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Decimals

Decimals are another way of representing numbers and are commonly used in real-life contexts such as money and measurements. They are closely linked to fractions and percentages, and accuracy when calculating with decimals is essential in GCSE Maths.

Overview

Decimals are another way of writing parts of a whole.

They are based on place value, so each digit has a value depending on its position.

\( 3.47 = 3 + 0.4 + 0.07 \)

In GCSE Maths, decimals appear in place value, ordering, rounding, money, measures and calculator work. You need to be confident reading them and calculating accurately.

What you should understand after this topic

  • Understand decimal place value
  • Compare and order decimals
  • Round decimals correctly
  • Add, subtract, multiply and divide decimals
  • Link decimals to fractions and percentages

Key Definitions

Decimal

A number written using a decimal point.

Decimal Point

The point that separates whole numbers from parts of a whole.

Tenths

The first place after the decimal point.

Hundredths

The second place after the decimal point.

Thousandths

The third place after the decimal point.

Round

Write a number to a given level of accuracy.

Key Rules

Line up decimal points

When adding or subtracting, decimal points must stay in the same column.

Use zeros as placeholders

\( 2.5 = 2.50 \) if that helps line up columns.

Multiply first, then place the decimal

Count total decimal places in the question.

Make division easier

Multiply both numbers by 10, 100 or 1000 to remove decimal places.

Quick Decimal Facts

\(0.1\)

One tenth

\(0.01\)

One hundredth

\(0.001\)

One thousandth

\(1.0\)

Same value as 1

How to Solve

What is a decimal?

A decimal represents whole numbers and parts of a whole in a single number. The digits after the decimal point show tenths, hundredths, thousandths and beyond.

\( 5.268 = 5 + \frac{2}{10} + \frac{6}{100} + \frac{8}{1000} \)
Exam tip: Each place value is 10 times smaller than the one before it.

Decimal place value

In \(14.372\), the 3 represents \(\frac{3}{10}\), the 7 represents \(\frac{7}{100}\), and the 2 represents \(\frac{2}{1000}\).

Reading and writing decimals

\(0.4\)

Four tenths

\(0.09\)

Nine hundredths

\(2.35\)

Two and thirty-five hundredths

\(7.008\)

Seven and eight thousandths

Comparing and ordering decimals

Compare digits from left to right. If needed, add zeros so numbers have the same number of decimal places.

\( 0.7 = 0.70 \)
Compare \(0.7\) and \(0.65\): write them as \(0.70\) and \(0.65\). Since 70 hundredths is greater than 65 hundredths, \(0.7 > 0.65\).
Exam tip: Adding zeros does not change the value.

Rounding decimals

To round, look at the digit after the place value you want. If it is 5 or more, round up. If it is 4 or less, keep it the same.

\( 4.786 \approx 4.79 \text{ (to 2 decimal places)} \)
To round \(4.786\) to 2 decimal places, look at the third decimal place. Since the next digit is 6, round the hundredths up to \(4.79\).
Rounding is covered further in place value and rounding.

Adding and subtracting decimals

Write numbers in columns and line up the decimal points.

\( 3.45 + 1.2 = 3.45 + 1.20 = 4.65 \)
Use zeros where needed so each place value lines up correctly.

Multiplying decimals

Multiply as if there are no decimal points, then place the decimal point in the answer.

\( 0.4 \times 0.3 = 0.12 \)
Ignore decimals first: \(4 \times 3 = 12\). There are 2 decimal places in total, so the answer is \(0.12\).

Dividing decimals

If the divisor is a decimal, multiply both numbers by the same power of 10 first.

\( 4.8 \div 0.6 = 48 \div 6 = 8 \)
Multiplying both numbers by 10 keeps the value the same but removes the decimal from the divisor.

Decimals, fractions and percentages

Decimals can be converted into fractions and percentages.

These conversions are covered further in fractions and percentages.

\(0.5\)

\(\frac{1}{2}\) and 50%

\(0.25\)

\(\frac{1}{4}\) and 25%

\(0.75\)

\(\frac{3}{4}\) and 75%

\(0.2\)

\(\frac{1}{5}\) and 20%

Example Questions

Edexcel

Exam-style questions inspired by Edexcel GCSE Mathematics.

Edexcel

Write 0.375 as a fraction in its simplest form.

Edexcel

Write \( \frac{7}{8} \) as a decimal.

Edexcel

Work out \( 3.6 + 4.75 \).

Edexcel

Work out \( 8.2 - 3.47 \).

Edexcel

Work out \( 0.4 \times 0.6 \).

AQA

Exam-style questions based on the AQA GCSE Mathematics specification, focusing on decimal operations and numerical fluency.

AQA

Work out \( 7.2 \div 0.3 \).

AQA

Write these numbers in order of size, starting with the smallest.

\( 0.5,\; 0.05,\; 0.505,\; 0.55 \)

AQA

Round 6.748 to 2 decimal places.

AQA

A student says that 0.4 \times 0.3 = 0.12.

AQA

Tick one box. Yes ☐     No ☐

AQA

Give a reason for your answer.

OCR

Exam-style questions aligned with OCR GCSE Mathematics, emphasising reasoning, accuracy, and confident use of decimal notation.

OCR

Write \( 5.03 \times 10 \).

OCR

Write \( 84.6 \div 100 \).

OCR

Work out \( 2.5^2 \).

OCR

Write 0.0072 in the form \( a \times 10^{-n} \).

OCR

Explain why 0.09 is smaller than 0.9.

Exam Checklist

Step 1

Check the place value carefully.

Step 2

Line up decimal points for addition and subtraction.

Step 3

Count decimal places when multiplying.

Step 4

Round only at the end unless told otherwise.

Most common exam mistakes

Ordering

Thinking \(0.9\) is smaller than \(0.35\) because 9 is smaller than 35.

Rounding

Looking at the wrong digit when rounding.

Multiplication

Forgetting how many decimal places to put in the answer.

Division

Removing the decimal from only one number instead of both.

Common Mistakes

These are common mistakes students make when working with decimals in GCSE Maths.

Thinking a longer decimal is always bigger

Incorrect

A student says 0.45 is larger than 0.5 because it has more digits.

Correct

The number of digits does not determine size. Compare place values: 0.5 is greater than 0.45 because 5 tenths is more than 4 tenths.

Not lining up decimal points

Incorrect

A student adds or subtracts decimals without aligning the decimal points.

Correct

In column methods, decimal points must be lined up so that each place value (units, tenths, hundredths) is added or subtracted correctly.

Placing the decimal point incorrectly after multiplying

Incorrect

A student multiplies decimals but puts the decimal point in the wrong place.

Correct

Multiply as if the numbers are whole, then count the total number of decimal places in the original numbers and place the decimal point correctly in the final answer.

Rounding to the wrong place value

Incorrect

A student rounds to the wrong digit or ignores the next digit.

Correct

Identify the correct place value (e.g. tenths, hundredths) and check the digit to the right to decide whether to round up or down.

Forgetting to adjust both numbers in division

Incorrect

A student removes the decimal from only one number when dividing.

Correct

When dividing decimals, multiply both numbers by the same power of 10 to make the divisor a whole number before dividing.

Try It Yourself

Practise performing calculations and conversions with decimals.

Questions coming soon
Foundation

Foundation Practice

Add, subtract and understand decimal numbers.

Question 1

What is the value of the 7 in 3.74?

Games

Practise this topic with interactive games.

Games coming soon.

Decimals Video Tutorial

Frequently Asked Questions

How do I add decimals?

Line up the decimal points before adding.

How do I multiply decimals?

Multiply normally, then count decimal places in the final answer.

Why are decimals important?

They are widely used in real-life contexts like money and measurements.