How to Use This Page
1. Attempt every problem on your own first — use your notebook and work through each step. 2. Compare your method with the step-by-step solution shown here. Focus on how the answer is reached, not just the final result. 3. Mark mistakes and redo them — if your answer differs, rework the problem before moving on. 4. Use the Key Concepts section as a quick revision sheet before exams.
Key Concepts (Quick Recap)
- Decimal place values: Tenths (1/10), Hundredths (1/100), Thousandths (1/1000).
- Rounding rule: If the digit to the right of the required place is ≥ 5, add 1 to the last retained digit; otherwise, leave it unchanged.
- Addition/Subtraction of decimals: Align decimal points, fill in trailing zeros, then operate as with whole numbers.
- Multiplication of decimals: Multiply as whole numbers, then place the decimal point so the product has as many decimal places as the sum of decimal places in the factors.
- Multiplication by 10, 100, 1000: Shift the decimal point right by 1, 2, or 3 places respectively.
- Division by 10, 100, 1000: Shift the decimal point left by 1, 2, or 3 places respectively.
- Division of decimals: Convert to fraction form (p/q), then divide. Alternatively, multiply both dividend and divisor by a power of 10 to make the divisor a whole number.
- Area formulas: Parallelogram = base × height; Rectangle = length × breadth.
- Perimeter of equilateral triangle = 3 × side.
Exercise 1.1 — Rounding Decimals
Q1: Round each decimal to the nearest whole number
(i) 8.71
The digit in ones place is 8.
The digit after the decimal point (tenths) is 7, which is ≥ 5.
So, add 1 to the ones digit: 8 + 1 = 9.
Answer: 9
(ii) 26.01
The digit in ones place is 6.
The tenths digit is 0, which is < 5.
So, the ones digit remains 6.
Answer: 26
(iii) 69.48
The digit in ones place is 9.
The tenths digit is 4, which is < 5.
So, the ones digit remains 9.
Answer: 69
(iv) 103.72
The digit in ones place is 3.
The tenths digit is 7, which is ≥ 5.
So, add 1: 3 + 1 = 4.
Answer: 104
(v) 49.84
The digit in ones place is 9.
The tenths digit is 8, which is ≥ 5.
So, add 1: 9 + 1 = 10. This carries over.
49 → 50
Answer: 50
(vi) 101.35
The digit in ones place is 1.
The tenths digit is 3, which is < 5.
So, the ones digit remains 1.
Answer: 101
(vii) 39.814
The digit in ones place is 9.
The tenths digit is 8, which is ≥ 5.
So, add 1: 9 + 1 = 10. Carry over.
39 → 40
Answer: 40
(viii) 1.23
The digit in ones place is 1.
The tenths digit is 2, which is < 5.
So, the ones digit remains 1.
Answer: 1
Q2: Round each decimal to the given place value
(i) 5.992 → tenths place
Tenths digit = 9. Next digit (hundredths) = 9, which is ≥ 5.
Add 1: 9 + 1 = 10. Carry over to ones place.
5.992 → 6.0
Answer: 6.0
(ii) 21.805 → hundredths place
Hundredths digit = 0. Next digit (thousandths) = 5, which is ≥ 5.
Add 1: 0 + 1 = 1.
21.805 → 21.81
Answer: 21.81
(iii) 35.0014 → thousandths place
Thousandths digit = 1. Next digit = 4, which is < 5.
Keep the thousandths digit unchanged.
35.0014 → 35.001
Answer: 35.001
Q3: Round to 1 decimal place
(i) 123.37
Tenths digit = 3. Hundredths digit = 7, which is ≥ 5.
Add 1: 3 + 1 = 4.
Answer: 123.4
(ii) 19.99
Tenths digit = 9. Hundredths digit = 9, which is ≥ 5.
Add 1: 9 + 1 = 10. Carry over.
19.99 → 20.0
Answer: 20.0
(iii) 910.546
Tenths digit = 5. Hundredths digit = 4, which is < 5.
Keep the tenths digit unchanged.
Answer: 910.5
Q4: Round to 2 decimal places
(i) 87.755
Hundredths digit = 5. Thousandths digit = 5, which is ≥ 5.
Add 1: 5 + 1 = 6.
Answer: 87.76
(ii) 301.513
Hundredths digit = 1. Thousandths digit = 3, which is < 5.
Keep the hundredths digit unchanged.
Answer: 301.51
(iii) 79.997
Hundredths digit = 9. Thousandths digit = 7, which is ≥ 5.
Add 1: 9 + 1 = 10. Carry over.
79.997 → 80.00
Answer: 80.00
Q5: Round to 3 decimal places
(a) 24.4003
Thousandths digit = 0. Next digit = 3, which is < 5.
Keep the thousandths digit unchanged.
Answer: 24.400
(b) 1251.2345
Thousandths digit = 4. Next digit = 5, which is ≥ 5.
Add 1: 4 + 1 = 5.
Answer: 1251.235
(c) 61.00203
Thousandths digit = 2. Next digit = 0, which is < 5.
Keep the thousandths digit unchanged.
Answer: 61.002
Exercise 1.2 — Addition and Subtraction of Decimals
Q1: Add by using grid: 0.51 + 0.25
0.51 = 51/100 and 0.25 = 25/100
On a 10 × 10 grid (100 squares), shade 51 squares for 0.51
and an additional 25 squares for 0.25.
Total shaded = 51 + 25 = 76 squares
76/100 = 0.76
Answer: 0.51 + 0.25 = 0.76
Q2: Add by using place value grid
(i) 25.8 + 18.53
Align decimal points:
25.80
+ 18.53
-------
44.33
Answer: 44.33
(ii) 17.4 + 23.435
Align decimal points:
17.400
+ 23.435
--------
40.835
Answer: 40.835
Q3: Find the value of 0.46 − 0.13 by grid model
0.46 = 46/100 and 0.13 = 13/100
Shade 46 squares on the 100-grid.
Cross out 13 of the shaded squares.
Remaining shaded = 46 − 13 = 33 squares
33/100 = 0.33
Answer: 0.46 − 0.13 = 0.33
Q4: Subtract by using place value grid
(i) 6.567 from 9.231
9.231
− 6.567
-------
2.664
Answer: 9.231 − 6.567 = 2.664
(ii) 3.235 from 7
7.000
− 3.235
-------
3.765
Answer: 7 − 3.235 = 3.765
Q5: Simplify: 23.5 − 27.89 + 35.4 − 17
Group positive and negative terms:
= (23.5 + 35.4) − (27.89 + 17)
= 58.9 − 44.89
= 14.01
Answer: 14.01
Q6: Sulaiman bought 3.350 kg of Potato, 2.250 kg of Tomato and some onions. If total weight is 10.250 kg, find weight of onions.
Given:
Weight of Potato = 3.350 kg
Weight of Tomato = 2.250 kg
Total weight = 10.250 kg
Find: Weight of onions
Weight of Potato + Tomato = 3.350 + 2.250 = 5.600 kg
Weight of onions = 10.250 − 5.600 = 4.650 kg
Answer: Weight of onions = 4.650 kg
Q7: What should be subtracted from 7.1 to get 0.713?
Let the number to be subtracted be x.
7.1 − x = 0.713
x = 7.1 − 0.713 = 6.387
Answer: 6.387
Q8: How much is 35.6 km less than 53.7 km?
Difference = 53.7 − 35.6 = 18.1
Answer: 35.6 km is 18.1 km less than 53.7 km
Q9: Akilan purchased a geometry box for ₹25.75, a pencil for ₹3.75 and a pen for ₹17.90. He gave ₹50 to the shopkeeper. What amount did he get back?
Given:
Cost of geometry box = ₹25.75
Cost of pencil = ₹ 3.75
Cost of pen = ₹17.90
Amount given = ₹50.00
Total cost = 25.75 + 3.75 + 17.90 = ₹47.40
Amount returned = 50.00 − 47.40 = ₹2.60
Answer: ₹2.60
Q10: Find the perimeter of an equilateral triangle with side 3.8 cm.
Given: Side = 3.8 cm
Formula: Perimeter of equilateral triangle = 3 × side
Perimeter = 3 × 3.8 = 11.4 cm
Answer: 11.4 cm
Objective Type Questions
1. 1.0 + 0.83 = ? → (iii) 1.83 2. 7.0 − 2.83 = ? → (ii) 4.17 3. Subtract 1.35 from 3.51 → (iv) 2.16 4. Sum of two decimals is 4.78; one is 3.21. Find the other. → (i) 1.57 5. Difference of two decimals is 86.58; one is 42.31. Find the other. → (i) 128.89
Exercise 1.3 — Multiplication of Decimals
Q1: Find the product of the following
(i) 0.5 × 3
5 × 3 = 15
Decimal places in 0.5 = 1
Place decimal: 1.5
Answer: 1.5
(ii) 3.75 × 6
375 × 6 = 2250
Decimal places in 3.75 = 2
Place decimal: 22.50
Answer: 22.50
(iii) 50.2 × 4
502 × 4 = 2008
Decimal places in 50.2 = 1
Place decimal: 200.8
Answer: 200.8
(iv) 0.03 × 9
3 × 9 = 27
Decimal places in 0.03 = 2
Place decimal: 0.27
Answer: 0.27
(v) 453.03 × 7
45303 × 7 = 317121
Decimal places in 453.03 = 2
Place decimal: 3171.21
Answer: 3171.21
(vi) 4 × 0.7
4 × 7 = 28
Decimal places in 0.7 = 1
Place decimal: 2.8
Answer: 2.8
Q2: Find the area of the parallelogram whose base is 6.8 cm and height is 3.5 cm
Given: Base (b) = 6.8 cm, Height (h) = 3.5 cm
Formula: Area = b × h
Area = 6.8 × 3.5
68 × 35 = 2380
Decimal places: 1 + 1 = 2
Answer: 23.80 cm²
Q3: Find the area of the rectangle whose length is 23.7 cm and breadth is 15.2 cm
Given: Length (l) = 23.7 cm, Breadth (b) = 15.2 cm
Formula: Area = l × b
Area = 23.7 × 15.2
237 × 152 = 36024
Decimal places: 1 + 1 = 2
Answer: 360.24 cm²
Q4: Multiply the following
| # | Problem | Calculation | Answer |
|---|---|---|---|
| (i) | 2.57 × 10 | Shift decimal right 1 place | 25.7 |
| (ii) | 0.51 × 10 | Shift decimal right 1 place | 5.1 |
| (iii) | 125.367 × 100 | Shift decimal right 2 places | 12536.7 |
| (iv) | 34.51 × 100 | Shift decimal right 2 places | 3451 |
| (v) | 62.735 × 100 | Shift decimal right 2 places | 6273.5 |
| (vi) | 0.7 × 10 | Shift decimal right 1 place | 7.0 |
| (vii) | 0.03 × 100 | Shift decimal right 2 places | 3 |
| (viii) | 0.4 × 1000 | Shift decimal right 3 places | 400 |
Q5: A wheel of a baby cycle covers 49.7 cm in one rotation. Find the distance covered in 10 rotations.
Given: Distance in 1 rotation = 49.7 cm
Find: Distance in 10 rotations
Distance = 49.7 × 10 = 497 cm
Answer: 497 cm
Q6: A picture chart costs ₹1.50. Radha wants to buy 20 charts. How much does she have to pay?
Given: Cost of 1 chart = ₹1.50
Find: Cost of 20 charts
Cost = 1.50 × 20 = ₹30
Answer: ₹30
Q7: Find the product of the following (decimal × decimal)
(i) 3.6 × 0.3
36 × 3 = 108
Decimal places: 1 + 1 = 2
Place decimal: 1.08
Answer: 1.08
(ii) 52.3 × 0.1
523 × 1 = 523
Decimal places: 1 + 1 = 2
Place decimal: 5.23
Answer: 5.23
(iii) 537.4 × 0.2
5374 × 2 = 10748
Decimal places: 1 + 1 = 2
Place decimal: 107.48
Answer: 107.48
(iv) 0.6 × 0.06
6 × 6 = 36
Decimal places: 1 + 2 = 3
Place decimal: 0.036
Answer: 0.036
(v) 62.2 × 0.23
622 × 23 = 14306
Decimal places: 1 + 2 = 3
Place decimal: 14.306
Answer: 14.306
(vi) 1.02 × 0.05
102 × 5 = 510
Decimal places: 2 + 2 = 4
Place decimal: 0.0510
Answer: 0.0510
(vii) 10.05 × 1.05
1005 × 105 = 105525
Decimal places: 2 + 2 = 4
Place decimal: 10.5525
Answer: 10.5525
(viii) 101.01 × 0.01
10101 × 1 = 10101
Decimal places: 2 + 2 = 4
Place decimal: 1.0101
Answer: 1.0101
(ix) 100.01 × 1.1
1001 × 11 = 110011
Decimal places: 2 + 1 = 3
Place decimal: 110.011
Answer: 110.011
Objective Type Questions
8. 1.07 × 0.1 = ? → (ii) 0.107 9. 2.08 × 10 = ? → (i) 20.8 10. A frog jumps 5.3 cm in one jump. Distance in 10 jumps? → 5.3 × 10 = (iii) 53.0 cm
Exercise 1.4 — Division of Decimals
Q1: Simplify the following (decimal ÷ whole number)
(i) 0.6 ÷ 3
0.6 ÷ 3 = (6/10) × (1/3) = 6/30 = 2/10 = 0.2
Answer: 0.2
(ii) 0.90 ÷ 5
0.90 ÷ 5 = (90/100) × (1/5) = 90/500 = 18/100 = 0.18
Answer: 0.18
(iii) 4.08 ÷ 4
4.08 ÷ 4 = (408/100) × (1/4) = 408/400 = 102/100 = 1.02
Answer: 1.02
(iv) 21.56 ÷ 7
21.56 ÷ 7 = (2156/100) × (1/7) = 2156/700 = 308/100 = 3.08
Answer: 3.08
(v) 0.564 ÷ 6
0.564 ÷ 6 = (564/1000) × (1/6) = 564/6000 = 94/1000 = 0.094
Answer: 0.094
(vi) 41.36 ÷ 4
41.36 ÷ 4 = (4136/100) × (1/4) = 4136/400 = 1034/100 = 10.34
Answer: 10.34
(vii) 298.2 ÷ 3
298.2 ÷ 3 = (2982/10) × (1/3) = 2982/30 = 994/10 = 99.4
Answer: 99.4
Q2: Simplify the following (÷ 10)
(i) 5.7 ÷ 10 → 0.57 (ii) 93.7 ÷ 10 → 9.37 (iii) 0.9 ÷ 10 → 0.09 (iv) 301.301 ÷ 10 → 30.1301 (v) 0.83 ÷ 10 → 0.083 (vi) 0.062 ÷ 10 → 0.0062
Q3: Simplify the following (÷ 100)
(i) 0.7 ÷ 100 → 0.007 (ii) 3.8 ÷ 100 → 0.038 (iii) 49.3 ÷ 100 → 0.493 (iv) 463.85 ÷ 100 → 4.6385 (v) 0.3 ÷ 100 → 0.003 (vi) 27.4 ÷ 100 → 0.274
Q4: Simplify the following (÷ 1000)
(i) 18.9 ÷ 1000 → 0.0189 (ii) 0.87 ÷ 1000 → 0.00087 (iii) 49.3 ÷ 1000 → 0.0493 (iv) 0.3 ÷ 1000 → 0.0003 (v) 382.4 ÷ 1000 → 0.3824 (vi) 93.8 ÷ 1000 → 0.0938
Q5: Simplify the following (decimal ÷ decimal)
(i) 19.2 ÷ 2.4
19.2 ÷ 2.4 = 192/24 = 8
Answer: 8
(ii) 4.95 ÷ 0.5
4.95 ÷ 0.5 = 495/50 = 9.9
Answer: 9.9
(iii) 19.11 ÷ 1.3
19.11 ÷ 1.3 = 191.1/13 = 14.7
Answer: 14.7
(iv) 0.399 ÷ 2.1
0.399 ÷ 2.1 = 3.99/21 = 0.19
Answer: 0.19
(v) 5.4 ÷ 0.6
5.4 ÷ 0.6 = 54/6 = 9
Answer: 9
(vi) 2.197 ÷ 1.3
2.197 ÷ 1.3 = 21.97/13 = 1.69
Answer: 1.69
Q6: Divide 9.55 kg of sweet among 5 children
Given: Total sweet = 9.55 kg, Number of children = 5
Find: Sweet per child
Each child gets = 9.55 ÷ 5 = 955/500 = 1.91 kg
Answer: Each child gets 1.91 kg
Q7: A vehicle covers 76.8 km for 1.2 litres of petrol. Distance for 1 litre?
Given: 76.8 km for 1.2 litres
Find: Distance for 1 litre
Distance per litre = 76.8 ÷ 1.2 = 768/12 = 64 km
Answer: 64 km per litre
Q8: Cost of levelling at ₹15.50/sq. ft is ₹10,075. Find area.
Given: Rate = ₹15.50 per sq. ft, Total cost = ₹10,075
Find: Area
Area = Total cost ÷ Rate = 10075 ÷ 15.50
= 10075 ÷ 15.5 = 100750/155 = 650 sq. ft
Answer: 650 sq. ft
Q9: Cost of 28 books is ₹1506.4. Find cost of one book.
Given: Total cost of 28 books = ₹1506.4
Find: Cost of 1 book
Cost of 1 book = 1506.4 ÷ 28 = 53.80
Answer: ₹53.80
Q10: Product of two numbers is 40.376. One number is 14.42. Find the other.
Given: Product = 40.376, One number = 14.42
Find: Other number
Other number = 40.376 ÷ 14.42 = 40376/14420 = 2.8
Answer: 2.8
Objective Type Questions
1. 5.6 ÷ 0.5 = ? → (iv) 11.2 2. 2.01 ÷ 0.03 = ? → (ii) 67.0 3. 0.05 ÷ 0.5 = ? → (ii) 0.1
Exercise 1.5 — Miscellaneous and Challenge Problems
Miscellaneous Practice Problems
Q1: Malini bought three ribbons of lengths 13.92 m, 11.5 m and 10.64 m. Find the total length.
Given: Ribbon lengths = 13.92 m, 11.5 m, 10.64 m
Find: Total length
Total = 13.92 + 11.50 + 10.64
13.92
+ 11.50
+ 10.64
-------
36.06
Answer: 36.06 m
Q2: Chitra bought 10 kg 35 g of ghee and used 8 kg 59 g. How much ghee is left?
Given: Total ghee = 10 kg 35 g = 10.35 kg
Used ghee = 8 kg 59 g = 8.59 kg
Find: Ghee left
Ghee left = 10.35 − 8.59
10.35
− 8.59
-------
1.76
Answer: 1.76 kg (or 1 kg 76 g)
Q3: If the capacity of a milk can is 2.53 l, how much milk is required to fill 8 such cans?
Given: Capacity of 1 can = 2.53 l
Find: Milk for 8 cans
Total = 2.53 × 8
253 × 8 = 2024
Decimal places = 2
= 20.24 l
Answer: 20.24 l
Q4: A basket of orange weighs 22.5 kg. If each family requires 2.5 kg, how many families can share?
Given: Total weight = 22.5 kg, Each family needs = 2.5 kg
Find: Number of families
Number = 22.5 ÷ 2.5 = 225/25 = 9
Answer: 9 families
Q5: A baker uses 3.924 kg of sugar to bake 10 cakes of equal size. How much sugar per cake?
Given: Sugar for 10 cakes = 3.924 kg
Find: Sugar per cake
Sugar per cake = 3.924 ÷ 10 = 0.3924 kg
Answer: 0.3924 kg
Q6: Evaluate
Q7: Each bag of vegetables weighs 20.55 kg. Total weight is 308.25 kg. How many bags?
Given: Weight per bag = 20.55 kg, Total weight = 308.25 kg
Find: Number of bags
Number = 308.25 ÷ 20.55 = 30825/2055 = 15
Answer: 15 bags
Q8: A man walks around a circular park of distance 23.761 m. How much distance in 100 rounds?
Given: Distance per round = 23.761 m
Find: Distance for 100 rounds
Total = 23.761 × 100 = 2376.1 m
Answer: 2376.1 m
Q9: How much is 0.0543 greater than 0.002?
Difference = 0.0543 − 0.002 = 0.0523
Answer: 0.0523
Q10: A printer prints 15 pages per minute. How many pages in 4.6 minutes?
Given: 15 pages/minute, Time = 4.6 minutes
Find: Total pages
Pages = 15 × 4.6 = 69
Answer: 69 pages
Challenge Problems
Q1: Prabhu travels 102 m from home to Yoga centre and 165 m from Yoga centre to school. Total distance in km (decimal form)?
Given: Home → Yoga centre = 102 m, Yoga centre → School = 165 m
Find: Total distance in km
Total distance = 102 + 165 = 267 m
Convert to km: 267/1000 = 0.267 km
Answer: 0.267 km
Q2: Anbu travels A→B (8.3 km) then B→C (15.6 km). Mala travels A→D (7.5 km) then D→C (16.9 km). Who travelled more and by how much?
Given: Anbu's route = 8.3 + 15.6 km, Mala's route = 7.5 + 16.9 km
Find: Who travelled more and by how much
Anbu's distance = 8.3 + 15.6 = 23.9 km
Mala's distance = 7.5 + 16.9 = 24.4 km
Difference = 24.4 − 23.9 = 0.5 km
Mala travelled more by 0.5 km
Answer: Mala by 0.5 km
Q3: Ramesh paid ₹97.75 per hour for a taxi, used 35 hours in a week. Total fare?
Given: Rate = ₹97.75/hour, Time = 35 hours
Find: Total fare
Total = 97.75 × 35 = ₹3421.25
Answer: ₹3421.25
Q4: An aeroplane travelled 2781.20 km in 6 hours. Find average speed in km/hr.
Given: Distance = 2781.20 km, Time = 6 hours
Find: Average speed
Speed = 2781.20 ÷ 6 = 463.53 km/hr
Answer: 463.53 km/hr
Q5: Kumar's car gives 12.6 km per litre mileage. Fuel tank holds 25.8 litres. How far can he travel?
Given: Mileage = 12.6 km/l, Fuel = 25.8 l
Find: Maximum distance
Distance = 12.6 × 25.8
126 × 258 = 32508
Decimal places: 1 + 1 = 2
= 325.08 km
Answer: 325.08 km
Common Mistakes
1. Forgetting to align decimal points in addition/subtraction — Always line up decimal points and pad with zeros (e.g., write 25.8 as 25.80 before adding to 18.53).
2. Misplacing the decimal point after multiplication — Count the total decimal places in both factors and ensure the product has that many places. Example: 0.6 × 0.06 has 1 + 2 = 3 decimal places, so the answer is 0.036, not 0.36.
3. Rounding errors — When rounding, look only at the digit immediately to the right of the required place. Do not round sequentially (e.g., for 5.992 → tenths, do not round hundredths first then tenths).
4. Confusing shift directions for ×10 and ÷10 — Multiplying by 10 shifts the decimal right; dividing by 10 shifts it left. A quick check: 0.5 × 10 = 5 (bigger), not 0.05.
5. Dividing decimals without converting the divisor — When dividing by a decimal (e.g., 5.4 ÷ 0.6), multiply both numbers by the same power of 10 to make the divisor a whole number first: 54 ÷ 6 = 9.
6. Dropping trailing zeros incorrectly — In money or measurement problems, 2.50 and 2.5 are numerically equal, but in context (₹2.50), the zero matters. Keep zeros where precision is needed.
7. Wrong carry-over in rounding — When 9 + 1 = 10 in rounding, carry over to the next left digit. For example, 79.997 rounded to 2 decimal places becomes 80.00, not 79.100.
Exam Checklist
- [ ] I can round decimals to any place value (whole number, tenths, hundredths, thousandths)
- [ ] I can add and subtract decimals by aligning decimal points
- [ ] I can multiply decimals and correctly place the decimal point in the product
- [ ] I can multiply/divide decimals by 10, 100, and 1000 by shifting the decimal point
- [ ] I can divide a decimal by a whole number using fraction method
- [ ] I can divide a decimal by another decimal by converting the divisor to a whole number
- [ ] I can solve word problems involving decimals (cost, distance, weight, area, perimeter)
- [ ] I can convert between units (e.g., m to km, kg and g to decimal kg)
- [ ] I remember to include correct units in word problem answers
- [ ] I double-check my decimal point placement in every calculation
Frequently Asked Questions
Rational numbers can be expressed as p/q where p and q are integers and q ≠ 0. Irrational numbers cannot be expressed as such fractions — their decimal expansions are non-terminating and non-recurring (e.g., √2, π).