Decimals

This lesson covers DfE content statements L1.10, L1.11 and L1.12 — reading, writing, ordering and comparing decimals up to three decimal places; adding, subtracting, multiplying and dividing decimals (up to two decimal places); and rounding decimals.

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Understanding Decimals

A decimal is another way of writing a fraction. The decimal point separates the whole number part from the fractional part.

Ones	.	Tenths	Hundredths	Thousandths
3	.	4	7	5

The number 3.475 means:

• 3 ones
• 4 tenths (4/10)
• 7 hundredths (7/100)
• 5 thousandths (5/1000)

Place Value After the Decimal Point

Place	Value	As a fraction
Tenths	0.1	1/10
Hundredths	0.01	1/100
Thousandths	0.001	1/1000

Worked Example — Reading Decimals

Scenario: A parcel weighs 2.35 kg. What does each digit represent?

• 2 = 2 whole kilograms
• 3 = 3 tenths of a kilogram (0.3 kg = 300 g)
• 5 = 5 hundredths of a kilogram (0.05 kg = 50 g)

So 2.35 kg = 2 kg and 350 g.

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Ordering and Comparing Decimals

The key rule: compare digit by digit from left to right, just like whole numbers.

Common mistake: Thinking 0.35 is bigger than 0.4 because 35 > 4. This is wrong! Compare the tenths first: 0.35 vs 0.4 — the 4 tenths wins.

Worked Example — Ordering

Scenario: Put these lengths in order from shortest to longest: 1.25 m, 1.3 m, 1.205 m, 1.35 m.

Tip: Write them with the same number of decimal places by adding trailing zeros:

Number	Rewritten
1.25	1.250
1.3	1.300
1.205	1.205
1.35	1.350

Now compare: 1.205 < 1.250 < 1.300 < 1.350

Answer: 1.205 m, 1.25 m, 1.3 m, 1.35 m

Exam Tip: When ordering decimals, always add trailing zeros so all numbers have the same number of decimal places. This trick makes comparison much easier and prevents the most common decimal mistake.

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Adding and Subtracting Decimals

The golden rule: line up the decimal points.

Worked Example — Addition

Scenario: You buy items costing £4.85 and £12.60. What is the total?

   4.85
+ 12.60
------
  17.45

Answer: £17.45

Worked Example — Subtraction

Scenario: You have £20.00 and spend £7.35. How much change do you get?

  20.00
-  7.35
------
  12.65

Answer: £12.65

Worked Example — Adding Decimals with Different Decimal Places

Scenario: A runner completes three laps in 4.7 minutes, 5.35 minutes and 4.125 minutes. What is the total time?

Add trailing zeros to line everything up:

   4.700
   5.350
+  4.125
-------
  14.175

Answer: 14.175 minutes

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Multiplying Decimals

Method 1 — Ignore the Decimal, Then Put It Back

1. Remove the decimal points and multiply the whole numbers.
2. Count the total number of decimal places in the original numbers.
3. Put that many decimal places back into the answer.

Worked Example — Multiplying Decimals

Scenario: A shelf is 1.2 m long. You need 4 shelves. What is the total length?

1.2 × 4 = ?

Step 1: Ignore decimal: 12 × 4 = 48 Step 2: The original number (1.2) has 1 decimal place. Step 3: Put 1 decimal place back: 4.8 m

Worked Example — Multiplying Two Decimals

Scenario: Fabric costs £3.50 per metre. You buy 2.5 metres. What is the total cost?

3.50 × 2.5 = ?

Step 1: Ignore decimals: 350 × 25 = 8,750 Step 2: Total decimal places: 2 (from 3.50) + 1 (from 2.5) = 3 Step 3: Put 3 decimal places back: 8.750 = £8.75

Check by estimation: £3.50 × 2.5 ≈ £4 × 2.5 = £10. Our answer of £8.75 is reasonable. ✓

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Dividing Decimals

Dividing a Decimal by a Whole Number

Just do normal division — keep the decimal point in the same position.

Worked Example — Dividing a Decimal

Scenario: Three friends share a restaurant bill of £47.10 equally. How much does each person pay?

£47.10 ÷ 3 = ?

     15.70
3 )  47.10
     3       ← 3 × 1
     --
     17      ← bring down 7
     15      ← 3 × 5
     --
      21     ← bring down 1
      21     ← 3 × 7
      --
       00    ← bring down 0

Answer: Each person pays £15.70

Dividing by a Decimal