Converting decimals, fractions and percentages | AQA GCSE Maths Notes

Visual model

Decimals, fractions and percentages can show the same amount

0.25

\frac{1}{4}

25\%

convert to one form before comparing

Convert to one form before comparing.

Fractions, decimals and percentages can show the same value.

Use the form that makes the calculation easiest.

Gold-standard guide

20 mins

What you will learn

Move confidently between equivalent forms.

Use a clear step-by-step method for converting decimals, fractions and percentages.

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

Decimal → Fraction: count the decimal places to find the denominator (1 d.p

Step 1

Convert to a fraction using the place value denominator

0.325 has 3 decimal places, so write it as $32\frac{5}{1}000$

Step 2

Simplify the fraction by dividing by common factors

$32\frac{5}{1}000$ : both divide by 5 → $6\frac{5}{2}00$ ; divide by 5 again → $1\frac{3}{4}0$

Step 3

Convert to a percentage

0.325 × 100 = 32.5%

Watch out

Students forget to simplify fractions, leaving $32\frac{5}{1}000$ as their answer

Percent

$p% = \frac{p}{1}00.$

Decimal to percent

$decimal \times 100 = percentage.$

Worked example

Write 0.325 as a fraction in its simplest form and as a percentage

325 has 3 decimal places, so write it as $32\frac{5}{1}000$ .We use 1000 because the final digit (5) is in the thousandths column.

Simplify the fraction by dividing by common factors: $32\frac{5}{1}000$ : both divide by 5 → $6\frac{5}{2}00$ ; divide by 5 again → $1\frac{3}{4}0$ .Check: HCF(13, 40) = 1, so $1\frac{3}{4}0$ is fully simplified.

Convert to a percentage: 0.325 × 100 = 32.5%. Multiplying by 100 shifts every digit two places to the left.

Final answer

$1\frac{3}{4}0$ and 32.5%

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

Write 0.325 as a fraction in its simplest form and as a percentage

3 marks4 minsdecimals-fractions-and-percentages-worked

Show solution

Worked solution

1.325 has 3 decimal places, so write it as $32\frac{5}{1}000$ .We use 1000 because the final digit (5) is in the thousandths column.
2.Simplify the fraction by dividing by common factors: $32\frac{5}{1}000$ : both divide by 5 → $6\frac{5}{2}00$ ; divide by 5 again → $1\frac{3}{4}0$ .Check: HCF(13, 40) = 1, so $1\frac{3}{4}0$ is fully simplified.
3.Convert to a percentage: 0.325 × 100 = 32.5%. Multiplying by 100 shifts every digit two places to the left.

Final answer

$1\frac{3}{4}0$ and 32.5%

Mark points

M1: convert to a fraction using the place value denominator
M1: simplify the fraction by dividing by common factors
M1: convert to a percentage
A1: $1\frac{3}{4}0$ and 32.5%

Watch out

Students forget to simplify fractions, leaving $32\frac{5}{1}000$ as their answer.Always check whether the numerator and denominator share a common factor.A quick check: if both numbers are even, you can divide by 2; if both end in 0 or 5, divide by 5.

Diagnosticrecall

Write 0.6 as a fraction and a percentage

1 mark2 minsdecimals-fractions-and-percentages-q1

Show solution

Worked solution

1.Spot the skill: Decimal → Fraction: count the decimal places to find the denominator (1 d.p.
2.Use the convert to a fraction using the place value denominator stage first, then simplify the fraction by dividing by common factors.
3.Keep the final answer visible: $\frac{3}{5}$ and 60%.

Final answer

$\frac{3}{5}$ and 60%

Mark points

p. p. p. → /1000). decimal → percentage: multiply by 100. remember the chain: decimal ↔ fraction ↔ percentage.
A1: $\frac{3}{5}$ and 60%

Watch out

Easyprocedure

Write 45% as a decimal and a fraction in its simplest form

2 marks3 minsdecimals-fractions-and-percentages-q2

Show solution

Worked solution

1.Spot the skill: Decimal → Fraction: count the decimal places to find the denominator (1 d.p.
2.Use the simplify the fraction by dividing by common factors stage first, then convert to a percentage.
3.Keep the final answer visible: 0.45 and $\frac{9}{2}0$ .

Final answer

0.45 and $\frac{9}{2}0$

Mark points

p. p. p. → /1000). decimal → percentage: multiply by 100. remember the chain: decimal ↔ fraction ↔ percentage.
A1: 0.45 and $\frac{9}{2}0$

Watch out

Mediumreasoning

Order from smallest to largest: 0.4, $\frac{3}{8}$ , 37%

3 marks4 minsdecimals-fractions-and-percentages-q3

Show solution

Worked solution

1.Spot the skill: Decimal → Fraction: count the decimal places to find the denominator (1 d.p.
2.Use the convert to a percentage stage first, then convert to a fraction using the place value denominator.
3.Keep the final answer visible: $\frac{3}{8}$ (0.375), 37% (0.37), 0.4.

Final answer

$\frac{3}{8}$ (0.375), 37% (0.37), 0.4

Mark points

p. p. p. → /1000). decimal → percentage: multiply by 100. remember the chain: decimal ↔ fraction ↔ percentage.
A1: $\frac{3}{8}$ (0.375), 37% (0.37), 0.4

Watch out

Hardproblem solving

A student scores 36 out of 48. Write this as a percentage.

3 marks5 minsdecimals-fractions-and-percentages-q4

Show solution

Worked solution

1.Spot the skill: Decimal → Fraction: count the decimal places to find the denominator (1 d.p.
2.Use the convert to a fraction using the place value denominator stage first, then simplify the fraction by dividing by common factors.
3.Keep the final answer visible: 75%.

Final answer

75%

Mark points

p. p. p. → /1000). decimal → percentage: multiply by 100. remember the chain: decimal ↔ fraction ↔ percentage.
A1: 75%

Watch out

Exam-stylemulti-step

Which is greater: $\frac{5}{8}$ or 62%? Show your working.

4 marks6 minsdecimals-fractions-and-percentages-q5

Show solution

Worked solution

1.Spot the skill: Decimal → Fraction: count the decimal places to find the denominator (1 d.p.
2.Use the simplify the fraction by dividing by common factors stage first, then convert to a percentage.
3.Keep the final answer visible: $\frac{5}{8}$ = 62.5%, so $\frac{5}{8}$ is greater.

Final answer

$\frac{5}{8}$ = 62.5%, so $\frac{5}{8}$ is greater

Mark points

p. p. p. → /1000). decimal → percentage: multiply by 100. remember the chain: decimal ↔ fraction ↔ percentage.
A1: $\frac{5}{8}$ = 62.5%, so $\frac{5}{8}$ is greater

Watch out

Grade 9 stretchproblem solving

Write 0.375 as a fraction in its simplest form and as a percentage.

4 marks7 minsconversion-g9

Show solution

Worked solution

1.Write the decimal as $37\frac{5}{1}000$ .
2.Simplify the fraction.
3.Multiply the decimal by 100 for the percentage.

Final answer

$\frac{3}{8}$ and 37.5%

Mark points

M1: use $37\frac{5}{1}000$
A1: $\frac{3}{8}$
A1: 37.5%

Watch out

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

Hard exam-stylemulti-step problem

Put 0.42, 43% and $\frac{2}{5}$ in ascending order.

3 marks6 minsconversion-paper

Show solution

Worked solution

1.Convert all three values into the same form.
2.Compare the equivalent decimals carefully.

Final answer

$\frac{2}{5}$ , 0.42, 43%

Mark points

M1: use $\frac{2}{5}$ = 0.4 and 43% = 0.43
A1: give the correct order

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.

1Converting decimals, fractions and percentages - 2 marksWrite 0.6 as a fraction and a percentageMark answer

Answer

$\frac{3}{5}$ and 60%

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

Answer

3Integers, decimals and place value - 2 marksOrder these from smallest to largest: 0.3, 0.03, 0.303, 0.033Mark answer

Answer

0.03, 0.033, 0.3, 0.303

4Fractions - 3 marksA recipe needs ¾ cup of sugar. How much sugar is needed for 2½ batches?Mark answer

Answer

1⅞ cups

Mastery check

I can explain the method for converting decimals, fractions and percentages.
I can show clear working without skipping key steps.
I can avoid this mistake: Students forget to simplify fractions, leaving $32\frac{5}{1}000$ as their answer.Always check whether the numerator and denominator share a common factor.A quick check: if both numbers are even, you can divide by 2; if both end in 0 or 5, divide by 5.

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