Number · Number Operations · Lesson 1
Multiplying and Dividing by Powers of 10
Shopkeepers, bankers and engineers multiply and divide by 10, 100 and 1,000 all day long, usually in their heads. In this lesson you will discover the simple rule behind it: the digits slide across the place value chart while the decimal point stays still. You will also connect benchmark fractions such as , and to their decimals and percentages.
By the end you will be able to
- ✔Recall multiplication facts up to 12 × 12 = 144 and the related division facts.
- ✔Convert benchmark fractions to decimals and percentages, and back again.
- ✔Multiply whole numbers and decimals by 10, 100 and 1,000 by moving digits to the left.
- ✔Divide by 10, 100 and 1,000, and multiply by , and , by moving digits to the right.
Key words
- Power of 10
- A number made by multiplying 10 by itself, such as 10, 100 = and 1,000 = .
- Place value
- The value a digit has because of its position, e.g. in 3.45 the 4 is worth 4 tenths.
- Decimal point
- The dot that separates the whole-number part from the fractional part of a number.
- Benchmark fraction
- A common, friendly fraction such as , , , or that we use to compare and estimate.
- Placeholder zero
- A zero written to keep the other digits in their correct places, as in 470 or 0.063.
Curriculum reference
Content standard: B7.1.2.1 Apply mental mathematics strategies and number properties used to solve problems.
Indicator: Multiply and divide given numbers by powers of 10 including decimals and benchmark fractions.
Starter
A thousand sachets of water
Auntie Esi sells sachet water at Kejetia for 35 pesewas (GH₵0.35) each. On a busy Saturday she sells 1,000 sachets. How much money is that? She does not reach for a calculator — she answers 'GH₵350' in one second. By the end of this lesson you will know her secret too.
1. Facts you must know by heart
Fast mental work stands on two foundations. The first is your multiplication facts up to 12 × 12 = 144. If you know 7 × 8 = 56 without counting, you also know three division facts for free: 56 ÷ 8 = 7, 56 ÷ 7 = 8 and, with powers of 10, 560 ÷ 8 = 70. Practise the facts you find hardest (often the 7s, 8s and 12s) until they are automatic.
The second foundation is the benchmark fractions. These are the fractions we meet so often in shops and in news reports that we should know their decimal and percentage names instantly. A percentage simply means 'out of 100', so to change a decimal to a percentage you multiply by 100.
| Fraction | Decimal | Percentage |
|---|---|---|
| 0.5 | 50% | |
| 0.25 | 25% | |
| 0.75 | 75% | |
| 0.2 | 20% | |
| 0.125 | 12.5% | |
| 0.1 | 10% | |
| 0.01 | 1% | |
| 0.001 | 0.1% |
2. Multiplying by 10, 100 and 1,000
Our number system is base ten: every place is 10 times the place on its right. So when you multiply a number by 10, every digit becomes ten times bigger — and the only way a digit can become ten times bigger is to move one place to the left. Multiplying by 100 moves every digit two places left, and multiplying by 1,000 moves every digit three places left.
Notice what does not move: the decimal point. It is like the edge of the market stall — fixed in place while the goods (the digits) are moved around it. Step through the animation to see 3.45 × 100.
▶ Watch the digits slide two places to the left while the decimal point stays exactly where it is.
With whole numbers the empty places on the right must be filled with placeholder zeros: 47 × 100 = 4,700 because the 4 moves to the thousands, the 7 to the hundreds, and the tens and ones are now empty. With decimals you often need no new zeros at all: 105.25 × 1,000 = 105,250 (the digits 1, 0, 5, 2, 5 move three places left, and one zero fills the ones place).
3. Dividing by 10, 100 and 1,000 — and multiplying by tenths
Division is the reverse of multiplication, so dividing by 10 makes each digit ten times smaller: every digit moves one place to the right. Dividing by 100 moves digits two places right, and dividing by 1,000 moves them three places right. If you run out of digits before the decimal point, put placeholder zeros in front: 63 ÷ 1,000 = 0.063.
Now look at multiplying by a fraction. Taking of something means splitting it into 10 equal parts and keeping one, which is exactly dividing by 10. So multiplying by , or is the same as dividing by 10, 100 or 1,000. The curriculum example 105.25 × is simply 105.25 ÷ 100.
▶ Multiplying by one hundredth: the digits slide two places to the RIGHT and a new decimal place appears.
| Operation | Digits move | Example |
|---|---|---|
| × 10 | 1 place left | 2.5 × 10 = 25 |
| × 100 | 2 places left | 0.36 × 100 = 36 |
| × 1,000 | 3 places left | 105.25 × 1,000 = 105,250 |
| ÷ 10 or × | 1 place right | 470 ÷ 10 = 47 |
| ÷ 100 or × | 2 places right | 46.8 × = 0.468 |
| ÷ 1,000 or × | 3 places right | 63 ÷ 1,000 = 0.063 |
4. Putting benchmark fractions and powers of 10 together
Benchmark fractions and powers of 10 work beautifully together. Because 0.25 is , the calculation 0.25 × 400 is just a quarter of 400, which is 100. Because 0.5 is , 0.5 × 1,000 is half of 1,000, which is 500. Knowing the fraction behind a decimal often turns a hard-looking sum into an easy one.
Back to Auntie Esi's sachet water. One sachet costs GH₵0.35 and she sells 1,000 of them, so she earns 0.35 × 1,000. Move the digits three places left: 0.35 → 3.5 → 35 → 350. She earns GH₵350. In the same way, 1,000 grams make 1 kilogram, so 2,500 g ÷ 1,000 = 2.5 kg, and 1 cedi is 100 pesewas, so 450 pesewas ÷ 100 = GH₵4.50.
Worked examples
Try each one yourself first, then open the solution and compare every step.
Example 1
Multiplying a decimal by 100
Show the solution
- Step 1× 100 means every digit moves 2 places to the left.
- Step 2The 4 (ones) moves to the hundreds; the 7 (tenths) moves to the tens.
- Step 3The ones place is now empty, so fill it with a placeholder zero.
- Step 4Check: 4.7 is about 5, and 5 × 100 = 500, so 470 is sensible ✓
Answer: 470
Example 2
The curriculum example: 105.25 × 1,000
Show the solution
- Step 1× 1,000 moves every digit 3 places to the left.
- Step 21 hundred → 1 hundred-thousand; 0 tens → 0 ten-thousands; 5 ones → 5 thousands; 2 tenths → 2 hundreds; 5 hundredths → 5 tens.
- Step 3The ones place is empty, so write a placeholder zero.
- Step 4105.25 × 1,000 = 105,250.
Answer: 105,250
Example 3
Multiplying by one hundredth
Show the solution
- Step 1Multiplying by is the same as dividing by 100.
- Step 2÷ 100 moves every digit 2 places to the right.
- Step 3The 1 (hundreds) moves to the ones, the 0 to the tenths, the 5 to the hundredths, the 2 to the thousandths and the last 5 to the ten-thousandths.
- Step 4Check: 105.25 is about 100, and 100 ÷ 100 = 1, so 1.0525 is sensible ✓
Answer: 1.0525
Example 4
Dividing a whole number by 1,000
Show the solution
- Step 1Litres = 63 ÷ 1,000.
- Step 2÷ 1,000 moves every digit 3 places to the right: 63 → 6.3 → 0.63 → 0.063.
- Step 3Placeholder zeros keep the 6 in the hundredths place and the 3 in the thousandths place.
Answer: 0.063 litres
Example 5
Benchmark fraction and power of 10 together
Show the solution
- Step 10.25 = , so 0.25 × 1,000 = of 1,000 = 250 bags.
- Step 2Mass = 250 × 64 kg. Think 64 × 25 × 10: 64 × 25 = 1,600 (because 4 × 25 = 100, and 64 = 16 × 4), so 1,600 × 10 = 16,000 kg.
- Step 3Change to tonnes: 16,000 ÷ 1,000 = 16 tonnes.
Answer: 250 bags, which is 16,000 kg (16 tonnes)
Summary
- Know the multiplication facts up to 12 × 12 = 144 and their division facts by heart.
- Know the benchmark fractions: = 0.5 = 50%, = 0.25 = 25%, = 0.75 = 75%, = 0.1 = 10%.
- × 10, × 100, × 1,000: digits move 1, 2 or 3 places LEFT. The decimal point stays still.
- ÷ 10, ÷ 100, ÷ 1,000 (or × , × , × ): digits move 1, 2 or 3 places RIGHT.
- Use placeholder zeros to fill empty places, and always estimate to check your answer.
Practice: 15 questions
5 easy, 5 medium, 5 hard. Each one is marked straight away, with a full worked solution.
You can practise without an account, but sign up free or sign in to save your progress.