Fractions

Video coming soon3 worked examples

Theory

The Golden Rule: Fractions in National 5 Applications often involve adding and subtracting parts of a whole to find a missing amount, or calculating leftover amounts from multiple wholes (such as multiple cakes or pizzas). Always ensure you have a common denominator before doing any addition or subtraction!

1. Adding & Subtracting Fractions

You cannot add or subtract fractions unless the bottom numbers (denominators) are the same.

Step 1: Find a common denominator (the lowest common multiple of the bottom numbers).
Step 2: Multiply the top numbers (numerators) by the same amount you multiplied the bottom numbers by.
Step 3: Add or subtract the numerators. The denominator stays the same.

2. Finding "The Rest" (Fractions of a Whole)

If a question gives you several fractions of a group and asks for "the rest" or "the remaining", you must:

Add the given fractions together.
Subtract your total from 1 (a whole).

Tip: The number 1 can be written as any fraction where the top and bottom numbers are identical (e.g., $1 = \frac{15}{15}$ or $1 = \frac{24}{24}$ ).

3. Top-Heavy (Improper) Fractions & Mixed Numbers

You are expected to be able to work with and convert between mixed numbers (e.g., $3 \frac{1}{2}$ ) and top-heavy fractions (e.g., $\frac{7}{2}$ ).

Converting to Top-Heavy: Multiply the whole number by the denominator, then add the numerator (e.g., for $2 \frac{3}{4}$ , calculate $2 \times 4 + 3 = 11$ , so the fraction is $\frac{11}{4}$ ).
Always convert mixed numbers into top-heavy fractions before trying to add or subtract them.

4. Simplifying

Always check your final answer to see if the top and bottom numbers can be divided by a common factor.

⚠️ Common Examiner Traps

The "Adding Denominators" Trap: A classic error under exam pressure is adding both the numerators AND the denominators together (e.g., deciding that $\frac{1}{3} + \frac{1}{4} = \frac{2}{7}$ ). You must find a common denominator first!
The "Multiple Wholes" Trap: When a question states that someone bought 2 identical cakes or pizzas and gives you the fractions eaten from each, candidates frequently just add the fractions together and subtract from 1, completely forgetting that there were 2 wholes to begin with.
The Unsimplified Final Answer: Failing to express your final fraction in its simplest form can occasionally cost you the final communication mark. Always check if both numbers halve, or divide by 3 or 5.

Worked examples

Example 1

A school is organising a summer trip for the S0 year group.

$\frac{2}{5}$ of the pupils voted to go to a theme park.
$\frac{1}{3}$ of the pupils voted to go to the zoo.
The remaining pupils voted to go to the cinema.

Calculate the fraction of pupils who voted to go to the cinema. (3 marks)

Step 1: Add the two known fractions together by finding a common denominator (15).

\frac{2}{5} + \frac{1}{3} = \frac{6}{15} + \frac{5}{15} = \frac{11}{15}

Step 2: Subtract this total from 1 (the whole year group) to find the remainder.

1 - \frac{11}{15} \Rightarrow \frac{15}{15} - \frac{11}{15} = \frac{4}{15}

Final Answer: $\frac{4}{15}$ of the pupils voted for the cinema.

Example 2

A carpenter has a wooden board measuring $5 \frac{1}{4}$ metres in length. He cuts off a piece measuring $2 \frac{2}{3}$ metres to build a shelf. Calculate the length of the remaining wooden board. (3 marks)

Step 1: Convert both mixed numbers into top-heavy fractions.

5 \frac{1}{4} = \frac{(5 \times 4) + 1}{4} = \frac{21}{4}

2 \frac{2}{3} = \frac{(2 \times 3) + 2}{3} = \frac{8}{3}

Step 2: Find a common denominator (12) and subtract the cut piece from the total.

\frac{21}{4} - \frac{8}{3} = \frac{63}{12} - \frac{32}{12} = \frac{31}{12}

Step 3: Convert the top-heavy fraction back into a mixed number (how many 12s go into 31?).

Answer: $2 \frac{7}{12}$ metres.

Example 3

David bought 2 identical large pizzas for a family movie night.

His children ate $\frac{5}{6}$ of the first pizza.
The adults ate $\frac{3}{8}$ of the second pizza.

Calculate the total amount of pizza left over. Give your answer as a fraction of a single pizza. (3 marks)

Step 1: Calculate how much is left over from the first pizza.

1 - \frac{5}{6} = \frac{1}{6} \text{ left over}

Step 2: Calculate how much is left over from the second pizza.

1 - \frac{3}{8} = \frac{5}{8} \text{ left over}

Step 3: Add the two leftover amounts together by finding a common denominator (24).

\frac{1}{6} + \frac{5}{8} = \frac{4}{24} + \frac{15}{24} = \frac{19}{24}

Final Answer: There is $\frac{19}{24}$ of a pizza left over.

Ready to practise?

11 Fractions questions with answers and video solutions