Decimals — Class 6 Mathematics
'Decimals make the in-between numbers visible — from the price of a pen to the length of a pencil.'
1. Introduction
We already know fractions like 1/10, 1/100, 1/1000. Decimals are another way to write these fractions — using a decimal point (.). The decimal system is based on powers of 10, just like our number system.
Why Decimals Matter
- Money: Rs. 45.75 (rupees and paise)
- Length: 5.6 cm, 2.35 m
- Weight: 1.5 kg of rice, 250.5 g of dal
- AP context: Prices at the market — 'Rs. 28.50 per kg of mangoes'
Fraction to Decimal — Quick Look
| Fraction | Decimal | Read As |
|---|---|---|
| 1/10 | 0.1 | Zero point one |
| 1/100 | 0.01 | Zero point zero one |
| 1/1000 | 0.001 | Zero point zero zero one |
| 3/10 | 0.3 | Zero point three |
| 23/100 | 0.23 | Zero point two three |
2. Tenths
One-tenth = 1/10 = 0.1 = one part out of ten equal parts.
Representing Tenths
□□□□□□□□□□ 1 whole = 10 tenths
■□□□□□□□□□ 0.1 (one-tenth shaded)
■■□□□□□□□□ 0.2
■■■■■■■■■□ 0.9
■■■■■■■■■■ 1.0 (ten-tenths)
Converting Fractions to Decimals (Tenths)
| Fraction | Decimal | Read As |
|---|---|---|
| 1/10 | 0.1 | Zero point one |
| 2/10 | 0.2 | Zero point two |
| 5/10 | 0.5 | Zero point five |
| 9/10 | 0.9 | Zero point nine |
| 10/10 | 1.0 | One point zero |
Converting Decimals to Fractions
0.7 = 7/10 0.9 = 9/10 3.2 = 32/10 = 3 + 2/10
3. Hundredths
One-hundredth = 1/100 = 0.01 = one part out of one hundred equal parts.
Understanding 0.01
A square divided into 100 equal parts:
- 1 small part = 0.01
- 10 small parts = 0.10 = 0.1
- 100 small parts = 1.00 = 1
Converting Fractions to Decimals (Hundredths)
| Fraction | Decimal | Read As |
|---|---|---|
| 1/100 | 0.01 | Zero point zero one |
| 12/100 | 0.12 | Zero point one two |
| 25/100 | 0.25 | Zero point two five |
| 50/100 | 0.50 | Zero point five zero |
| 99/100 | 0.99 | Zero point nine nine |
Important Relationship
10 hundredths = 1 tenth = 0.1 0.10 = 0.1 (trailing zero does NOT change value)
4. Thousandths
One-thousandth = 1/1000 = 0.001 = one part out of one thousand equal parts.
Converting
| Fraction | Decimal | Read As |
|---|---|---|
| 1/1000 | 0.001 | Zero point zero zero one |
| 7/1000 | 0.007 | Zero point zero zero seven |
| 25/1000 | 0.025 | Zero point zero two five |
| 125/1000 | 0.125 | Zero point one two five |
| 999/1000 | 0.999 | Zero point nine nine nine |
Relationship Chain
1 = 10 tenths = 100 hundredths = 1000 thousandths 0.1 = 10 hundredths = 100 thousandths 0.01 = 10 thousandths
5. Place Value in Decimals
The place value system extends to the RIGHT of the decimal point.
Place Value Chart
| ... | Hundreds | Tens | Ones | . | Tenths | Hundredths | Thousandths |
|---|---|---|---|---|---|---|---|
| 100 | 10 | 1 | . | 1/10 | 1/100 | 1/1000 |
Example: 345.678
| Digit | Place | Value |
|---|---|---|
| 3 | Hundreds | 300 |
| 4 | Tens | 40 |
| 5 | Ones | 5 |
| . | Decimal point | |
| 6 | Tenths | 0.6 |
| 7 | Hundredths | 0.07 |
| 8 | Thousandths | 0.008 |
Value: 300 + 40 + 5 + 0.6 + 0.07 + 0.008 = 345.678
Expanded Form
345.678 = 3×100 + 4×10 + 5×1 + 6×1/10 + 7×1/100 + 8×1/1000 = 300 + 40 + 5 + 0.6 + 0.07 + 0.008
'Each move to the right divides the place value by 10. Each move to the left multiplies by 10.'
6. Comparing Decimals
Rule 1: Compare the Whole Part First
The decimal with the larger whole part is larger.
Example: 12.56 > 9.99 (12 > 9)
Rule 2: Same Whole Part — Compare Tenths
Example: 12.56 vs 12.78
- Whole parts equal (12 = 12)
- Tenths: 5 < 7
- Therefore 12.56 < 12.78
Rule 3: Compare Hundredths Then Thousandths
Example: 12.56 vs 12.59
- Whole parts equal, tenths equal (5 = 5)
- Hundredths: 6 < 9
- Therefore 12.56 < 12.59
Trick: Line Up Decimal Points
12.560
12.568
- Line up decimal points vertically
- Add trailing zeros to make same length
- Compare digit by digit from left
Worked Examples
| Comparison | Steps | Result |
|---|---|---|
| 0.5 vs 0.25 | 0.5 = 0.50 > 0.25 (tenths: 5 > 2) | 0.5 > 0.25 |
| 3.14 vs 3.140 | 3.14 = 3.140 (trailing zero doesn't change) | 3.14 = 3.140 |
| 0.07 vs 0.1 | 0.07 = 0.07, 0.1 = 0.10 (tenths: 0 < 1) | 0.07 < 0.1 |
'Many students think 0.07 > 0.1 because 7 > 1. But 0.1 = 0.10 = 10 hundredths, which is more than 7 hundredths.'
7. Using Decimals in Daily Life
Length
- 1 cm = 1/100 m = 0.01 m
- 5 cm = 5/100 m = 0.05 m
- 1 m 25 cm = 1.25 m
- 3 m 8 cm = 3.08 m
Example: A table is 2 m 35 cm long. Write in metres. Answer: 2 m + 35/100 m = 2.35 m
Weight
- 1 g = 1/1000 kg = 0.001 kg
- 500 g = 500/1000 kg = 0.5 kg
- 2 kg 250 g = 2.250 kg
- 150 g = 0.15 kg
Example: A bag of rice weighs 5 kg 750 g. Write in kg. Answer: 5 kg + 750/1000 kg = 5.750 kg = 5.75 kg
Money
- 1 paisa = 1/100 rupee = Rs. 0.01
- 50 paise = Rs. 0.50
- Rs. 7 and 25 paise = Rs. 7.25
- Rs. 125 and 75 paise = Rs. 125.75
Example: Sita bought vegetables for Rs. 45.50 and gave Rs. 100. How much change? Answer: Rs. 100.00 − Rs. 45.50 = Rs. 54.50
8. Addition of Decimals
Steps
- Write the numbers vertically, ALIGNING the decimal points
- Add trailing zeros if needed (optional, but helps)
- Add column by column from right to left
- Place the decimal point in the answer
Example 1: 23.45 + 7.8
23.45
+ 7.80 (added trailing zero)
--------
31.25
Example 2: 123.456 + 78.9 + 5.67
123.456
+ 78.900
+ 5.670
----------
208.026
Example 3 (Money): Rs. 245.50 + Rs. 378.75
245.50
+ 378.75
--------
624.25
Answer: Rs. 624.25
9. Subtraction of Decimals
Steps
- Align decimal points vertically
- Add trailing zeros
- Subtract column by column (borrow if needed)
- Place decimal point
Example 1: 56.78 − 23.45
56.78
− 23.45
--------
33.33
Example 2: 100 − 45.67
100.00
− 45.67
---------
54.33
Example 3 (Length): 5.6 m − 2.35 m
5.60
− 2.35
--------
3.25
Answer: 3.25 m
10. Word Problems
Problem 1 (Market Shopping)
Raju bought 2.5 kg of mangoes at Rs. 60 per kg. How much did he pay?
Solution: Cost = 2.5 × 60 = 150.0 Answer: Rs. 150.00
Problem 2 (AP Context — Mango Weight)
A farmer in Krishna district harvested 125.75 kg of mangoes on Monday and 98.50 kg on Tuesday. What is the total harvest?
Solution: 125.75 + 98.50 = 224.25 kg Answer: 224.25 kg
Problem 3 (Length)
A ribbon is 3.75 m long. If 1.25 m is cut off, how much is left?
Solution: 3.75 − 1.25 = 2.50 m Answer: 2.50 m
Problem 4 (Money — Savings)
Priya saves Rs. 45.50 per week. How much does she save in 4 weeks?
Solution: 45.50 × 4 = 182.00 Answer: Rs. 182.00
11. Common Mistakes — Fix Them Now
| # | Mistake | Correction |
|---|---|---|
| 1 | 0.3 < 0.25 (thinking 25 > 3) | Add zero: 0.30 > 0.25. Tenths: 3 > 2 |
| 2 | Not aligning decimal points: 3.5 + 12.75 = 4.775 | Align: 3.50 + 12.75 = 16.25 |
| 3 | 0.07 = 7/10 | 0.07 = 7/100 (two decimal places = hundredths) |
| 4 | Writing Rs. 45.5 instead of Rs. 45.50 | For money, always use two decimal places |
| 5 | 5.0 = 5 but not equal to 5.00 | 5.0 = 5.00 = 5. Trailing zeros don't change value |
12. Exam Focus
Marks Blueprint
| Question Type | Marks | Topic |
|---|---|---|
| MCQ | 1 | Comparison of decimals |
| Conversion | 1 | Fraction to decimal |
| Short answer | 2 | Addition/subtraction |
| Word problem | 3 | Real-life application |
| Place value | 2 | Expanded form |
Quick Self-Test (5 Questions)
Q1: Write 3/8 as a decimal.
<details><summary>Answer</summary>3÷8 = 0.375</details>Q2: Which is larger: 0.6 or 0.59?
<details><summary>Answer</summary>0.6 = 0.60 > 0.59</details>Q3: Add: 34.56 + 7.845
<details><summary>Answer</summary>34.560 + 7.845 = 42.405</details>Q4: Subtract: 50 − 23.75
<details><summary>Answer</summary>50.00 − 23.75 = 26.25</details>Q5: Write 5 m 35 cm in metres using decimal.
<details><summary>Answer</summary>5.35 m</details>13. Chapter Summary
- Decimals: fractions with denominators 10, 100, 1000 written with a decimal point
- Place value: ones. tenths (1/10) | hundredths (1/100) | thousandths (1/1000)
- Tenths: 1 decimal place; Hundredths: 2 places; Thousandths: 3 places
- Comparing: align points, compare left to right
- Real uses: length (m, cm), weight (kg, g), money (Rs, paise)
- Addition/Subtraction: align decimal points, add/subtract, place decimal
'Decimals bridge fractions and whole numbers — they are the language of measurement, money, and precision.'
