What is a fraction?
A fraction represents a part of a whole, written as one number over another: numerator (top, the number of parts you have) over denominator (bottom, the number of equal parts the whole is split into).
Equivalent fractions represent the same value written differently — you get one from another by multiplying or dividing the numerator and denominator by the same number.
1/2 and 2/4 shade the same total area — they are equivalent fractions.
Simplifying fractions
Worked Example 1
Simplify 18/24 to its lowest terms.
1
Find the HCF of 18 and 24 (see the HCF and LCM page): HCF = 6
2
Divide both numerator and denominator by 6: 18 ÷ 6 = 3, and 24 ÷ 6 = 4
Answer3/4
Adding and subtracting fractions
Fractions can only be added or subtracted once they share the same denominator. Use the LCM of the two denominators as the common denominator.
Worked Example 2
Work out 2/3 + 1/4
1
Find the LCM of 3 and 4: 12
2
Convert both fractions to twelfths: 2/3 = 8/12, and 1/4 = 3/12
3
Add the numerators, keep the denominator: 8/12 + 3/12 = 11/12
Answer11/12
Multiplying and dividing fractions
Multiplying fractions is direct — multiply the numerators together and the denominators together. Dividing uses "keep, flip, multiply": keep the first fraction, flip the second, then multiply.
Worked Example 3
Work out (a) 2/5 × 3/7 and (b) 3/4 ÷ 2/9
1
(a) Multiply straight across: (2×3)/(5×7) = 6/35
2
(b) Flip the second fraction and multiply: 3/4 × 9/2 = (3×9)/(4×2) = 27/8 = 3⅜
Answer(a) 6/35 (b) 3⅜
Mixed numbers and improper fractions
Worked Example 4
Convert 2⅗ to an improper fraction, then work out 2⅗ + 1⅘
1
Convert: multiply the whole number by the denominator, then add the numerator: (2×5)+3 = 13, so 2⅗ = 13/5
2
Convert 1⅘ the same way: (1×5)+4 = 9, so 1⅘ = 9/5
3
Add: 13/5 + 9/5 = 22/5, then convert back: 22 ÷ 5 = 4 remainder 2, so 4⅖
Answer4⅖
Practice questions
Work through each question before checking the answers.
Foundation (Grade 3–5)
Q1Simplify 12/16 to its lowest terms.Foundation
Q2Work out 1/4 + 1/4Foundation
Q3Work out 3/5 − 1/5Foundation
Q4Work out 2/3 × 1/4Foundation
Q5Convert 3¼ to an improper fraction.Foundation
Higher (Grade 5–7)
Q6Work out 3/4 + 2/5Higher
Q7Work out 5/6 − 3/8Higher
Q8Work out 4/9 ÷ 2/3Higher
Q9Work out 1⅔ + 2¾, giving your answer as a mixed number.Higher
Q10Work out 2⅖ × 1½, giving your answer as a mixed number.Higher
Higher — Hard (Grade 8–9)
Q11Work out 3⅕ ÷ 2⅗, giving your answer as a fraction in its simplest form.Grade 8–9
Q12A tank is ⅜ full. After adding 15 litres it becomes ¾ full. Find the capacity of the tank.Grade 8–9
Q13Work out (2/3 + 1/4) ÷ (1/2 − 1/3), giving your answer as a fraction in its simplest form.Grade 8–9
Q14Prove algebraically that 1/n + 1/(n+1) can be written as a single fraction (2n+1)/(n(n+1)).Grade 8–9
Q15Two fractions have a sum of 1 and a product of 3/16. Find the two fractions.Grade 8–9
Answers
Foundation (Q1–Q5)
Q13/4
Q21/2
Q32/5
Q41/6
Q513/4
Higher (Q6–Q10)
Q623/20 = 1³/₂₀
Q711/24
Q82/3(4/9 × 3/2 = 12/18 = 2/3)
Q94⁵⁄₁₂(5/3 + 11/4 = 20/12 + 33/12 = 53/12)
Q103⁶⁄₁₀ = 3⅗(12/5 × 3/2 = 36/10)
Higher — Hard (Q11–Q15)
Q1116/13(3⅕ = 16/5, 2⅗ = 13/5; 16/5 ÷ 13/5 = 16/5 × 5/13 = 16/13)
Q1240 litres(¾ − ⅜ = ⅜ of tank = 15 litres, so full tank = 15 ÷ ⅜ = 40)
Q1311/2 = 5½(2/3 + 1/4 = 11/12; 1/2 − 1/3 = 1/6; (11/12) ÷ (1/6) = 11/12 × 6 = 66/12 = 11/2)
Q141/n + 1/(n+1) = (n+1)/(n(n+1)) + n/(n(n+1)) = (2n+1)/(n(n+1)), as required.
Q151/4 and 3/4(sum 1, product 3/16)
Common mistakes
Common Mistake 1
Adding or subtracting fractions without a common denominator
1/2 + 1/3 is not 2/5. Always convert to a common denominator first (LCM of 2 and 3 is 6): 3/6 + 2/6 = 5/6.
Common Mistake 2
Forgetting to flip the second fraction when dividing
Dividing fractions needs the second fraction flipped before multiplying — 3/4 ÷ 2/9 becomes 3/4 × 9/2, not 3/4 × 2/9.
Common Mistake 3
Converting mixed numbers incorrectly
To convert 2⅗ to an improper fraction, multiply the whole number by the denominator then add the numerator: (2×5)+3 = 13, giving 13/5 — not just placing the whole number in front of the fraction.
Common Mistake 4
Not simplifying the final answer
GCSE marks are often lost for leaving a correct fraction unsimplified. Always check whether the numerator and denominator share a common factor before writing your final answer.
Exam tips
💡 Exam Tip 1
Convert mixed numbers before multiplying or dividing
Always turn mixed numbers into improper fractions first — multiplying or dividing whole-number and fraction parts separately is a common source of errors.
💡 Exam Tip 2
Use the LCM, not just any common denominator
Any common denominator works, but using the LCM keeps the numbers smaller and reduces the chance of arithmetic slips — and often means less simplifying at the end.
💡 Exam Tip 3
Sketch a bar or pie diagram for word problems
For problems describing a fraction of an amount (like the tank question), a quick bar diagram showing what fraction is known can make the structure of the problem much clearer.
💡 Exam Tip 4
Check your answer makes sense
Adding two fractions less than 1 should never give an answer of 2 or more. Multiplying two fractions less than 1 should always make the result smaller. Use this as a quick sanity check.
Want to improve your grade faster?
If fractions are still causing problems, Alamin's diagnostic approach identifies exactly which skills are missing and builds a targeted plan to address them — with AI-powered practice between sessions.
Book an Assessment Session (£60)No upfront payment required — payment is taken after confirmation.