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This lesson covers DfE content statements L2.9, L2.10 and L2.12 — ordering decimals, performing all four operations with decimals (to three decimal places), understanding and using indices, and applying the correct order of operations.
Decimals extend the place value system to the right of the decimal point.
| Position | Place Value | Example in 45.378 |
|---|---|---|
| Tens | 10 | 4 |
| Units | 1 | 5 |
| . | decimal point | . |
| Tenths | 0.1 | 3 |
| Hundredths | 0.01 | 7 |
| Thousandths | 0.001 | 8 |
To order decimals, compare digit by digit from left to right, just as with whole numbers. It helps to write all numbers with the same number of decimal places by adding trailing zeros.
Scenario: A purchasing manager compares prices per unit from four suppliers: £3.45, £3.405, £3.5, and £3.045. Order from cheapest to most expensive.
Rewrite with 3 decimal places: 3.450, 3.405, 3.500, 3.045
Compare: 3.045 < 3.405 < 3.450 < 3.500
Answer: £3.045, £3.405, £3.45, £3.50
Exam Tip: A very common mistake is thinking that 3.45 is less than 3.405 because "45 < 405." Always compare place by place — or add trailing zeros so all numbers have the same number of decimal places.
Line up the decimal points and use the column method — exactly the same as with whole numbers.
Scenario: You buy items costing £14.75, £3.08, and £27.50. What is the total?
14.75
3.08
+ 27.50
------
45.33
Answer: £45.33
Scenario: A shelf measures 2.450 m long. You cut off a piece measuring 0.875 m. How long is the remaining piece?
2.450
- 0.875
-------
1.575
Answer: 1.575 m
Method 1 — Ignore the decimal, multiply, then reposition:
Scenario: You buy 47.5 litres of diesel at £1.42 per litre. What is the total cost?
Step 1: Total decimal places = 1 + 2 = 3 Step 2: 475 × 142 = 67,450 Step 3: Place decimal point 3 places from the right → 67.450
Answer: £67.45
Scenario: An employee works 37.5 hours at £12.80 per hour. What are their gross weekly earnings?
375 × 1280 = 480,000 → 3 decimal places → £480.00
Dividing by a whole number: Use short or long division, keeping the decimal point aligned.
Dividing by a decimal: Make the divisor a whole number by multiplying both numbers by 10, 100, etc.
Scenario: Three colleagues share a lunch bill of £47.85 equally. How much does each person pay?
47.85 ÷ 3:
15.95
3)47.85
3 → 4-3=1, bring down 7 → 17
15 → 17-15=2, bring down 8 → 28
27 → 28-27=1, bring down 5 → 15
15 → 15-15=0
Answer: £15.95
Scenario: A pack of 2.5 kg of rice costs £3.75. What is the price per kilogram?
£3.75 ÷ 2.5 → multiply both by 10 → £37.5 ÷ 25
37.5 ÷ 25 = £1.50 per kg
Exam Tip: When dividing by a decimal, multiply both numbers by 10 (or 100) to make the divisor a whole number. This is perfectly valid and makes the division much easier.
| Round to | Look at | Example: round 4.3672 |
|---|---|---|
| Nearest whole number | Tenths digit (3) | 4 |
| 1 decimal place | Hundredths digit (6) | 4.4 |
| 2 decimal places | Thousandths digit (7) | 4.37 |
| 3 decimal places | Ten-thousandths digit (2) | 4.367 |
Scenario: A calculator shows that your share of a £50 meal between 3 people is 16.6666... What should you pay?
Round to 2 decimal places (nearest penny): look at the third decimal place (6 ≥ 5, round up) → £16.67
An index (plural: indices) tells you how many times to multiply a number by itself.
| Expression | Meaning | Value |
|---|---|---|
| 5² | 5 × 5 | 25 |
| 2³ | 2 × 2 × 2 | 8 |
| 10⁴ | 10 × 10 × 10 × 10 | 10,000 |
| 3¹ | 3 | 3 |
| 7⁰ | (any number)⁰ | 1 |
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