Integers, decimals and place value | AQA GCSE Maths Notes

Visual model

Place value shows what each digit is worth

10s1s.tenthshundredths

4

7

3

5

left multiplies by 10, right divides by 10

Gold-standard guide

20 mins

What you will learn

Read, compare, order and calculate with integers and decimals.

Use a clear step-by-step method for integers, decimals and place value.

Check your answer and avoid the most common exam mistake.

Useful before you start

Core number skillsEarlier number skillsShowing clear working

Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

Line up the decimal points — not the digits at the end

Step 1

Write the numbers vertically, aligning decimal points

14.070 and 5.938 share a decimal point column

Step 2

Subtract from right to left, borrowing as needed

Thousandths: 0 − 8, need to borrow → 10 − 8 = 2

Step 3

Write the answer with the decimal point in line

14.070 − 5.938 = 8.132

Watch out

Students drop or misplace the decimal point, especially when one number has fewer decimal places

Place value

Each column is 10 times the column to its right.

Decimal operations

Line up decimal points before adding or subtracting.

Worked example

Work out 14.07 − 5.938

938 share a decimal point column. 938.

Subtract from right to left, borrowing as needed: Thousandths: 0 − 8, need to borrow → 10 − 8 = 2.Hundredths: 6 − 3 = 3 (after borrow). Tenths: 0 − 9, need to borrow → 10 − 9 = 1.Units: 3 − 5, need to borrow → 13 − 5 = 8. Tens: 0 − 0 = 0 (after borrow from 1).

Write the answer with the decimal point in line: 14.070 − 5.938 = 8.132

Final answer

8.132

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

Work out 14.07 − 5.938

3 marks4 minsintegers-decimals-and-place-value-worked

Show solution

Worked solution

1.938 share a decimal point column. 938.
2.Subtract from right to left, borrowing as needed: Thousandths: 0 − 8, need to borrow → 10 − 8 = 2.Hundredths: 6 − 3 = 3 (after borrow). Tenths: 0 − 9, need to borrow → 10 − 9 = 1.Units: 3 − 5, need to borrow → 13 − 5 = 8. Tens: 0 − 0 = 0 (after borrow from 1).
3.Write the answer with the decimal point in line: 14.070 − 5.938 = 8.132

Final answer

8.132

Mark points

M1: write the numbers vertically, aligning decimal points
M1: subtract from right to left, borrowing as needed
M1: write the answer with the decimal point in line
A1: 8.132

Watch out

Students drop or misplace the decimal point, especially when one number has fewer decimal places.Always add zero placeholders so both numbers have the same length after the decimal point before subtracting.132 is reasonable.

Diagnosticrecall

Work out 3.6 + 1.45

1 mark2 minsintegers-decimals-and-place-value-q1

Show solution

Worked solution

1.Spot the skill: Line up the decimal points — not the digits at the end.
2.Use the write the numbers vertically, aligning decimal points stage first, then subtract from right to left, borrowing as needed.
3.Keep the final answer visible: 5.05.

Final answer

5.05

Mark points

M1: use the correct line up the decimal points — not the digits at the end.every digit lives in a column (hundreds, tens, units, tenths, hundredths, thousandths).misaligning columns is the source of almost all decimal arithmetic errors.
A1: 5.05

Watch out

Easyprocedure

Work out 10.5 − 3.72

2 marks3 minsintegers-decimals-and-place-value-q2

Show solution

Worked solution

1.Spot the skill: Line up the decimal points — not the digits at the end.
2.Use the subtract from right to left, borrowing as needed stage first, then write the answer with the decimal point in line.
3.Keep the final answer visible: 6.78.

Final answer

6.78

Mark points

M1: use the correct line up the decimal points — not the digits at the end.every digit lives in a column (hundreds, tens, units, tenths, hundredths, thousandths).misaligning columns is the source of almost all decimal arithmetic errors.
A1: 6.78

Watch out

Mediumreasoning

Order these from smallest to largest: 0.3, 0.03, 0.303, 0.033

3 marks4 minsintegers-decimals-and-place-value-q3

Show solution

Worked solution

1.Spot the skill: Line up the decimal points — not the digits at the end.
2.Use the write the answer with the decimal point in line stage first, then write the numbers vertically, aligning decimal points.
3.Keep the final answer visible: 0.03, 0.033, 0.3, 0.303.

Final answer

0.03, 0.033, 0.3, 0.303

Mark points

M1: use the correct line up the decimal points — not the digits at the end.every digit lives in a column (hundreds, tens, units, tenths, hundredths, thousandths).misaligning columns is the source of almost all decimal arithmetic errors.
A1: 0.03, 0.033, 0.3, 0.303

Watch out

Hardproblem solving

A plank is 2.4 m long. Three $pie$ ces of lengths 0.75 m, 0.68 m and 0.5 m are cut off. How much remains?

3 marks5 minsintegers-decimals-and-place-value-q4

Show solution

Worked solution

1.Spot the skill: Line up the decimal points — not the digits at the end.
2.Use the write the numbers vertically, aligning decimal points stage first, then subtract from right to left, borrowing as needed.
3.Keep the final answer visible: 0.47 m.

Final answer

0.47 m

Mark points

M1: use the correct line up the decimal points — not the digits at the end.every digit lives in a column (hundreds, tens, units, tenths, hundredths, thousandths).misaligning columns is the source of almost all decimal arithmetic errors.
A1: 0.47 m

Watch out

Exam-stylemulti-step

Write the value of the digit 7 in the number 3.0074

4 marks6 minsintegers-decimals-and-place-value-q5

Show solution

Worked solution

1.Spot the skill: Line up the decimal points — not the digits at the end.
2.Use the subtract from right to left, borrowing as needed stage first, then write the answer with the decimal point in line.
3.Keep the final answer visible: 7 ten-thousandths (0.0007).

Final answer

7 ten-thousandths (0.0007)

Mark points

M1: use the correct line up the decimal points — not the digits at the end.every digit lives in a column (hundreds, tens, units, tenths, hundredths, thousandths).misaligning columns is the source of almost all decimal arithmetic errors.
A1: 7 ten-thousandths (0.0007)

Watch out

Grade 9 stretchproblem solving

Work out 3.6 / 0.012 without a calculator.

4 marks7 minsdecimal-g9

Show solution

Worked solution

1.Multiply both numbers by 1000.
2.Calculate 3600 / 12.

Final answer

300

Mark points

M1: form an equivalent division
A1: 300

Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Hard exam-stylemulti-step problem

A tank contains 18.75 litres. 3.68 litres are used and then 0.925 litres are added. How much water is in the tank now?

3 marks6 minsdecimal-paper

Show solution

Worked solution

1.Subtract the amount used.
2.Add the new amount.
3.Keep the decimal points aligned.

Final answer

15.995 litres

Mark points

M1: calculate 18.75 - 3.68
M1: add 0.925
A1: obtain 15.995 litres

Watch out

Read the full question before calculating. Keep each stage of your working visible.

Timed checkpoint

12 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Integers, decimals and place value - 2 marksWork out 3.6 + 1.45Mark answer

Answer

5.05

2Calculations and order of operations - 2 marksWork out (6 + 2) × 3 − 5Mark answer

Answer

3Fractions - 2 marksWork out 2⅓ + 1½Mark answer

Answer

3⅚

4Converting decimals, fractions and percentages - 3 marksA student scores 36 out of 48. Write this as a percentage.Mark answer

Answer

75%

Mastery check

I can explain the method for integers, decimals and place value.
I can show clear working without skipping key steps.
I can avoid this mistake: Students drop or misplace the decimal point, especially when one number has fewer decimal places.Always add zero placeholders so both numbers have the same length after the decimal point before subtracting.132 is reasonable.

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