Decimals and Order of Operations

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.

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Understanding Decimal Place Value

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

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Ordering Decimals

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.

Worked Example — Comparing Product Prices

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.

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Adding and Subtracting Decimals

Line up the decimal points and use the column method — exactly the same as with whole numbers.

Worked Example — Shopping

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

Worked Example — Measuring

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

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Multiplying Decimals

Method 1 — Ignore the decimal, multiply, then reposition:

1. Count the total number of decimal places in the question.
2. Multiply as if the numbers were whole numbers.
3. Put the decimal point in the answer so it has the same total number of decimal places.

Worked Example — Fuel Cost

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

Worked Example — Wages

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

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Dividing Decimals

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.

Worked Example — Splitting a Bill

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

Worked Example — Unit Price

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.

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Rounding Decimals

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

Worked Example

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

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Indices (Powers)

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

Square Numbers and Cube Numbers