Fractions and Converting to Decimals and Percentages — AQA GCSE Maths Revision Notes
What you'll learn
This topic covers calculating with fractions and moving between fractions, decimals and percentages. By the end of this guide you should be able to add, subtract, multiply and divide fractions, work with mixed numbers, and simplify a fraction to its lowest terms.
You should also be able to find a fraction of an amount, compare fractions, convert confidently between the three forms, and recognise which of them a question actually wants.
The organising idea is that the denominator names the unit. A fraction with 5 on the bottom is counting fifths, exactly as a measurement might count metres. You can only add or subtract quantities measured in the same unit — three fifths plus two fifths is five fifths, just as 3 m plus 2 m is 5 m — and that single fact explains why addition and subtraction need a common denominator. Multiplication and division are different because they change the unit rather than combining within it, which is why they need no common denominator at all. Students who find fractions arbitrary usually have not been shown this difference.
Key terms and definitions
Numerator — the number on top. It counts how many of the unit you have.
Denominator — the number on the bottom. It names the unit: fifths, eighths, and so on.
Equivalent fractions — different-looking fractions of the same value, such as 1/2 and 2/4.
Simplest form — a fraction whose numerator and denominator share no common factor.
Mixed number — a whole number with a fraction, such as 2¾.
Improper fraction — a fraction whose numerator is larger than its denominator, such as 11/4.
Reciprocal — a fraction turned upside down. The reciprocal of 3/4 is 4/3.
Core concepts
Equivalent fractions and simplifying
Multiplying or dividing both the top and the bottom by the same number leaves the value unchanged, because you are only renaming the unit.
So 1/2, 2/4 and 5/10 are all the same quantity. Going the other way, 12/18 simplifies by dividing both by 6 to give 2/3.
A fraction is in its simplest form when no number divides into both parts. Simplifying at the end of every calculation is expected, and marks are often reserved for it.
Adding and subtracting
Because the denominator names the unit, the two fractions must be converted to the same unit before they can be combined.
Find a common denominator — the lowest common multiple of the two is neatest — convert each fraction, then add or subtract the numerators only.
For 1/2 + 1/3, the common denominator is 6, giving 3/6 + 2/6 = 5/6.
The denominators are never added. Writing 1/2 + 1/3 = 2/5 is the classic error, and a sanity check kills it instantly: 2/5 is smaller than 1/2, yet something was added.
Multiplying
Multiply the numerators together and the denominators together. No common denominator is needed.
So 2/3 × 3/4 = 6/12, which simplifies to 1/2.
Cancelling before multiplying keeps the numbers small: in 2/3 × 3/4 the 3s cancel, leaving 2/4 = 1/2 with almost no arithmetic.
The word of means multiply. Three quarters of 20 is 3/4 × 20 = 15.
Multiplying by a fraction less than 1 makes a number smaller, which is worth noticing because it contradicts what multiplication usually does.
Dividing
Dividing by a fraction is the same as multiplying by its reciprocal: keep the first fraction, change the sign to multiply, and flip the second.
So 2/3 ÷ 3/4 = 2/3 × 4/3 = 8/9.
Dividing by a fraction less than 1 makes a number larger. That is not a trick: asking how many quarters fit into 3 gives 12, and 3 ÷ 1/4 = 12 says exactly that.
Mixed numbers
Convert a mixed number into an improper fraction before multiplying or dividing.
For 2¾: multiply the whole number by the denominator and add the numerator, giving (2 × 4) + 3 = 11, so 2¾ = 11/4.
Convert back at the end if the question was asked in mixed numbers. To convert 11/4 back, divide: 4 goes into 11 twice with 3 left over, giving 2¾.
Adding and subtracting mixed numbers can be done by dealing with the whole parts and the fraction parts separately, but converting to improper fractions first is more reliable when the subtraction requires borrowing.
Comparing fractions
Convert to a common denominator, then compare numerators. To compare 3/5 and 5/8, use fortieths: 24/40 and 25/40, so 5/8 is larger.
Converting both to decimals also works and is often quicker with a calculator.
Take care with fractions close together. 3/5 and 5/8 differ by only one fortieth, so estimating by eye is unreliable and the conversion is worth doing properly.
A fraction of an amount, and working backwards
To find a fraction of an amount, divide by the denominator and multiply by the numerator.
For 3/4 of 60: dividing by 4 gives 15, and multiplying by 3 gives 45.
Doing it in that order keeps the numbers small. Multiplying first gives 180 and then dividing gives 45 as well, but the arithmetic is heavier.
Many questions run this backwards, giving the result and asking for the original. Here the key is to work out what one part is worth.
If 2/5 of a number is 18, then 2 parts are worth 18, so one part is 9, and the whole number is 5 parts, which is 45.
Check it forwards: 2/5 of 45 is indeed 18. ✓ That check takes seconds and catches the usual error of multiplying when you should divide.
Recurring decimals
A fraction whose denominator contains only the factors 2 and 5 converts to a decimal that stops. Any other denominator produces one that recurs forever.
So 7/8 gives 0.875, because 8 is made only of 2s, while 1/3 gives 0.333… and 1/7 gives 0.142857142857…
Recurring decimals are written with a dot over the repeating digit, or over the first and last digits of a repeating block. Knowing that 1/3 = 0.333… and 2/3 = 0.666… saves time, and it explains why 1/3 as a percentage is written 33.3% rather than exactly.
Fractions, decimals and percentages
These are three notations for the same quantity, and moving between them freely saves a great deal of time.
Fraction to decimal: divide the top by the bottom. 3/4 = 3 ÷ 4 = 0.75.
Decimal to percentage: multiply by 100. 0.75 = 75%.
Percentage to fraction: write it over 100 and simplify. 40% = 40/100 = 2/5.
Some conversions are worth knowing without calculation: 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 3/4 = 0.75 = 75%, 1/10 = 0.1 = 10%, 1/5 = 0.2 = 20%, and 1/3 = 0.333… ≈ 33.3%.
A fraction whose denominator has only 2s and 5s as factors gives a terminating decimal; any other denominator gives a recurring one, which is why 1/3 and 1/7 never stop.
Worked examples
Example 1: Subtracting with different denominators
Work out 5/6 − 1/4.
The lowest common multiple of 6 and 4 is 12, so convert both to twelfths.
5/6 becomes 10/12, and 1/4 becomes 3/12.
Subtract the numerators only: 10/12 − 3/12 = 7/12.
Check for sense: the answer should be a little over a half, and 7/12 is. ✓
Example 2: Dividing mixed numbers
Work out 2½ ÷ 1¼.
Convert both to improper fractions first: 2½ = 5/2 and 1¼ = 5/4.
Keep, change, flip: 5/2 ÷ 5/4 = 5/2 × 4/5.
Cancel the 5s and simplify: 4/2 = 2.
The answer makes sense, since 1¼ fits into 2½ exactly twice.
Example 3: Converting between all three forms
Write 7/8 as a decimal and as a percentage.
Divide the top by the bottom: 7 ÷ 8 = 0.875.
Multiply by 100 for the percentage: 87.5%.
The decimal terminates because 8 is made only of 2s. Compare 7/9, where the 9 guarantees a recurring decimal of 0.777…
Example 4: Working backwards from a fraction of an amount
Three eighths of a number is 21. What is the number?
Three parts out of eight are worth 21, so one part is 21 ÷ 3 = 7.
The whole number is all eight parts: 8 × 7 = 56.
Check forwards: 3/8 of 56 is 56 ÷ 8 × 3 = 21. ✓
Multiplying 21 by 3/8 instead would give 7.875, which is the error to watch for — the given figure is a part, not the whole.
Common mistakes and how to avoid them
Adding the denominators. Only the numerators are added, once the units match. Check the answer is bigger than what you started with.
Finding a common denominator for multiplication. It is not needed — that requirement belongs to addition and subtraction only.
Forgetting to flip when dividing. Keep, change, flip. Dividing by a fraction less than 1 should make the answer bigger.
Multiplying mixed numbers directly. Convert to improper fractions first, or the whole-number parts get lost.
Leaving the answer unsimplified. Simplify every final answer; marks are frequently held back for it.
Converting a percentage by dividing by 10. Percent means out of 100, so a percentage goes over 100.
Assuming multiplying always makes things bigger. Multiplying by a fraction below 1 makes a number smaller.
Exam technique for "Fractions"
Show the converted fractions as a separate line before adding or subtracting. That line carries the method mark even if the arithmetic afterwards slips.
Cancel before multiplying rather than simplifying a large answer afterwards. It is quicker and much less error-prone.
Convert mixed numbers to improper fractions as your first written step, and convert back only at the very end.
Sense-check every answer against the size you expect: adding should increase, and dividing by a fraction below 1 should increase too.
Read which form the question wants. "Give your answer as a fraction in its simplest form" and "give your answer as a percentage" are different instructions, and the final mark depends on following them.
On a calculator paper, use the fraction key rather than converting to decimals, so exact values are kept.
Quick revision summary
The denominator names the unit. That is why adding and subtracting need a common denominator — you can only combine like with like — and why multiplying and dividing do not.
To add or subtract: convert to a common denominator, then combine the numerators only. So 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
To multiply: multiply tops and bottoms, cancelling first where possible. The word of means multiply.
To divide: keep, change, flip — multiply by the reciprocal. Dividing by a fraction below 1 makes the answer larger.
Convert mixed numbers to improper fractions before multiplying or dividing, and back again at the end.
To find a fraction of an amount, divide by the denominator then multiply by the numerator. Working backwards from a part to the whole means finding what one part is worth first.
Fraction to decimal: divide top by bottom. Decimal to percentage: multiply by 100. Percentage to fraction: put it over 100 and simplify.
A denominator built only from 2s and 5s gives a terminating decimal; anything else recurs.
Always simplify the final answer, and check its size against what you would expect.