Multiplying decimals by whole numbers and 10, 100, 1000
4 packets of 2.5 kg flour weigh 4 × 2.5 = 10 kg. Shop bills mix whole numbers and decimals every day.
To multiply a decimal by a whole number, multiply as if both are whole numbers, then put back the same number of decimal places. Multiplying by 10, 100, 1000 moves the decimal point right by 1, 2, 3 places.
In the figure: Each 0 in the multiplier moves the point one place right.
Multiplying decimals by whole numbers and 10, 100, 1000
In the app you can move this figure with your finger.
Open in the appRule 1: By 10, 100, 1000 — Move the point right; add zeros if needed.
- Easy: 3.45 × 10 = 34.5.
- Medium: 0.07 × 100 = 7.
- Exam-type: 12.6 × 1000 = 12600.
Rule 2: By any whole number — Ignore the point, multiply, then count the decimal places.
- Easy: 4.2 × 6: 42 × 6 = 252 → 25.2.
- Medium: 0.35 × 8: 35 × 8 = 280 → 2.80 = 2.8.
- Exam-type: 15.25 × 12 = 183.
Rule 3: Trailing zeros — Zeros at the end of the decimal part can be dropped.
- Easy: 2.50 = 2.5.
- Medium: 0.125 × 16 = 2.000 = 2.
- Exam-type: 9.6 × 25 = 240.
Decimal places in answer = decimal places in the decimal factor.
Moving the point left when multiplying by 10. It moves right.
Estimate first: 4.2 × 6 is close to 4 × 6 = 24. Memory clue: "Multiply bigger → point goes right."
Spot the difference
| Operation | Point moves |
|---|---|
| × 10, 100, 1000 | right |
| ÷ 10, 100, 1000 | left |
Practice questions
- 1Find 2.75 × 10.
- 2Find 0.09 × 100.
- 3Find 5.34 × 1000.
- 4Find 3.6 × 7.
- 5Find 0.45 × 12.
- 6Find 8.25 × 40.
- 7Find 0.0125 × 80.
- 8Find 16.4 × 250.
Show answers (try them yourself first)
- 127.5
- 29
- 35340
- 425.2
- 55.4
- 6330
- 71
- 84100
Multiplying decimals by decimals
A garden bed is 1.5 m long and 0.8 m wide. Its area is 1.5 × 0.8 square metres, a product of two decimals.
Remove both points, multiply the whole numbers, then give the answer as many decimal places as the two numbers have together. A product of two numbers less than 1 is smaller than both.
In the figure: The shaded part of the 10 × 10 grid is 0.3 of the width and 0.4 of the height: 12 small squares = 0.12.
Rule 1: Count the places — Places in the answer = places in first + places in second.
- Easy: 0.3 × 0.4: 3 × 4 = 12, two places → 0.12.
- Medium: 1.5 × 0.2 = 0.30 = 0.3.
- Exam-type: 0.6 × 0.07 = 0.042 (three places).
Rule 2: Longer numbers — Multiply whole numbers carefully, then place the point.
- Easy: 3.2 × 1.5: 32 × 15 = 480 → 4.80 = 4.8.
- Medium: 2.5 × 2.5 = 6.25.
- Exam-type: 0.12 × 0.15 = 0.018.
Rule 3: Three factors — Multiply two at a time, left to right.
- Easy: 0.075 × 0.8 × 2.5: 0.06 × 2.5 = 0.15.
- Medium: 1.25 × 0.64 = 0.8.
- Exam-type: 12.5 × 0.04 = 0.5.
Total decimal places of both numbers go to the answer.
Writing 0.3 × 0.4 = 1.2. Count two places: 0.12.
If both numbers are less than 1, the answer is smaller than both. Memory clue: "Count the places, then place the point."
Spot the difference
| Numbers | Product vs factors |
|---|---|
| both > 1 | bigger than both |
| both < 1 | smaller than both |
Practice questions
- 1Find 0.2 × 0.5.
- 2Find 1.2 × 0.3.
- 3Find 0.8 × 0.8.
- 4Find 2.5 × 0.4.
- 5Find 4.5 × 1.2.
- 6Find 0.15 × 0.6.
- 7Find 3.6 × 0.25.
- 8Find 0.04 × 2.5 × 1.5.
Show answers (try them yourself first)
- 10.1
- 20.36
- 30.64
- 41
- 55.4
- 60.09
- 70.9
- 80.15
Dividing decimals by whole numbers
₹8.40 shared equally among 4 children gives each ₹2.10. Sharing money is a decimal division.
Divide as whole numbers and put the point in the answer straight above the point in the dividend. Dividing by 10, 100, 1000 moves the point left. If a remainder is left, add zeros after the point and continue.
In the figure: 84 tenths shared among 4 gives 21 tenths, that is 2.1.
Rule 1: Point above point — Line up the points and divide as usual.
- Easy: 7.5 ÷ 5 = 1.5.
- Medium: 0.84 ÷ 4 = 0.21.
- Exam-type: 9.6 ÷ 3 = 3.2.
Rule 2: By 10, 100, 1000 — The point moves left one place for each zero.
- Easy: 36.4 ÷ 10 = 3.64.
- Medium: 5.2 ÷ 100 = 0.052.
- Exam-type: 123.5 ÷ 1000 = 0.1235.
Rule 3: Remainder? Add zeros — Write 0s after the last decimal digit and keep dividing.
- Easy: 2.5 ÷ 8: 2.500 ÷ 8 = 0.3125.
- Medium: 0.5 ÷ 8 = 0.0625.
- Exam-type: 100.1 ÷ 77 = 1.3.
Keep the answer's point right above the dividend's point.
Forgetting the zero in 0.045 ÷ 15 = 0.003. Add placeholder zeros.
Check by multiplying the answer back. Memory clue: "Point on point."
Spot the difference
| Divisor | Point in the answer |
|---|---|
| 10 | 1 place left |
| 100 | 2 places left |
Practice questions
- 1Find 8.4 ÷ 4.
- 2Find 0.65 ÷ 5.
- 3Find 52.3 ÷ 10.
- 4Find 6.3 ÷ 9.
- 5Find 14.4 ÷ 12.
- 6Find 0.096 ÷ 16.
- 7Find 2.5 ÷ 8.
- 8Find 0.9 ÷ 72.
Show answers (try them yourself first)
- 12.1
- 20.13
- 35.23
- 40.7
- 51.2
- 60.006
- 70.3125
- 80.0125
Dividing by decimals
How many 0.4 L glasses can be filled from 7.2 L of juice? We need 7.2 ÷ 0.4.
Make the divisor a whole number by multiplying both numbers by 10, 100, … equally; the answer does not change. Then divide as usual.
In the figure: 7.2 ÷ 0.4 and 72 ÷ 4 give the same answer, 18.
Rule 1: Shift both equally — Multiply both by 10 for one decimal place, by 100 for two.
- Easy: 4.5 ÷ 0.5 = 45 ÷ 5 = 9.
- Medium: 7.2 ÷ 0.4 = 72 ÷ 4 = 18.
- Exam-type: 1.8 ÷ 0.06 = 180 ÷ 6 = 30.
Rule 2: Whole number ÷ decimal — Add zeros to the dividend while shifting.
- Easy: 9 ÷ 0.3 = 90 ÷ 3 = 30.
- Medium: 8 ÷ 0.5 = 80 ÷ 5 = 16.
- Exam-type: 6.3 ÷ 0.9 = 63 ÷ 9 = 7.
Rule 3: Answer less than 1 — When the divisor is bigger, the answer is below 1.
- Easy: 0.72 ÷ 0.9 = 72 ÷ 90 = 0.8.
- Medium: 0.063 ÷ 0.21 = 6.3 ÷ 21 = 0.3.
- Exam-type: 0.144 ÷ 0.12 = 1.2.
Shift both numbers by the same amount.
Shifting only the divisor. Shift the dividend as well.
Multiply by 10, 100 until the divisor is whole. Memory clue: "Whole divisor first."
Spot the difference
| Divisor | Multiply both by |
|---|---|
| 0.4 | 10 |
| 0.06 | 100 |
Practice questions
- 1Find 3.6 ÷ 0.4.
- 2Find 8 ÷ 0.5.
- 3Find 0.84 ÷ 0.07.
- 4Find 6.3 ÷ 0.9.
- 5Find 1.44 ÷ 0.12.
- 6Find 0.085 ÷ 0.017.
- 7Find 9.75 ÷ 1.25.
- 8Find 0.0072 ÷ 0.0018.
Show answers (try them yourself first)
- 19
- 216
- 312
- 47
- 512
- 65
- 77.8
- 84
Estimation and decimal word problems
At a shop you round prices in your head: ₹19.90 is about ₹20. A quick estimate warns you if the final answer is far off.
To estimate, round the numbers to easy values, then calculate. Always compare the exact answer with the estimate. In word problems decide the operation, then work with decimals carefully.
In the figure: 4.78 is closest to 5 on the number line.
Estimation and decimal word problems
In the app you can move this figure with your finger.
Open in the appRule 1: Estimate by rounding — Round each number to the nearest whole or ten.
- Easy: 3.9 × 5.1 ≈ 4 × 5 = 20.
- Medium: 49.8 ÷ 10.1 ≈ 50 ÷ 10 = 5.
- Exam-type: 29.7 × 4.1 ÷ 5.9 ≈ 30 × 4 ÷ 6 = 20.
Rule 2: Cost and quantity problems — Total = price per unit × quantity; units = total ÷ size of one.
- Easy: Cloth ₹40 a metre, 2.35 m costs 2.35 × 40 = ₹94.
- Medium: 3.5 kg at ₹24.60: 3.5 × 24.60 = ₹86.10.
- Exam-type: 48.6 m ribbon into 0.6 m pieces: 486 ÷ 6 = 81 pieces.
Rule 3: Speed and sharing — Speed = distance ÷ time; equal share = total ÷ number of people.
- Easy: 14.4 km in 0.2 h: 14.4 ÷ 0.2 = 72 km/h.
- Medium: 7.5 L shared by 5: 1.5 L each.
- Exam-type: 12.6 L into 0.35 L bottles: 36 bottles.
Estimate first, then calculate, then compare.
Trusting an answer like 9.4 when the estimate is 94. Recheck the point.
Round to numbers you can do in your head. Memory clue: "Round, compute, compare."
Spot the difference
| Question asks | Do |
|---|---|
| total cost | price × quantity |
| how many pieces | total ÷ piece |
Practice questions
- 1Estimate 5.98 × 3.02.
- 24 notebooks at ₹14.75 each. Total (₹)?
- 30.75 kg costs ₹60. Price per kg (₹)?
- 4A 9.6 m rope is cut into 0.8 m pieces. How many pieces?
- 57.2 L lasts 6 days equally. Litres per day?
- 6Estimate 80.3 ÷ 3.9.
- 745 km in 0.75 h. Speed (km/h)?
- 812 pens at ₹18.50 each, pay ₹250. Change (₹)?
Show answers (try them yourself first)
- 118
- 259
- 380
- 412
- 51.2
- 620
- 760
- 828