Fractions

Video lesson5 worked examples

Theory

Candidates must be able to perform operations and combinations of operations on fractions, including mixed numbers (e.g., $3\tfrac{1}{2}$ , $1\tfrac{1}{3}$ , $1\tfrac{1}{4}$ ).

You must be able to add and subtract simple fractions (e.g., $\tfrac{1}{2} + \tfrac{1}{4}$ and $\tfrac{2}{3} - \tfrac{1}{3}$ ).

A key skill is understanding the interrelationships between fractions, decimal fractions, and percentages to choose an efficient route to a solution.

The Golden Rule: before you multiply or divide, turn every mixed number into a top-heavy (improper) fraction. Before you add or subtract, find a common denominator. Get those set up first and the rest is routine.

⚠️ Common Examiner Traps

Adding the denominators: $\frac{1}{3} + \frac{1}{4}$ is not $\frac{2}{7}$ . Only the numerators are added, once the denominators match.
Multiplying mixed numbers directly: $2\frac{1}{2} \times 1\frac{1}{3}$ is not $2\frac{1}{6}$ . Convert to $\frac{5}{2} \times \frac{4}{3}$ first.
Forgetting to simplify: an unsimplified final fraction can cost the last mark — always check whether the top and bottom share a factor.
“Of” means multiply: $\frac{2}{3}$ of a quantity is $\frac{2}{3} \times$ it, not a division.

Worked examples

Example 1

Adding Mixed Numbers

Calculate $2\tfrac{1}{3} + 1\tfrac{1}{4}$ .

Step 1: Add the whole numbers: $2 + 1 = 3$ .

Step 2: Find a common denominator for the fractions (12): $\tfrac{4}{12} + \tfrac{3}{12} = \tfrac{7}{12}$ .

Answer: $3\tfrac{7}{12}$ .

Example 2

Subtracting Mixed Numbers

Calculate $5\tfrac{1}{4} - 2\tfrac{2}{3}$ .

Step 1: Borrowing across whole numbers is fiddly, so change both mixed numbers to top-heavy fractions: $5\tfrac{1}{4} = \tfrac{21}{4}$ and $2\tfrac{2}{3} = \tfrac{8}{3}$ .

Step 2: Use a common denominator of 12: $\tfrac{63}{12} - \tfrac{32}{12} = \tfrac{31}{12}$ .

Step 3: Convert back to a mixed number.

Answer: $2\tfrac{7}{12}$ .

Example 3

Multiplying Mixed Numbers

Calculate $2\tfrac{1}{5} \times 1\tfrac{1}{4}$ .

Step 1: Change both to top-heavy fractions first — never multiply the whole numbers and fractions separately: $2\tfrac{1}{5} = \tfrac{11}{5}$ and $1\tfrac{1}{4} = \tfrac{5}{4}$ .

Step 2: Multiply straight across the tops and bottoms: $\tfrac{11}{5} \times \tfrac{5}{4} = \tfrac{55}{20}$ .

Step 3: Simplify by dividing top and bottom by 5, then convert to a mixed number: $\tfrac{55}{20} = \tfrac{11}{4}$ .

Answer: $2\tfrac{3}{4}$ .

Example 4

Dividing Fractions

Calculate $\tfrac{3}{4} \div \tfrac{2}{5}$ .

Step 1: Flip the second fraction and multiply (keep, change, flip): $\tfrac{3}{4} \times \tfrac{5}{2}$ .

Step 2: Multiply numerators and denominators: $\tfrac{15}{8}$ .

Answer: $1\tfrac{7}{8}$ .

Example 5

🔗 Bringing it together

Calculate $1\tfrac{1}{2} + \tfrac{2}{3} \times \tfrac{3}{4}$ , giving your answer in its simplest form.

Step 1: Order of operations still applies to fractions — do the multiplication before the addition: $\tfrac{2}{3} \times \tfrac{3}{4} = \tfrac{6}{12} = \tfrac{1}{2}$ .

Step 2: Now add: $1\tfrac{1}{2} + \tfrac{1}{2}$ .

Step 3: The two halves make a whole: $1\tfrac{1}{2} + \tfrac{1}{2} = 2$ .

Answer: 2.

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19 Fractions questions with answers and video solutions