NCERT Solutions for Class 6 Maths Chapter 8 — Decimals
Key Concepts
| Concept | Definition | Example |
|---|---|---|
| Decimal | A number written with a decimal point to represent fractions with denominators 10, 100, 1000, etc. | 0.3, 2.75, 15.6 |
| Tenths | The first digit after the decimal point represents tenths (1/10) | 0.7 means 7 tenths = 7/10 |
| Hundredths | The second digit after the decimal point represents hundredths (1/100) | 0.05 means 5 hundredths = 5/100 |
| Decimal Place Value | Each position after the decimal point has a value 1/10 of the position to its left | 3.47 = 3 + 4/10 + 7/100 |
| Converting Fractions to Decimals | Write the numerator with the appropriate decimal places based on the denominator | 7/10 = 0.7; 23/100 = 0.23 |
| Converting Decimals to Fractions | Write the decimal as a fraction with denominator 10, 100, or 1000 and simplify | 0.75 = 75/100 = 3/4 |
| Comparing Decimals | Compare whole number parts first, then tenths, then hundredths | 3.45 < 3.54 because 4 < 5 in tenths place |
| Addition of Decimals | Align decimal points, then add digit by digit from right to left | 2.35 + 1.42 = 3.77 |
| Subtraction of Decimals | Align decimal points, then subtract digit by digit from right to left | 5.80 − 2.35 = 3.45 |
Exercise 8.1 — Tenths (Introduction to Decimals)
Q1. Write each of the following as decimals.
(i) Seven-tenths
- Seven-tenths = 7/10 = 0.7
(ii) Two ones and three-tenths
- Two ones = 2
- Three-tenths = 3/10 = 0.3
- Combined: 2 + 3/10 = 2.3
(iii) Fourteen point six
- Written directly as 14.6
- 14 is the whole number part, 6 is in the tenths place.
(iv) Six hundred point eight
- Written directly as 600.8
- 600 is the whole number part, 8 is in the tenths place.
Q2. Write the following decimals as fractions.
(i) 0.6
- 0.6 = 6/10
- Simplify: HCF(6, 10) = 2
- 6/10 = (6 ÷ 2)/(10 ÷ 2) = 3/5
(ii) 2.5
- 2.5 = 2 + 0.5 = 2 + 5/10 = 2 + 1/2 = 5/2
- Alternatively: 2.5 = 25/10 = (25 ÷ 5)/(10 ÷ 5) = 5/2
(iii) 1.0
- 1.0 = 1 (or 10/10 = 1)
(iv) 13.7
- 13.7 = 13 + 7/10 = (130 + 7)/10 = 137/10
Q3. Express the following as rupees using decimals.
(i) 5 paise
- 100 paise = 1 rupee
- 5 paise = 5/100 rupees = ₹0.05
(ii) 75 paise
- 75 paise = 75/100 rupees = ₹0.75
(iii) 20 rupees 50 paise
- 20 rupees + 50 paise = 20 + 50/100 = ₹20.50
Q4. Express the following as cm using decimals.
(i) 2 mm
- 10 mm = 1 cm
- 2 mm = 2/10 cm = 0.2 cm
(ii) 15 mm
- 15 mm = 15/10 cm = 1.5 cm
(iii) 5 cm 3 mm
- 5 cm + 3 mm = 5 + 3/10 = 5.3 cm
Q5. Write the following decimals in the place value table.
Number: 0.4
| Ones | Tenths |
|---|---|
| 0 | 4 |
Number: 2.7
| Ones | Tenths |
|---|---|
| 2 | 7 |
Number: 13.9
| Tens | Ones | Tenths |
|---|---|---|
| 1 | 3 | 9 |
Exercise 8.2 — Hundredths
Q1. Write as decimals.
(i) Eight-hundredths
- 8/100 = 0.08
(ii) Two hundred fifty-three hundredths
- 253/100 = 2.53
(iii) Six and twenty-five hundredths
- 6 + 25/100 = 6.25
(iv) Ninety and nine hundredths
- 90 + 9/100 = 90.09
Q2. Write the following as fractions in lowest terms.
(i) 0.60
- 0.60 = 60/100
- HCF(60, 100) = 20
- 60/100 = (60 ÷ 20)/(100 ÷ 20) = 3/5
(ii) 0.05
- 0.05 = 5/100
- HCF(5, 100) = 5
- 5/100 = (5 ÷ 5)/(100 ÷ 5) = 1/20
(iii) 0.25
- 0.25 = 25/100
- HCF(25, 100) = 25
- 25/100 = (25 ÷ 25)/(100 ÷ 25) = 1/4
(iv) 0.18
- 0.18 = 18/100
- HCF(18, 100) = 2
- 18/100 = (18 ÷ 2)/(100 ÷ 2) = 9/50
Q3. Express the following as rupees using decimals.
(i) 3 rupees 5 paise
- 3 + 5/100 = ₹3.05
(ii)) 25 paise
- 25/100 = ₹0.25
(iii) 8 rupees 75 paise
- 8 + 75/100 = ₹8.75
Q4. Express the following as metres using decimals.
(i) 15 cm
- 100 cm = 1 m
- 15 cm = 15/100 m = 0.15 m
(ii) 6 cm
- 6/100 m = 0.06 m
(iii) 2 m 45 cm
- 2 + 45/100 = 2.45 m
Q5. Write the following decimals in the place value table.
Number: 0.39
| Ones | Tenths | Hundredths |
|---|---|---|
| 0 | 3 | 9 |
Number: 2.05
| Ones | Tenths | Hundredths |
|---|---|---|
| 2 | 0 | 5 |
Number: 14.87
| Tens | Ones | Tenths | Hundredths |
|---|---|---|---|
| 1 | 4 | 8 | 7 |
Exercise 8.3 — Comparing and Ordering Decimals
Q1. Which is greater?
(i) 0.3 or 0.5
- Compare tenths place: 3 < 5
- 0.5 is greater
(ii) 0.07 or 0.09
- Whole number parts are equal (both 0).
- Compare tenths: both have 0.
- Compare hundredths: 7 < 9
- 0.09 is greater
(iii) 3.0 or 0.3
- Compare whole number parts: 3 > 0
- 3.0 is greater
(iv) 0.8 or 0.88
- Write 0.8 as 0.80
- Compare: 0.80 < 0.88 (since hundredths: 0 < 8)
- 0.88 is greater
(v) 5.64 or 5.06
- Whole number parts are equal (both 5).
- Compare tenths: 6 > 0
- 5.64 is greater
Q2. Arrange in ascending order: 0.3, 0.03, 0.33, 0.1
- 0.30, 0.03, 0.33, 0.10
Step 2: Compare:
- 0.03 < 0.10 < 0.30 < 0.33
Ascending order: 0.03, 0.1, 0.3, 0.33
Q3. Arrange in descending order: 2.05, 2.50, 0.25, 5.02
- 5.02 has the largest whole number part (5).
- 2.05, 2.50 have whole number 2.
- 0.25 has whole number 0.
Step 2: Among 2.05 and 2.50:
- Compare tenths: 0 < 5, so 2.50 > 2.05
Descending order: 5.02, 2.50, 2.05, 0.25
Q4. Which is the largest: 0.7, 0.07, 0.007, 0.0007?
- 0.7 = 7 tenths
- 0.07 = 7 hundredths
- 0.007 = 7 thousandths
- 0.0007 = 7 ten-thousandths
0.7 is the largest because 7 tenths > 7 hundredths > 7 thousandths > 7 ten-thousandths.
Q5. Write the following in ascending order: 1.11, 1.01, 1.10, 1.00
- Compare whole number: all are 1.
- Compare tenths: 0 (in 1.01 and 1.00), 1 (in 1.11 and 1.10).
- Among those with 0 in tenths: compare hundredths: 1.00 < 1.01
- Among those with 1 in tenths: compare hundredths: 1.10 < 1.11
Ascending order: 1.00, 1.01, 1.10, 1.11
Exercise 8.4 — Addition of Decimals
Q1. Add: 0.5 + 0.3
Q2. Add: 1.75 + 2.25
Q3. Add: 3.42 + 5.67
Q4. Add: 12.5 + 3.86
Q5. Nasreen bought 3 m 50 cm cloth for her shirt and 2 m 25 cm cloth for her trouser. Find the total length of cloth she bought.
- 3 m 50 cm = 3.50 m
- 2 m 25 cm = 2.25 m
Total cloth bought = 5.75 m
Q6. Add: 0.6 + 1.85 + 3.4
- 0.6 = 0.60, 1.85 = 1.85, 3.4 = 3.40
Q7. Samson bought a notebook for ₹15.75 and a pen for ₹8.50. How much did he spend in total?
Total spent = ₹24.25
Exercise 8.5 — Subtraction of Decimals
Q1. Subtract: 1.5 − 0.3
Q2. Subtract: 3.75 − 2.25
Q3. Subtract: 8.00 − 3.56
Q4. Subtract: 10.5 − 6.82
Q5. Urmila's school is at a distance of 7.62 km from her house. She walked 3.45 km and then took a bus. How much distance did she travel by bus?
Distance by bus = 4.17 km
Q6. Subtract: 25.00 − 13.87
Q7. Namita travels 20 km 50 m every day. Out of this she travels 10 km 200 m by bus and the rest by auto. How much distance does she travel by auto?
- 20 km 50 m = 20.050 km
- 10 km 200 m = 10.200 km
Distance by auto = 9.850 km = 9 km 850 m
Exercise 8.6 — Word Problems and Applications
Q1. Raju bought a book for ₹35.65 and a pen for ₹12.40. He gave ₹50 to the shopkeeper. How much money did he get back?
Total cost = ₹48.05
Step 2: Find the change.
Raju got back ₹1.95
Q2. A tin holds 16.5 litres of oil. If 4.25 litres of oil is used, how much oil is left?
Oil left = 12.25 litres
Q3. Sujata had ₹100. She bought a notebook for ₹24.50, a pencil box for ₹32.75 and an eraser for ₹5.00. How much money is left with her?
Step 2: Find remaining money.
Money left = ₹37.75
Q4. A cricket bat costs ₹285.75 and a ball costs ₹35.50. What is the total cost of one bat and two balls?
Step 2: Total cost.
Total cost = ₹356.75
Q5. The temperature at 12 noon was 15.5°C. It decreased by 3.2°C in the first hour and by 2.7°C in the second hour. What was the temperature at 2 pm?
Step 2: Temperature at 2 pm.
Temperature at 2 pm = 9.6°C
Q6. Rani had ₹18.50. She bought one chocolate for ₹11.75. How much money does she have now?
Rani has ₹6.75 left.
Common Mistakes
1. Misplacing the decimal point during addition/subtraction — Always align the decimal points vertically before adding or subtracting. Write extra zeros if needed (e.g., write 2.5 as 2.50 when adding to 3.75).
2. Confusing tenths and hundredths — In 0.5, the digit 5 is in the tenths place (5/10), NOT hundredths. In 0.05, the digit 5 is in the hundredths place (5/100).
3. Forgetting to write trailing zeros when comparing decimals — To compare 0.5 and 0.55, write 0.5 as 0.50 first. Then it becomes clear that 0.50 < 0.55.
4. Incorrect conversion between fractions and decimals — Remember: denominator 10 → one decimal place, denominator 100 → two decimal places. For example, 3/100 = 0.03, NOT 0.3.
5. Ignoring the decimal point in word problems — When a problem says "3 m 50 cm," convert it to 3.50 m, not 350 m or 3.5 cm. Pay attention to the unit.
6. Borrowing errors in subtraction — When subtracting decimals like 5.00 − 2.37, remember to borrow correctly across the decimal point, just as you do with whole numbers.
Exam Checklist
- [ ] I can write fractions with denominator 10, 100 as decimals (e.g., 7/10 = 0.7, 3/100 = 0.03)
- [ ] I can identify the place value of each digit in a decimal number (ones, tenths, hundredths)
- [ ] I can convert decimals to fractions and simplify them
- [ ] I can convert money (paise to rupees), length (cm to m), and weight (g to kg) using decimals
- [ ] I can compare two decimal numbers by comparing digits place by place
- [ ] I can arrange decimals in ascending and descending order
- [ ] I can add decimals by aligning decimal points
- [ ] I can subtract decimals by aligning decimal points and borrowing correctly
- [ ] I can solve word problems involving decimals for money, length, and weight
- [ ] I always align decimal points before adding or subtracting
- [ ] I simplify my final answer and remove unnecessary trailing zeros where appropriate