P5 Decimal Scaling

Scaling a decimal changes the value of every place. When you multiply by a power of 10, each digit is worth more; when you divide, each digit is worth less. A multiple such as 30 or 300 can be split into a small factor and a power of 10, so you can work without a calculator.

Parents: let your child choose a prediction first, then ask which place-value change makes the answer larger or smaller.

By the end you will be able to multiply and divide decimals through three decimal places by 10, 100, 1000 and their multiples, explain the place-value change, and check your answer by reversing the operation.

Multiply by a power of 10

Multiplying by 10, 100 or 1000 makes a decimal number 10, 100 or 1000 times as large. The digits keep their order, but their place values shift left. For example, 2.375 has three decimal places, and multiplying by 100 makes each digit two places more valuable.

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Decimal scaling

🤔 Predict first: What is 2.375 × 100?

23.75237.52,375

Show the worked solution

2.375 × 100

$$100=10\times10$$

$$2.375\times10=23.75$$

$$23.75\times10=237.5$$

$$2.375\times100=237.5$$

The decimal point has not moved by itself. The digits have changed place value because the number was multiplied by two factors of 10.

Multiply by a multiple such as 30

Split a multiple into a small factor and a power of 10. With 30, multiply by 3 first and then by 10. This keeps the factor decomposition visible and works for decimals with one, two or three decimal places.

Show the worked solution

4.8 × 30

$$30=3\times10$$

$$4.8\times3=14.4$$

$$14.4\times10=144$$

$$4.8\times30=144$$

Check by reversing the multiplication: 144 ÷ 30 = 4.8. The check returns to the starting value.

Divide by powers and multiples of 10

Dividing by 10, 100 or 1000 makes a decimal number smaller. A multiple such as 300 can be split into 3 × 100. Divide by 3 first, then divide by 100, and keep zero placeholders when a place is empty.

Show the worked solution

48.6 ÷ 300

$$300=3\times100$$

$$48.6\div3=16.2$$

$$16.2\div100=0.162$$

$$48.6\div300=0.162$$

Check by reversing the division: 0.162 × 300 = 48.6. The inverse check is especially useful when the answer has a zero before the decimal point.

Practise the factor method

Try each complete question before opening its solution. Write the factorisation of every multiple and check by reversing the operation.

Practice 1 — start here. Calculate 6.25 × 1000.

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Practice 1

$$1{,}000=10\times10\times10$$

$$6.25\times10=62.5$$

$$62.5\times10=625$$

$$625\times10=6{,}250$$

$$6.25\times1{,}000=6{,}250$$

$$6{,}250\div1{,}000=6.25\text{ (check)}$$

Practice 2 — use a multiple. Calculate 3.06 × 40.

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Practice 2

$$40=4\times10$$

$$3.06\times4=12.24$$

$$12.24\times10=122.4$$

$$3.06\times40=122.4$$

$$122.4\div40=3.06\text{ (check)}$$

Practice 3 — divide by a multiple. Calculate 72.9 ÷ 900.

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Practice 3

$$900=9\times100$$

$$72.9\div9=8.1$$

$$8.1\div100=0.081$$

$$72.9\div900=0.081$$

$$0.081\times900=72.9\text{ (check)}$$

Watch out — easily mixed up

Quick recap

🎯 Mastery check

Answer all 7 — your progress is saved on this device.

1. What is 2.375 × 100?

   23.75237.52,375

2. What is 4.8 × 30?

   14.41441,440

3. What is 48.6 ÷ 300?

   16.20.1620.0162

4. What is 0.408 × 1000?

   4.0840840.8

5. What is 6.25 × 1000?

   6256,25062.5

6. What is 3.06 × 40?

   12.24122.41,224

7. Which check confirms that 72.9 ÷ 900 = 0.081?

   0.081 × 900 = 72.90.081 ÷ 900 = 72.972.9 × 900 = 0.081

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