Decimals
1. Understanding Tenths, Hundredths, Thousandths
A decimal is another way to write a fraction where the denominator is 10, 100, or 1000.
Tenths
1/10 = 0.1 (one-tenth) 3/10 = 0.3 (three-tenths) 9/10 = 0.9 (nine-tenths)
'One whole = 10 tenths. The FIRST place after the decimal point is TENTHS.'
Hundredths
1/100 = 0.01 (one-hundredth) 35/100 = 0.35 (thirty-five hundredths)
| Fraction | Decimal | Words |
|---|---|---|
| 1/10 | 0.1 | One-tenth |
| 1/100 | 0.01 | One-hundredth |
| 25/100 | 0.25 | Twenty-five hundredths |
| 99/100 | 0.99 | Ninety-nine hundredths |
Thousandths
1/1000 = 0.001 (one-thousandth) 375/1000 = 0.375 (three hundred seventy-five thousandths)
Place Value Chart
| H | T | O | . | Tenths (1/10) | Hundredths (1/100) | Thousandths (1/1000) |
|---|---|---|---|---|---|---|
| 2 | 3 | 4 | . | 5 | 6 | 7 |
This number is 234.567 = Two hundred thirty-four point five six seven.
'Every decimal digit has a PLACE VALUE one-tenth of the place to its left. The decimal point separates the WHOLE part from the FRACTIONAL part.'
Expanded Form
234.567 = 200 + 30 + 4 + 0.5 + 0.06 + 0.007
2. Converting Between Fractions and Decimals
Fraction to Decimal
Divide the numerator by the denominator.
| Fraction | Working | Decimal |
|---|---|---|
| 3/5 | 3 ÷ 5 = 0.6 | 0.6 |
| 7/20 | 7 ÷ 20 = 0.35 | 0.35 |
| 9/8 | 9 ÷ 8 = 1.125 | 1.125 |
| 2 3/4 | 2 + (3 ÷ 4) = 2 + 0.75 | 2.75 |
'To convert a fraction to a decimal, think of the fraction as a DIVISION problem. The numerator goes INSIDE the division box.'
Decimal to Fraction
Write the decimal as a fraction with denominator 10, 100, or 1000. Simplify.
| Decimal | Fraction (unsimplified) | Simplified |
|---|---|---|
| 0.7 | 7/10 | 7/10 |
| 0.45 | 45/100 | 9/20 |
| 0.125 | 125/1000 | 1/8 |
| 2.6 | 26/10 | 13/5 = 2 3/5 |
'Count the number of decimal places. That tells you the denominator — 1 decimal place = 10, 2 places = 100, 3 places = 1000.'
3. Comparing and Ordering Decimals
Comparing Decimals
| Step | Example: Compare 0.45 and 0.4 |
|---|---|
| 1. Make the decimal places equal | 0.45 and 0.40 |
| 2. Remove the decimal point | 45 and 40 |
| 3. Compare as whole numbers | 45 > 40, so 0.45 > 0.4 |
'ALWAYS add trailing zeros to make the number of decimal places equal BEFORE comparing.'
Ordering
Arrange in ascending order: 0.8, 0.75, 0.825, 0.7
Step 1: Make equal decimal places (3 places): 0.800, 0.750, 0.825, 0.700. Step 2: Compare: 0.700 < 0.750 < 0.800 < 0.825. Step 3: Ascending order: 0.7, 0.75, 0.8, 0.825.
4. Addition and Subtraction of Decimals
Addition
Align the decimal points. Add like whole numbers. Place the decimal point.
2 3 4.5 6
+ 1 2 3.4 5
-----------
3 5 8.0 1
'Line up the DECIMAL POINTS. If a number has fewer decimal places, add zeros to make them equal.'
Subtraction
5 6 7.8 0
− 3 4 5.6 7
-----------
2 2 2.1 3
Real-World Example: Money
₹250.50 + ₹175.75 = ₹426.25
'When working with money, decimals are NATURAL. Rupees are the whole part, paise are the decimal part (100 paise = ₹1).'
5. Multiplication of Decimals
Decimal × Whole Number
3.25 × 4
Step 1: Multiply without decimal: 325 × 4 = 1,300. Step 2: Count decimal places in original (2 decimal places in 3.25). Step 3: Place decimal: 13.00 = 13.
'Multiply as if there is NO decimal. Then count the total decimal places from the RIGHT and place the decimal point.'
Decimal × Decimal
2.5 × 1.3
Step 1: 25 × 13 = 325. Step 2: Total decimal places = 1 + 1 = 2. Step 3: 2.5 × 1.3 = 3.25.
Multiplying by 10, 100, 1000
| Multiply By | Move Decimal Point | Example |
|---|---|---|
| 10 | 1 place RIGHT | 0.45 × 10 = 4.5 |
| 100 | 2 places RIGHT | 0.45 × 100 = 45 |
| 1000 | 3 places RIGHT | 0.45 × 1000 = 450 |
'When multiplying a decimal by 10, 100, or 1000 — move the decimal point to the RIGHT by as many places as there are zeros.'
6. Applications — Money and Measurement
Money
| Rupees | Conversion |
|---|---|
| ₹1 | 100 paise |
| 50 paise | ₹0.50 |
| 75 paise 50 paisa | ₹0.755 |
| ₹5.75 | 575 paise |
Length
| Unit | Conversion | Decimal Form |
|---|---|---|
| 1 m | 100 cm | 1.00 m |
| 75 cm | 0.75 m | 0.75 m |
| 2 m 50 cm | 2.50 m | 2.50 m |
| 1 km | 1000 m | 1.000 km |
Mass
| Unit | Conversion | Decimal Form |
|---|---|---|
| 1 kg | 1000 g | 1.000 kg |
| 500 g | 0.500 kg | 0.500 kg |
| 2 kg 250 g | 2.250 kg | 2.250 kg |
Key Facts to Remember
- The decimal point separates the whole number part from the fractional part.
- Adding a zero AFTER the decimal does NOT change the value (0.5 = 0.50 = 0.500).
- 'Zeros to the LEFT of the first non-zero decimal digit DO change the value: 0.05 ≠ 0.5.'
- A decimal greater than 1 has a whole number part to the left of the decimal point.
- Every decimal can be written as a fraction.
Common Mistakes
| Mistake | Why It Is Wrong | Correct Approach |
|---|---|---|
| Misaligning decimal points | 3.45 + 12.3 = 4.68 (aligned 45+3) | Line up the decimal points: 3.45 + 12.30 = 15.75 |
| Thinking 0.5 = 0.05 | 0.5 is five-tenths, 0.05 is five-hundredths | 0.5 > 0.05 — 0.5 = 0.50 |
| Forgetting to count digits after decimal | 0.25 × 0.3 = 0.75 (wrong) | 25 × 3 = 75. 3 decimal places: 0.075 |
| Writing ₹5.75 as 5.75 paise | ₹1 = 100 paise, so ₹5.75 = 575 paise | ₹5.75 is 5 rupees 75 paise |
Exam Focus (ICSE Class 5)
| Topic | Marks (Typical) | Question Type |
|---|---|---|
| Place value in decimals | 2-3 marks | Identify the digit in a given place |
| Fraction ↔ Decimal conversion | 3-4 marks | Convert and simplify |
| Addition and subtraction | 4-5 marks | Computation and word problems |
| Multiplication by 10, 100, 1000 | 2 marks | Direct computation |
| Money and measurement applications | 4 marks | Word problems with ₹, m, kg |
Self-Test: 5 Questions
Q1. Write 7/8 as a decimal and 0.625 as a fraction in simplest form.
Q2. Arrange in descending order: 0.35, 0.3, 0.305, 0.35.
Q3. Simplify: 45.78 + 123.456 − 67.9.
Q4. Multiply: 0.75 × 0.8.
Q5. A shopkeeper bought 5.5 kg of apples at ₹85.50 per kg. How much did he pay?
Answers
A1. 7/8 = 0.875. 0.625 = 625/1000 = 5/8.
A2. 0.35 (0.350) = 0.35 (second 0.35), then 0.305, then 0.3 (0.300). So: 0.35 = 0.35 > 0.305 > 0.3.
A3. 45.78 + 123.456 = 169.236. 169.236 − 67.9 = 101.336.
A4. 75 × 8 = 600. Total decimal places = 2 + 1 = 3. 0.75 × 0.8 = 0.600 = 0.6.
A5. 5.5 × 85.50 = 55/10 × 855/10 = 47025/100 = ₹470.25.
