What is not flipping when dividing?

Division by a fraction means multiplying by its reciprocal.

SQA SQA National 5 2018-style practice: Calculate 214 + 123, giving your answer as a mixed number in its simplest form.

Convert to improper fractions: 214 = 94 and 123 = 53 (1 mark). The common denominator of 4 and 3 is 12: 2712 + 2012 = 4712 (1 mark). Convert back to a mixed number: 4712 = 31112 (1 mark). Markers reward the improper fractions, the common denominator, and the mixed-number answer.

ScotlandMathsSyllabus dot point

How do you add, subtract, multiply and divide fractions, including mixed numbers, without a calculator?

Calculating with fractions: adding, subtracting, multiplying and dividing fractions and mixed numbers, and finding a fraction of a quantity.

A focused answer to the SQA National 5 Mathematics fractions content, covering adding and subtracting fractions with a common denominator, multiplying and dividing fractions, working with mixed numbers, and finding a fraction of a quantity, all for non-calculator work.

Generated by Claude Opus 4.89 min answerUpdated 2026-06-14

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this dot point is asking
Adding and subtracting fractions
Multiplying fractions
Dividing fractions
Mixed numbers and fractions of a quantity
Subtracting mixed numbers
Examples in context
Try this

What this dot point is asking

The SQA wants you to add, subtract, multiply and divide fractions, including mixed numbers, by hand for non-calculator work, and to find a fraction of a quantity. The methods mirror algebraic fractions, but here the numbers are concrete.

Adding and subtracting fractions

Fractions can only be added or subtracted when they share a denominator. Find the common denominator (often the lowest common multiple of the two denominators), rewrite each fraction over it, then add or subtract the numerators.

Calculate

\tfrac{2}{3} + \tfrac{1}{4}

Fractions can only be added when they share a denominator. We need to rewrite both fractions over the same bottom number before the numerators can be combined.

Step 1: Find the common denominator

The denominators are $3$ and $4$ . The lowest common multiple of $3$ and $4$ is $12$ , because $12$ is the smallest number that both $3$ and $4$ divide into exactly.

Step 2: Rewrite each fraction over the common denominator

To convert $\tfrac{2}{3}$ to twelfths, multiply top and bottom by $4$ : $\tfrac{2}{3} = \tfrac{8}{12}$ . To convert $\tfrac{1}{4}$ , multiply top and bottom by $3$ : $\tfrac{1}{4} = \tfrac{3}{12}$ .

Step 3: Add the numerators

Now the denominators match, so add the numerators and keep the denominator:

\tfrac{8}{12} + \tfrac{3}{12} = \tfrac{11}{12}

Final answer: $\tfrac{2}{3} + \tfrac{1}{4} = \tfrac{11}{12}$ .

Multiplying fractions

Multiplying is the simplest operation: multiply the numerators together and the denominators together. Cancelling common factors before multiplying keeps the numbers small.

Calculate

\tfrac{4}{9} \times \tfrac{3}{8}

When multiplying fractions, we multiply the numerators together and the denominators together. Cancelling common factors before multiplying keeps the numbers small and avoids having to simplify at the end.

Step 1: Cancel common factors across numerators and denominators

Look for a factor shared by any numerator and any denominator. The $4$ in the first numerator and the $8$ in the second denominator share a common factor of $4$ : $4 \div 4 = 1$ and $8 \div 4 = 2$ . The $3$ in the second numerator and the $9$ in the first denominator share a factor of $3$ : $3 \div 3 = 1$ and $9 \div 3 = 3$ . After cancelling, the calculation becomes $\tfrac{1}{3} \times \tfrac{1}{2}$ .

Step 2: Multiply the remaining numerators and denominators

\tfrac{1 \times 1}{3 \times 2} = \tfrac{1}{6}

Final answer: $\tfrac{4}{9} \times \tfrac{3}{8} = \tfrac{1}{6}$ .

Dividing fractions

To divide by a fraction, multiply by its reciprocal: turn the second fraction upside down and multiply.

Mixed numbers and fractions of a quantity

A mixed number such as $2\tfrac{1}{3}$ should be converted to an improper fraction ( $\tfrac{7}{3}$ ) before any calculation: multiply the whole number by the denominator and add the numerator. To find a fraction of an amount, multiply: $\tfrac{3}{4}$ of $20$ is $\tfrac{3}{4} \times 20 = 15$ .

Subtracting mixed numbers

Subtracting mixed numbers is a common exam case. Convert both to improper fractions first, find a common denominator, then subtract, and convert back to a mixed number at the end. Working with improper fractions throughout avoids the trap of having to "borrow" from the whole number.

Calculate

3\tfrac{1}{2} - 1\tfrac{3}{4}

Working with mixed numbers directly risks confusion when the fractional part of the first number is smaller than that of the second. Converting to improper fractions first avoids this completely.

Step 1: Convert both mixed numbers to improper fractions

For $3\tfrac{1}{2}$ : multiply the whole number by the denominator and add the numerator, giving $3 \times 2 + 1 = 7$ , so $3\tfrac{1}{2} = \tfrac{7}{2}$ .

For $1\tfrac{3}{4}$ : $1 \times 4 + 3 = 7$ , so $1\tfrac{3}{4} = \tfrac{7}{4}$ .

Step 2: Find the common denominator and rewrite

The denominators are $2$ and $4$ ; the lowest common multiple is $4$ . Rewrite $\tfrac{7}{2}$ as $\tfrac{14}{4}$ (multiply top and bottom by $2$ ). The second fraction $\tfrac{7}{4}$ already has denominator $4$ .

Step 3: Subtract and convert back

\tfrac{14}{4} - \tfrac{7}{4} = \tfrac{7}{4} = 1\tfrac{3}{4}

Final answer: $3\tfrac{1}{2} - 1\tfrac{3}{4} = 1\tfrac{3}{4}$ .

This combines several skills, converting, finding a common denominator and simplifying, which is exactly how the harder Paper 1 fraction questions are built.

Examples in context

Fractions appear in recipes, measurements and time. Halving a recipe that needs $\tfrac{3}{4}$ of a cup of flour requires $\tfrac{1}{2} \times \tfrac{3}{4} = \tfrac{3}{8}$ of a cup. A joiner cutting boards into thirds and quarters adds fractions to check the total length used. Because Paper 1 is non-calculator, fluent fraction arithmetic is essential for these everyday calculations.

Try this

Q1. Calculate $\tfrac{1}{2} + \tfrac{2}{5}$ . [2 marks]

Cue. $\tfrac{5}{10} + \tfrac{4}{10} = \tfrac{9}{10}$ .

Q2. Calculate $\tfrac{2}{3} \times \tfrac{6}{7}$ . [2 marks]

Cue. $\tfrac{12}{21} = \tfrac{4}{7}$ .

Q3. Calculate $1\tfrac{1}{2} \div \tfrac{3}{4}$ . [3 marks]

Cue. $\tfrac{3}{2} \times \tfrac{4}{3} = 2$ .

Exam-style practice questions

Practice questions written in the style of SQA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

SQA National 5 20183 marksCalculate

2\tfrac{1}{4} + 1\tfrac{2}{3}

, giving your answer as a mixed number in its simplest form.

Show worked answer →

Convert to improper fractions: $2\tfrac{1}{4} = \tfrac{9}{4}$ and $1\tfrac{2}{3} = \tfrac{5}{3}$ (1 mark). The common denominator of $4$ and $3$ is $12$ : $\tfrac{27}{12} + \tfrac{20}{12} = \tfrac{47}{12}$ (1 mark). Convert back to a mixed number: $\tfrac{47}{12} = 3\tfrac{11}{12}$ (1 mark). Markers reward the improper fractions, the common denominator, and the mixed-number answer.

SQA National 5 20223 marksCalculate

\tfrac{3}{5} \times 2\tfrac{1}{2}

, giving your answer in its simplest form.

Show worked answer →

Convert the mixed number: $2\tfrac{1}{2} = \tfrac{5}{2}$ (1 mark). Multiply the fractions: $\tfrac{3}{5} \times \tfrac{5}{2} = \tfrac{15}{10}$ (1 mark). Simplify: $\tfrac{15}{10} = \tfrac{3}{2} = 1\tfrac{1}{2}$ (1 mark). Markers reward the improper fraction, multiplying tops and bottoms, and the simplified answer. Cancelling the $5$ s first gives $\tfrac{3}{2}$ directly.

Related dot points

Sources & how we know this

SQA National 5 Mathematics Course Specification — SQA (2018)