 decimal places	Thousandths digit	5,699	5,70
2 decimal places	Thousandths digit	4,453	4,45

Worked example: Round 81,899 to 2 decimal places.
Look at the third decimal: it's 9 (≥ 5), so round up the hundredths digit.
8 → 9 → \(\mathbf{81{,}90}\)

Check Your Understanding

1Round 29,452 to the nearest whole number.

Solution: Tenths digit = 4 (less than 5) → round down. Answer: 29

2Round 5,555 to one decimal place.

Solution: Look at hundredths digit = 5 (≥ 5) → round up tenths digit from 5 to 6. Answer: 5,6

3Round 7,394 to one decimal place.

Solution: Hundredths digit = 9 (≥ 5) → round up tenths digit. 3 becomes 4. Answer: 7,4

4Round 9,439 to two decimal places.

Solution: Thousandths digit = 9 (≥ 5) → round up hundredths digit. 3 becomes 4. Answer: 9,44

Operations with Decimal Fractions Day 4

Addition · Subtraction · Multiplication · Division

Addition & Subtraction

Line up the decimal commas. Tenths add to tenths, hundredths to hundredths.

Example: \(2{,}37 + 4{,}53 - 3{,}88\)

= (2+4-3) + (0{,}3+0{,}5-0{,}8) + (0{,}07+0{,}03-0{,}08) = 3 + 0 + 0{,}02 = \mathbf{3{,}02}

Multiplication

Convert to whole numbers (multiply by a power of 10), multiply, then divide back.

Shortcut for decimal × decimal: Count total decimal places; the answer has that many places.
\(0{,}2 \times 0{,}3 = 0{,}06\) (1 + 1 = 2 decimal places)
\(0{,}02 \times 0{,}3 = 0{,}006\) (2 + 1 = 3 decimal places)

Key rule: Multiplying by a number less than 1 gives a smaller answer.
e.g. \(2 \times 0{,}75 = 1{,}5\)

Division

Find the decimal with the most decimal places. Multiply both numbers by that power of 10 to make them whole numbers, then divide normally.

Example: \(6{,}25 \div 2{,}5\)
Multiply both by 10: \(62{,}5 \div 25 = 2{,}5\)
Or multiply both by 100: \(625 \div 250 = \mathbf{2{,}5}\)

Key rule: Dividing by a number between 0 and 1 gives a larger answer.
e.g. \(4 \div 0{,}5 = 8\)

Check Your Understanding

1Calculate: \(16{,}52 + 9{,}38\)

Solution: Line up decimals: 16,52 + 9,38
Units: 6+9=15 (write 5, carry 1)
Tens: 1+1=2
Tenths: 5+3=8
Hundredths: 2+8=10 (write 0, carry 1 to tenths → 8+1=9)
Answer: \(\mathbf{25{,}90}\)

2Calculate: \(0{,}12 \times 0{,}3\)

Solution: 2 + 1 = 3 decimal places total.
\(12 \times 3 = 36\), then place 3 decimal places: \(\mathbf{0{,}036}\)

3Calculate: \(62{,}5 \div 2{,}5\)

Solution: Multiply both by 10: \(625 \div 25 = \mathbf{25}\)

4True or False: \(3 \div 0{,}5 = 1{,}5\)

Solution: Dividing by a number between 0 and 1 gives a larger answer.
\(3 \div 0{,}5 = 3 \times \dfrac{1}{0{,}5} = 3 \times 2 = \mathbf{6}\). The statement is False.

Common & Decimal Fractions

Place Value Chart

How to Convert a Common Fraction → Decimal

The Three-Way Table

Strategy: Make Decimal Places Equal First

Symbols to know

Addition & Subtraction

Multiplication

Division

What this quiz covers

Example questions