Decimals
1. Understanding Decimals
A decimal is a fraction written in a special form using a decimal point. The decimal point separates the whole number part from the fractional part.
Example: In 34.67:
- 34 is the whole number part.
- 6 is the tenths place (6/10).
- 7 is the hundredths place (7/100).
2. Place Value in Decimals
| Thousands | Hundreds | Tens | Ones | . | Tenths | Hundredths | Thousandths |
|---|---|---|---|---|---|---|---|
| 1000 | 100 | 10 | 1 | . | 1/10 | 1/100 | 1/1000 |
Worked Example: Write the place value of each digit in 47.329.
4 = 4 tens (40), 7 = 7 ones (7), 3 = 3 tenths (3/10), 2 = 2 hundredths (2/100), 9 = 9 thousandths (9/1000).
Common Mistake: Thinking 0.3 and 0.30 are different. They are equivalent! 0.3 = 3/10 = 30/100 = 0.30.
Exam Focus (2 marks): 'Write 7.06 in expanded form.'
7.06 = 7 x 1 + 0 x 1/10 + 6 x 1/100 = 7 + 6/100.
3. Converting Fractions to Decimals
Denominator is 10, 100, or 1000
Count the number of zeros in the denominator and place the decimal point accordingly.
| Fraction | Decimal |
|---|---|
| 7/10 | 0.7 |
| 23/100 | 0.23 |
| 459/1000 | 0.459 |
| 3 7/100 | 3.07 |
Denominator is NOT 10, 100, or 1000
Convert to an equivalent fraction with denominator 10, 100, or 1000, then write the decimal.
Worked Example: Convert 3/5 to decimal.
3/5 = (3 x 2)/(5 x 2) = 6/10 = 0.6.
Worked Example: Convert 1/4 to decimal.
1/4 = (1 x 25)/(4 x 25) = 25/100 = 0.25.
4. Converting Decimals to Fractions
Write the decimal without the point as the numerator. Denominator is 1 followed by as many zeros as decimal places.
| Decimal | Fraction | Simplified |
|---|---|---|
| 0.7 | 7/10 | 7/10 |
| 0.35 | 35/100 | 7/20 |
| 2.4 | 24/10 | 12/5 = 2 2/5 |
| 0.125 | 125/1000 | 1/8 |
Common Mistake: Writing 0.05 as 5/10 instead of 5/100. Remember: two decimal places means denominator 100.
5. Comparing Decimals
Step 1: Compare the whole number parts first.
Step 2: If whole parts are equal, compare tenths.
Step 3: If tenths are equal, compare hundredths, and so on.
Worked Example: Arrange in ascending order: 0.7, 0.67, 0.607, 0.706.
Write them with the same number of decimal places:
0.700, 0.670, 0.607, 0.706.
Ascending: 0.607 < 0.670 < 0.700 < 0.706.
Common Mistake: Thinking 0.67 > 0.7 because 67 > 7. Correct: 0.7 = 0.70 so 0.70 > 0.67.
6. Addition and Subtraction of Decimals
Rule: Align the decimal points vertically, then add or subtract as with whole numbers.
Worked Example: Add 23.45 + 7.8 + 0.329.
23.450
+ 7.800
+ 0.329
--------
31.579
Worked Example: Subtract 9.54 from 15.2.
15.20
- 9.54
--------
5.66
Common Mistake: Not aligning decimal points, e.g., adding 23.45 + 7.8 as 23.45 + 7.80 = 31.25 (wrong). Always line up the decimal points.
7. Comparison Table: Fractions vs Decimals
| Aspect | Fractions | Decimals |
|---|---|---|
| Representation | a/b | With decimal point |
| Precision | Exact by nature | May involve rounding |
| Operations | LCM needed | Direct alignment |
| Everyday use | Cooking, sharing | Money, measurements |
8. Self-Test
- Write the decimal for: (a) 3/20 (b) 7/8 (c) 2 3/25.
- Convert to fraction in simplest form: (a) 0.45 (b) 3.6 (c) 0.008.
- Arrange in descending order: 0.5, 0.55, 0.505, 0.055.
- Add: 12.65 + 4.8 + 7.005.
- Subtract: 25.3 - 18.47.
- Write 38.207 in expanded form.
- Which is greater: 0.3 or 0.30? Explain.
9. Answers to Self-Test
- (a) 0.15 (b) 0.875 (c) 2.12.
- (a) 9/20 (b) 18/5 = 3 3/5 (c) 1/125.
- 0.55 > 0.505 > 0.5 > 0.055.
- 24.475.
- 6.83.
- 38.207 = 30 + 8 + 2/10 + 0/100 + 7/1000.
- They are equal. 0.30 = 0.3 (trailing zeros do not change value).
