Fractions, decimals and percentages — AQA GCSE Mathematics

Fractions, decimals and percentages are fundamental proportional reasoning tools used across all areas of mathematics and everyday life. This topic covers converting between these three forms, comparing and ordering them, and applying them to solve problems involving percentages of amounts, percentage increase and decrease, reverse percentages, and simple and compound interest. Mastery of these skills is essential for success in GCSE Mathematics and beyond.

In the AQA GCSE Mathematics specification, this topic appears in both the Foundation and Higher tiers, with Higher tier including more complex problems such as recurring decimals and compound interest. Understanding these concepts also supports work in ratio, proportion, probability, and statistics, making it a cornerstone of the curriculum.

• →Converting between fractions, decimals and percentages: divide numerator by denominator for decimals, multiply by 100 for percentages, and use place value or division for the reverse.
• →Finding a percentage of an amount using multipliers: for example, to find 35% of 80, multiply by 0.35.
• →Percentage change: calculate the change, divide by the original amount, and multiply by 100.
• →Reverse percentages: to find the original amount after a percentage change, divide by the multiplier (e.g., if after a 20% increase the value is 96, original = 96 / 1.20).
• →Simple and compound interest: simple interest is calculated on the original amount each year; compound interest uses a multiplier raised to the power of the number of years.

• A terminating decimal has a finite number of decimal places and can be written exactly as a fraction with denominator a power of 10 before simplifying.
• To convert a decimal to a fraction, write the decimal digits over the correct power of ten, then simplify by dividing numerator and denominator by their highest common factor.
• To convert a fraction to a decimal, divide the numerator by the denominator; for example 7 ÷ 2 = 3.5.
• Equivalent forms must be exact, not rounded; for example 0.375 = 3/8 exactly, whereas 0.38 is only an approximation.
• In a multiple-choice question, test each option by converting it to the other form or by cross-multiplying to check equality.
• A recurring decimal has a repeating digit or block of digits that continues indefinitely, indicated by dots or a bar over the repeating part.
• To convert a recurring decimal to a fraction, set x equal to the decimal, multiply by a power of ten that shifts one full repeating block, subtract the original equation, and solve for x.
• For a single repeating digit such as 0.7̇, multiplying by 10 gives 10x = 7.777…, and subtracting x = 0.777… gives 9x = 7, so x = 7/9.
• For a repeating block of two digits such as 0.4̇5̇, multiplying by 100 gives 100x = 45.4545…, and subtracting x = 0.4545… gives 99x = 45, so x = 45/99 = 5/11.
• To convert a fraction to a recurring decimal, divide the numerator by the denominator and identify the repeating block; for example 5 ÷ 11 = 0.454545… = 0.4̇5̇.
• In a multiple-choice question, check exact equivalence by converting the option to the other form or by substituting into the original equation.
• Recognises that in a ratio a : b the whole is a + b equal parts, so the shares are a/(a + b) and b/(a + b) of the total.
• Converts a ratio to fractions of the total by dividing each part by the sum of the parts, and converts fractions of a total back to a ratio using a common denominator.
• Uses one known share to find the value of one part, then scales to find the other share and the total.
• Checks that the fractions found sum to 1 and that the shares are in the stated ratio.
• Interprets a fraction as an operator by multiplying the quantity by the fraction, for example 3/4 of 240 = 240 × 3/4 = 180.
• Interprets a percentage as an operator by converting it to a decimal or fraction multiplier, for example 35% of £240 = 0.35 × £240 = £84.
• Uses a single multiplier for increase or decrease, for example multiplying by 1.2 for a 20% increase and by 0.8 for a 20% decrease.
• Reverses an operator to find an original amount, for example dividing a value after a 20% increase by 1.2.
• Selects the correct multiplier when the percentage is greater than 100 or when successive changes are applied.

• 💡For each option, convert it to the form used in the question and compare exactly; do not choose an option that is merely close.
• 💡When simplifying, divide numerator and denominator by their highest common factor in one step where possible, then check by multiplying back.
• 💡Use a quick reverse check: multiply the fraction by the denominator of the decimal's place value to see whether you recover the decimal digits.
• 💡Write the recurring decimal with a clear dot or bar notation before starting, so you can see exactly which digits repeat.
• 💡After solving for x, substitute back into the original recurring decimal to check that the fraction is exact.
• 💡In multiple-choice questions, eliminate options that are rounded or have the wrong repeating block before doing a full conversion.
• 💡Underline the word total in the question, because fractions in ratio problems are almost always fractions of the total, not of the other share.
• 💡Write the sum of the parts as the denominator before doing any arithmetic, so the fractions are formed correctly from the start.
• 💡Where a share is given in pounds or items, find the value of one part first, then multiply by the number of parts required.
• 💡Convert every percentage to a decimal multiplier before calculating, so increase, decrease and reverse problems use one consistent method.
• 💡For a reverse percentage, write the multiplier equation first, for example original × 1.2 = new value, then rearrange.
• 💡Check the size of the answer against the original: a percentage increase must give a larger value and a decrease a smaller value.
• 💡Always show your working, especially in percentage change and reverse percentage questions, as method marks are awarded even if the final answer is incorrect.
• 💡Use multipliers for percentage increase and decrease to avoid errors and save time. Write down the multiplier clearly before calculating.
• 💡When converting recurring decimals to fractions, show the algebraic steps clearly. Do not just state the answer, as you may lose method marks.

• Writing 0.375 as 375/100 instead of 375/1000; the correction is to count three decimal places and use denominator 10³ = 1000.
• Simplifying 375/1000 by dividing only the numerator or only the denominator; the correction is to divide both by the same highest common factor, 125, to reach 3/8.
• Treating 3.5 as 35/10 and then simplifying incorrectly to 7/5; the correction is 35/10 = 7/2, so 3.5 = 7/2.
• Assuming a fraction such as 7/2 cannot equal a decimal greater than 1; the correction is that improper fractions convert to decimals greater than 1, as 7 ÷ 2 = 3.5.
• Using a rounded decimal such as 0.78 for 0.7̇ instead of the exact recurring form; the correction is to keep the recurring notation or the exact fraction 7/9.
• Multiplying by the wrong power of ten when the repeating block has more than one digit; the correction is to multiply by 10ⁿ where n is the number of digits in the repeating block.
• Subtracting the equations in the wrong order and losing the sign; the correction is to subtract the smaller expression from the larger so the coefficient of x is positive.
• Forgetting to simplify the resulting fraction; the correction is to divide numerator and denominator by their highest common factor, as 45/99 simplifies to 5/11.
• Writing the ratio 3 : 5 as 3/5 and 5/3 instead of 3/8 and 5/8; the correction is to divide each part by the total number of parts, a + b.
• Treating the first number of the ratio as the numerator over the second number as the denominator; the correction is to remember that a fraction of the total uses the sum of the parts as the denominator.
• Forgetting to add the parts to find the whole before forming fractions; the correction is to calculate a + b first and use it as the denominator.
• Assuming a fraction of a total and a ratio share are the same thing without checking the whole; the correction is to state clearly what the whole represents before converting.
• Adding or subtracting the percentage of the original instead of multiplying by a single multiplier; the correction is to use 1 + r for an increase and 1 − r for a decrease, where r is the decimal rate.
• Dividing by the percentage instead of multiplying by its decimal equivalent; the correction is to convert the percentage to a decimal first, so 35% becomes 0.35.
• Reversing a percentage change by subtracting the same percentage from the new value; the correction is to divide by the multiplier, because the percentage was taken from the original amount.
• Treating a fraction operator as division only, for example reading 3/4 of 240 as 240 ÷ 3/4; the correction is to multiply by the fraction, 240 × 3/4.
• Students often think that increasing by 20% and then decreasing by 20% returns to the original amount. In fact, the final amount is 96% of the original (multipliers 1.20 and 0.80 give 0.96).
• When converting a fraction to a decimal, students may incorrectly divide the denominator by the numerator. The correct method is to divide the numerator by the denominator.
• Students sometimes believe that 0.5% is the same as 0.5. In fact, 0.5% is 0.005 as a decimal.

1. 1Week 1: Revise conversions between fractions, decimals and percentages. Practice converting common fractions (e.g., 1/2, 1/4, 1/5, 1/8, 1/10) to decimals and percentages, and vice versa.
2. 2Week 1: Learn and practice using multipliers for percentage increase and decrease. Complete exercises on finding percentages of amounts and calculating percentage change.
3. 3Week 2: Study reverse percentages and simple/compound interest. Work through examples and past paper questions, ensuring you understand the difference between simple and compound interest.
4. 4Week 2: Practice recurring decimals to fractions, especially for Higher tier. Use algebraic methods and simplify fractions fully.
5. 5Week 2: Complete a mixed set of exam-style questions under timed conditions to test your understanding and identify any remaining gaps.

• 📋Conversion questions: Convert a fraction to a decimal and percentage, or order a list of fractions, decimals and percentages. Advice: ensure you convert all to the same form for comparison.
• 📋Percentage of an amount: Calculate a percentage of a given quantity, often in a real-life context (e.g., VAT, discounts). Advice: use multipliers for efficiency.
• 📋Percentage change and reverse percentages: Calculate the percentage change between two values, or find the original amount after a percentage change. Advice: always identify the original amount and use the multiplier method.
• 📋Simple and compound interest: Calculate interest over a period, often comparing simple and compound. Advice: for compound interest, use the formula A = P(1 + r/100)^n.