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To multiply fractions, multiply across the top and bottom; to add or subtract, find the lowest common denominator first. Free notes and 14 questions answered.
Calculating Exactly with Fractions is part of Structure and calculation in GCSE AQA Higher Maths.
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GCSE AQA Higher
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To multiply fractions, multiply across the top and bottom; to add or subtract, find the lowest common denominator first. Free notes and 14 questions answered.
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Use this lesson to stay inside Structure and calculation while connecting the detail back to the wider Number topic.
Read the core explanations from Calculating Exactly with Fractions before testing yourself from memory.
Decimals are convenient, but they often lie. Write $\frac{1}{3}$ as a decimal and you get $0.333...$, which never ends. The moment you round it, your answer is wrong — and on a question without a calculator, that rounding error can cost you marks.
Calculating exactly with fractions means carrying the fraction all the way through to the final answer without converting to a decimal at any point. Some GCSE Higher exam questions do not allow a calculator, so this skill is tested directly. On calculator-allowed exam questions you may still be asked to give an exact answer, which means leaving it as a fraction.
Every fraction calculation uses one of four operations. Each has its own method, and mixing them up is the most common source of errors. Here is a clear breakdown before the worked examples.
You can only add or subtract fractions when they share the same denominator. Think of it like units: you cannot add 3 metres and 5 kilometres without converting first. The common denominator is your shared unit.
Worked example: Adding Fractions with Different Denominators
Calculate $\frac{3}{4} + \frac{5}{6}$. Give your answer as a fraction in its simplest form. (Without a Calculator)
Answer: $\frac{19}{12}$ (or equivalently $1\frac{7}{12}$)
Subtraction follows exactly the same process — find the LCD, convert, then subtract the numerators. The only extra trap is keeping track of negative results when the second fraction is larger.
Multiplication is the most straightforward operation: multiply across the top, multiply across the bottom. There is no need to find a common denominator. The key habit to build is cancelling common factors before you multiply — this keeps the numbers small and avoids a messy simplification step at the end.
Worked example: Multiplying Fractions with Cancelling
Calculate $\frac{4}{9} \times \frac{3}{8}$. Give your answer in its simplest form. (Without a Calculator)
Answer: $\frac{1}{6}$
Dividing by a fraction is where many students lose marks. The key insight: dividing by $\frac{c}{d}$ is the same as multiplying by $\frac{d}{c}$. This is because multiplying by the reciprocal undoes the division.
Notice what this means for numbers less than 1: if you divide by $\frac{1}{3}$, you are multiplying by 3, so the result is larger than what you started with. Dividing by a fraction less than 1 always gives a result further from zero, not closer.
Worked example: Dividing Fractions Using Keep-Change-Flip
Calculate $\frac{5}{6} \div \frac{2}{3}$. Give your answer in its simplest form. (Without a Calculator)
Answer: $\frac{5}{4}$ (or equivalently $1\frac{1}{4}$)
Exam questions often involve mixed numbers such as $2\frac{1}{3}$. Before applying any operation, convert the mixed number to an improper fraction. Multiply the whole number by the denominator, add the numerator, and place the result over the original denominator.
Worked example: Multi-Step Fraction Problem with Mixed Numbers
Calculate $1\frac{2}{3} \times 2\frac{1}{4}$. Give your answer as a mixed number in its simplest form. (Without a Calculator)
Answer: $3\frac{3}{4}$
Multi-step problems may chain two or more operations. The rule is the same throughout: keep everything as exact fractions, convert mixed numbers at the start, and only simplify at the very end.
You have now seen all four operations. These short problems test whether you can select the right method and carry it through accurately. Try each one before checking the solution.
The methods above are straightforward when applied carefully. Most marks are lost not through misunderstanding but through predictable slips. Knowing exactly where those slips happen is half the battle.
Calculating exactly with fractions — adding and subtracting via a common denominator, multiplying across, and dividing by flipping to the reciprocal — is the core skill tested in GCSE Higher 3.1.1. Keeping answers in exact fraction form throughout, and simplifying only at the final step, is what separates a full-mark response from a partial one.
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Move to the related lessons or back to the topic page when you need the wider sequence before exam-style practice.
To multiply fractions, multiply across the top and across the bottom, with no need for a common denominator. Cancelling common factors first keeps the numbers small, so 4/9 × 3/8 becomes 1/3 × 1/2 = 1/6.
Fractions are subtracted in exactly the same way as they are added: find the lowest common denominator, convert each fraction, then subtract the numerators while the denominator stays the same. The extra trap is keeping track of a negative result when the second fraction is larger.
To add fractions with different denominators, find the lowest common denominator, convert each fraction, then add the numerators. For example, 3/4 + 5/6 = 9/12 + 10/12 = 19/12, which is 1 7/12.
To convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and place the result over the original denominator. For example, 2 3/5 = (2 × 5 + 3)/5 = 13/5.
To multiply mixed numbers, convert them to improper fractions first, then cancel and multiply. For example, 1 2/3 × 2 1/4 = 5/3 × 9/4 = 5/1 × 3/4 = 15/4, which is 3 3/4.
The lowest common denominator (LCD) is the lowest common multiple of the denominators of the fractions being added or subtracted. For 4 and 6 it is 12, and using it keeps the numbers as small as possible.
5/6 ÷ 2/3 = 5/4, or 1 1/4. Keep the first fraction, change ÷ to ×, and flip 2/3 to 3/2, giving 5/6 × 3/2; cancelling the 3 and 6 then gives 5/2 × 1/2 = 5/4.
4/9 × 3/8 = 1/6. Cancelling diagonally, 4 and 8 share a factor of 4 and 3 and 9 share a factor of 3, giving 1/3 × 1/2 = 1/6.
3/4 + 5/6 = 19/12, or 1 7/12. The lowest common denominator of 4 and 6 is 12, so 3/4 = 9/12 and 5/6 = 10/12, and adding the numerators gives 19/12, which is already in its simplest form.
Cancelling common factors before multiplying fractions keeps the numbers small and avoids a messy simplification at the end. In 4/9 × 3/8, cancelling 4 with 8 and 3 with 9 gives 1/3 × 1/2 = 1/6, already fully simplified.
Keep change flip is a method for dividing fractions: keep the first fraction, change the division sign to multiplication, and flip the second fraction to its reciprocal. It works because dividing by c/d is the same as multiplying by d/c.
Exact fractions are used instead of decimals because some decimals never end, such as 1/3 = 0.333..., and rounding them makes the answer wrong. Calculating exactly means carrying the fraction all the way to the final answer and simplifying only at the end.
An improper fraction is converted to a mixed number by dividing the numerator by the denominator: the quotient is the whole number and the remainder goes over the denominator. For example, 15 ÷ 4 = 3 remainder 3, so 15/4 = 3 3/4.
1 2/3 × 2 1/4 = 3 3/4. Converting gives 5/3 × 9/4, cancelling the 9 and 3 gives 5/1 × 3/4 = 15/4, and 15/4 as a mixed number is 3 3/4.