1. Chapter at a Glance
Petrol costs ₹104.50 a litre — what do 12.5 litres cost? A 2.4 m rod is cut into 0.3 m pieces — how many pieces? Such questions need multiplication and division of decimals. This chapter extends your earlier work with decimals: multiplying and dividing by 10, 100 and 1000, multiplying and dividing decimals by whole numbers and by other decimals, and using these skills with money and measurement.
- Multiplying and dividing decimals by 10, 100, 1000 (shifting the point).
- Multiplying a decimal by a whole number and by another decimal.
- Dividing a decimal by a whole number and by a decimal.
- Connecting decimal operations to fractions.
- Estimating to place the decimal point correctly.
2. Multiplying and Dividing by 10, 100, 1000
Each place is ten times the place to its right, so multiplying by 10 moves every digit one place left — the decimal point appears to move one place right.
| Operation | Decimal point moves | Example |
|---|---|---|
| × 10 | 1 place right | 3.47 × 10 = 34.7 |
| × 100 | 2 places right | 3.47 × 100 = 347 |
| × 1000 | 3 places right | 3.47 × 1000 = 3470 |
| ÷ 10 | 1 place left | 58.2 ÷ 10 = 5.82 |
| ÷ 100 | 2 places left | 58.2 ÷ 100 = 0.582 |
| ÷ 1000 | 3 places left | 58.2 ÷ 1000 = 0.0582 |
Add zeros as place-holders when needed: 0.6 × 1000 = 600; 4 ÷ 1000 = 0.004.
3. Multiplying Decimals
Decimal × whole number
3.25 × 4 means 325 hundredths × 4 = 1300 hundredths = 13.00. In practice: multiply as whole numbers, then put the decimal point so the answer has the same number of decimal places as the decimal.
Decimal × decimal
0.3 × 0.2 = 3/10 × 2/10 = 6/100 = 0.06. Tenths × tenths gives hundredths.
Multiply the numbers ignoring the decimal points. The number of decimal places in the product = total number of decimal places in the two numbers.
Petrol costs ₹104.50 per litre. Find the cost of 12.5 L.
Solution: 104.50 × 12.5: 10450 × 125 = 13,06,250; decimal places 2 + 1 = 3 → 1306.250. Cost = ₹1,306.25. Estimate check: 100 × 12.5 = 1250 ✔ close.
4. Dividing Decimals
Decimal ÷ whole number
Divide as usual and place the decimal point in the quotient directly above the point in the dividend. 7.56 ÷ 4 = 1.89. If needed, add zeros: 3 ÷ 8 = 3.000 ÷ 8 = 0.375.
Dividing by a decimal
Change the divisor into a whole number by multiplying both numbers by the same power of 10 — the quotient does not change.
A rope 9.6 m long is cut into pieces of 0.8 m. How many pieces?
Solution: 9.6 ÷ 0.8 = 96 ÷ 8 = 12 pieces.
Notice: dividing by a decimal less than 1 gives a larger number, just as with fractions.
5. Decimals and Fractions Together
- Every operation can be checked with fractions: 0.25 × 0.4 = ¼ × 2/5 = 1/10 = 0.1.
- Fractions with denominators made of only 2s and 5s become terminating decimals: 3/8 = 0.375, 7/20 = 0.35.
- Others become repeating decimals: ⅓ = 0.333…, 2/11 = 0.1818…
6. Solved Problems (NCERT Style)
Find the area of a rectangle 6.5 m by 4.2 m.
Solution: 65 × 42 = 2730; places 1 + 1 = 2 → 27.30 m².
Five parcels weigh 2.35 kg, 1.8 kg, 3.05 kg, 2.6 kg and 1.2 kg. Find the average weight.
Solution: Total = 11.00 kg. Average = 11 ÷ 5 = 2.2 kg.
2.5 kg of apples cost ₹337.50. Find the cost of 1 kg and of 4 kg.
Solution: 1 kg = 337.50 ÷ 2.5 = 3375 ÷ 25 = ₹135. 4 kg = ₹540.
A scooter runs 52.8 km on 1.2 L of petrol. How far does it run on 1 L?
Solution: 52.8 ÷ 1.2 = 528 ÷ 12 = 44 km per litre.
7. Master Formula Table
| Operation | Rule | Example |
|---|---|---|
| × 10ⁿ | point moves n places right | 0.07 × 100 = 7 |
| ÷ 10ⁿ | point moves n places left | 7 ÷ 100 = 0.07 |
| Decimal × decimal | places add up | 0.5 × 0.03 = 0.015 |
| ÷ whole number | point stays above | 8.4 ÷ 4 = 2.1 |
| ÷ decimal | scale both to make divisor whole | 0.84 ÷ 0.4 = 8.4 ÷ 4 |
| × number < 1 | smaller result | 50 × 0.2 = 10 |
| ÷ number < 1 | larger result | 50 ÷ 0.2 = 250 |
8. Master Summary Table
| Topic | Key idea | Exam focus |
|---|---|---|
| Powers of 10 | Shift the point | Unit conversion |
| Multiplication | Count decimal places | Cost, area |
| Division | Make divisor whole | Pieces, unit price |
| Size of results | × or ÷ by numbers below 1 | Estimation |
| Fractions link | Terminating vs repeating | Conversions |
9. Mind Map
- Point moves right/left
- Zeros as place-holders
- Ignore points, then count places
- Grid model
- By whole numbers
- By decimals (scale)
- Money, fuel
- Area, average
10. Tips and Tricks
- Estimate first to check the position of the decimal point: 4.8 × 3.1 ≈ 5 × 3 = 15, so 14.88 is reasonable.
- Multiplying by 0.5 = halving; by 0.25 = quartering; by 0.1 = dividing by 10.
- Remove trailing zeros at the end: 27.30 = 27.3.
- Lining up decimal points in multiplication (needed only in addition/subtraction).
- Scaling only the divisor, not the dividend, when dividing by a decimal.
- Writing 0.2 × 0.3 = 0.6 (it is 0.06).
11. Important One-liners
- × 10 moves the decimal point one place right.
- ÷ 100 moves it two places left.
- Decimal places in a product = sum of decimal places of the factors.
- Tenths × tenths = hundredths.
- To divide by a decimal, make the divisor a whole number by scaling both numbers.
- Multiplying by a number less than 1 makes a positive number smaller.
- Dividing by a number less than 1 makes a positive number larger.
- 3/8 = 0.375 is a terminating decimal; ⅓ = 0.333… repeats.
12. Exam-pattern Questions with Detailed Answers
0.45 × 100 = ?
- 4.5
- 45
- 450
- 0.0045
Show answer
Move the point 2 places right: 45.
Answer: (B) 45
Find 0.7 × 0.08.
Show answer
7 × 8 = 56; places 1 + 2 = 3 → 0.056.
Answer: 0.056
Find 12.6 ÷ 3.
Show answer
126 tenths ÷ 3 = 42 tenths = 4.2.
Answer: 4.2
Find 3.75 ÷ 0.25.
Show answer
Scale by 100: 375 ÷ 25 = 15.
Answer: 15
A notebook costs ₹23.75. Find the cost of 16 notebooks.
Show answer
23.75 × 16 = 380.00.
Answer: ₹380
Which is equal to 0.6 ÷ 0.02?
- 0.3
- 3
- 30
- 0.03
Show answer
Scale by 100: 60 ÷ 2 = 30.
Answer: (C) 30
Convert 2.35 km to metres and 465 g to kilograms.
Show answer
2.35 × 1000 = 2350 m. 465 ÷ 1000 = 0.465 kg.
Answer: 2350 m; 0.465 kg
Write 5/8 as a decimal and then find 5/8 of 4.8.
Show answer
5/8 = 0.625. 0.625 × 4.8 = 3.000 = 3.
Answer: 0.625; 3
A car travels 45.6 km in 0.8 hour. Find its speed in km per hour and the distance covered in 2.5 hours at that speed.
Show answer
Speed = 45.6 ÷ 0.8 = 456 ÷ 8 = 57 km/h. Distance = 57 × 2.5 = 142.5 km.
Answer: 57 km/h; 142.5 km
Given 158 × 24 = 3792, find (a) 1.58 × 2.4 (b) 37.92 ÷ 2.4.
Show answer
(a) Places 2 + 1 = 3 → 3.792. (b) 37.92 ÷ 2.4 = 379.2 ÷ 24 = 15.8 (since 3792 ÷ 24 = 158).
Answer: (a) 3.792 (b) 15.8
A tank holds 37.5 L. How many 0.75 L bottles can be filled, and how much is left if bottles of 0.8 L are used instead?
Show answer
37.5 ÷ 0.75 = 3750 ÷ 75 = 50 bottles exactly. With 0.8 L: 37.5 ÷ 0.8 = 46.875, so 46 full bottles use 36.8 L, leaving 0.7 L.
Answer: 50 bottles; with 0.8 L bottles, 46 bottles and 0.7 L left
The product of two decimals is 0.144 and one of them is 0.36. Find the other.
Show answer
0.144 ÷ 0.36 = 14.4 ÷ 36 = 0.4.
Answer: 0.4
13. Quick Revision Summary
- × 10ⁿ → point moves n places right; ÷ 10ⁿ → n places left.
- Multiply as whole numbers, then give the product as many decimal places as the factors have in total.
- Divide by a decimal after scaling both numbers so the divisor is whole.
- Numbers below 1: × makes smaller, ÷ makes larger.
- Always estimate to check the decimal point.