Decimals — Tenths, Hundredths, and Money
1. Introduction to Decimals
A DECIMAL is another way to write a FRACTION that has a denominator of 10, 100, or 1000.
'A decimal point (.) separates the WHOLE part from the FRACTIONAL part.'
Examples:
- 0.5 = 5/10 (five-tenths)
- 0.25 = 25/100 (twenty-five hundredths)
- 1.5 = 1 + 5/10 (one and five-tenths)
2. Tenths
One-TENTH = 1 out of 10 equal parts of a whole.
| Fraction | Decimal | Words |
|---|---|---|
| 1/10 | 0.1 | One-tenth |
| 2/10 | 0.2 | Two-tenths |
| 3/10 | 0.3 | Three-tenths |
| 4/10 | 0.4 | Four-tenths |
| 5/10 | 0.5 | Five-tenths |
| 6/10 | 0.6 | Six-tenths |
| 7/10 | 0.7 | Seven-tenths |
| 8/10 | 0.8 | Eight-tenths |
| 9/10 | 0.9 | Nine-tenths |
| 10/10 | 1.0 | One |
'If you cut a chocolate bar into 10 equal pieces and eat 3, you have eaten 3/10 or 0.3 of the bar.'
3. Hundredths
One-HUNDREDTH = 1 out of 100 equal parts of a whole.
| Fraction | Decimal | Words |
|---|---|---|
| 1/100 | 0.01 | One-hundredth |
| 10/100 | 0.10 | Ten-hundredths |
| 25/100 | 0.25 | Twenty-five hundredths |
| 50/100 | 0.50 | Fifty-hundredths |
| 75/100 | 0.75 | Seventy-five hundredths |
| 99/100 | 0.99 | Ninety-nine hundredths |
| 100/100 | 1.00 | One |
Converting Tenths to Hundredths:
'1/10 = 10/100. So 0.1 = 0.10. They are equivalent!'
| Tenths | Hundredths |
|---|---|
| 0.1 | 0.10 |
| 0.2 | 0.20 |
| 0.5 | 0.50 |
| 0.7 | 0.70 |
| 0.9 | 0.90 |
4. Converting Fractions to Decimals
Denominator is 10:
Write the numerator after the decimal point in the TENTHS place.
- 3/10 = 0.3
- 7/10 = 0.7
- 9/10 = 0.9
Denominator is 100:
Write the numerator after the decimal point in the HUNDREDTHS place.
- 15/100 = 0.15
- 45/100 = 0.45
- 8/100 = 0.08 (use a ZERO in the tenths place!)
'When writing hundredths, always use TWO digits after the decimal point. If the numerator is less than 10, put a 0 in the tenths place.'
| Fraction | Decimal | Steps |
|---|---|---|
| 5/10 | 0.5 | 5 in tenths place |
| 5/100 | 0.05 | 0 in tenths, 5 in hundredths |
| 25/100 | 0.25 | 2 in tenths, 5 in hundredths |
| 3/100 | 0.03 | 0 in tenths, 3 in hundredths |
5. Place Value in Decimals
| ... | Hundreds | Tens | Ones | . | Tenths | Hundredths |
|---|---|---|---|---|---|---|
| ... | 100 | 10 | 1 | . | 1/10 | 1/100 |
Example:
Write 23.45 in expanded form.
23.45 = 20 + 3 + 4/10 + 5/100
| Digit | Place | Value |
|---|---|---|
| 2 | Tens | 20 |
| 3 | Ones | 3 |
| 4 | Tenths | 4/10 = 0.4 |
| 5 | Hundredths | 5/100 = 0.05 |
6. Addition of Decimals
Steps:
- Line up the DECIMAL POINTS.
- Add from the RIGHT — hundredths → tenths → ones → tens → etc.
- CARRY if needed.
Example:
Add 12.45 + 7.38
12.45
+ 7.38
--------
19.83
Step 1: Line up decimal points: 12.45 and 07.38 (add a leading zero) Step 2: 5 + 8 = 13 → write 3, carry 1 hundredth Step 3: 4 + 3 + 1 = 8 → write 8 tenths Step 4: 2 + 7 = 9 → write 9 ones Step 5: 1 + 0 = 1 → write 1 ten
Answer: 19.83
'Always line up the DECIMAL POINTS. If a number doesn't have a decimal part, add '.00' to help.'
7. Subtraction of Decimals
Steps:
- Line up the DECIMAL POINTS.
- Subtract from the RIGHT — hundredths → tenths → ones → tens → etc.
- BORROW if needed.
Example:
Subtract 7.38 from 15.50
15.50
- 7.38
--------
8.12
Another Example:
Subtract 3.25 from 8.00
8.00
- 3.25
--------
4.75
(Note: 8.00 = 8. We add .00 to help with borrowing.)
8. Money — Rupees and Paise
'In India, money is written in decimals. 1 Rupee = 100 Paise.'
| Money | In Paise | In Rupees (Decimal) |
|---|---|---|
| 1 Rupee | 100 paise | Re. 1.00 |
| 75 paise | 75 paise | Re. 0.75 |
| 1 Rupee 50 paise | 150 paise | Rs. 1.50 |
| 10 Rupees 25 paise | 1025 paise | Rs. 10.25 |
Converting:
Paise to Rupees: Divide by 100.
- 75 paise = 75 ÷ 100 = Rs. 0.75
- 250 paise = 250 ÷ 100 = Rs. 2.50
Rupees to Paise: Multiply by 100.
- Rs. 5.50 = 5.50 × 100 = 550 paise
- Rs. 12.05 = 12.05 × 100 = 1205 paise
Word Problem:
Riya buys a book for Rs. 156.75 and a pen for Rs. 25.50. How much does she spend?
Solution: 156.75 + 25.50 = Rs. 182.25
9. Common Mistakes
- Not lining up decimal points: 'When adding or subtracting decimals, the decimal points MUST be aligned. 12.45 + 7.3 should be 12.45 + 7.30, NOT 12.45 + 7.3 = 12.48!'
- Forgetting zeros in hundredths: '0.05 is five-hundredths. DON'T write 0.5 — that is five-tenths. They are NOT the same!'
- Paise conversion: '75 paise is Re. 0.75, NOT Rs. 75. Remember: 100 paise = 1 Rupee.'
- Adding unlike denominators: '0.5 (tenths) + 0.05 (hundredths) = 0.55, NOT 0.55. Actually 0.5 = 0.50, so 0.50 + 0.05 = 0.55. Think carefully!'
10. Key Facts to Remember
- 'The decimal point separates the whole number from the fractional part.'
- 'Adding a zero AFTER the decimal does NOT change the value (0.5 = 0.50 = 0.500).'
- '1 Rupee = 100 Paise. When converting, divide paise by 100 to get rupees.'
- 'The first digit after the decimal is TENTHS. The second digit is HUNDREDTHS.'
- 'A fraction with denominator 10 or 100 can be written as a decimal.'
11. Self-Test
Q1: Write as a decimal: (a) 7/10 (b) 23/100 (c) 4/100
Q2: Write as a fraction: (a) 0.3 (b) 0.45 (c) 0.07
Q3: Add: 23.45 + 56.78
Q4: Subtract: 89.50 - 34.25
Q5: Convert to rupees: (a) 450 paise (b) 75 paise (c) 1250 paise
Q6: Convert to paise: (a) Rs. 5.50 (b) Rs. 12.75
Q7: Which is larger — 0.5 or 0.05? Why?
Q8: Riya has Rs. 200. She buys a dress for Rs. 156.75. How much money is left?
Answers:
A1: (a) 0.7 (b) 0.23 (c) 0.04 A2: (a) 3/10 (b) 45/100 (c) 7/100 A3: 80.23 A4: 55.25 A5: (a) Rs. 4.50 (b) Re. 0.75 (c) Rs. 12.50 A6: (a) 550 paise (b) 1275 paise A7: 0.5 is larger. 0.5 = 0.50, 0.05 = 0.05. 50 hundredths > 5 hundredths. A8: Rs. 200.00 - Rs. 156.75 = Rs. 43.25
