GCSE Maths Practice: fractions

Question 4 of 12

This question tests your understanding of adding fractions with different denominators. Find a common denominator, rewrite each fraction, then add the numerators and simplify.

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\( \text{Calculate }\frac{2}{5}+\frac{1}{3}. \)

Choose one option:

Use the least common multiple (LCM) to create a common denominator. After adding, always check if the result can be simplified or expressed as a mixed number.

Understanding Addition of Fractions with Different Denominators

When fractions have different denominators, they represent parts of different sizes, so you cannot add them directly. Before adding, you must make the denominators the same — this ensures both fractions are measured in equal parts of a whole. Once denominators are equal, the addition works just like whole numbers: add the numerators and keep the denominator unchanged.

Step-by-Step Method

  1. Find the least common denominator (LCD): Determine the smallest number that both denominators divide into exactly. This is usually the least common multiple (LCM).
  2. Rewrite both fractions: Adjust each fraction so that the denominator becomes the LCD by multiplying both numerator and denominator by the same number.
  3. Add the numerators: Once denominators are the same, add the top numbers only.
  4. Keep the denominator the same: It represents the size of each part and remains unchanged.
  5. Simplify if possible: Check for common factors between numerator and denominator.

For this question, \(\frac{2}{5}+\frac{1}{3}\) → LCM(5,3)=15. Convert: \(\frac{2}{5}=\frac{6}{15}\) and \(\frac{1}{3}=\frac{5}{15}\). Add: \(6+5=11\). The answer is \(\frac{11}{15}\).

Worked Examples

  • Example 1: \(\frac{3}{4}+\frac{1}{6}\). LCD = 12 → \(\frac{3}{4}=\frac{9}{12}\), \(\frac{1}{6}=\frac{2}{12}\), so \(\frac{11}{12}\).
  • Example 2: \(\frac{5}{8}+\frac{3}{10}\). LCD = 40 → \(\frac{25}{40}+\frac{12}{40}=\frac{37}{40}\).
  • Example 3: \(\frac{7}{9}+\frac{2}{3}\). LCD = 9 → \(\frac{7}{9}+\frac{6}{9}=\frac{13}{9}=1\tfrac{4}{9}\).
  • Example 4: \(\frac{1}{2}+\frac{3}{5}=\frac{5}{10}+\frac{6}{10}=\frac{11}{10}=1\tfrac{1}{10}\).

Common Mistakes to Avoid

  • Adding denominators: \(\frac{2}{5}+\frac{1}{3}\neq\frac{3}{8}\). The denominator never adds — it only changes through finding a common multiple.
  • Forgetting to multiply both parts: When converting \(\frac{2}{5}\) to \(\frac{6}{15}\), multiply both numerator and denominator by the same number (3).
  • Leaving answers unsimplified: Always check if the result can be reduced to lowest terms or written as a mixed number.

Real-Life Applications

Adding fractions with different denominators is a common skill in everyday life. Suppose a recipe calls for \(\frac{2}{5}\) of a cup of sugar and \(\frac{1}{3}\) of a cup of honey. To find the total sweetness added, you need to calculate \(\frac{2}{5}+\frac{1}{3}=\frac{11}{15}\) cups. The same logic applies to combining probabilities, merging data ratios, or measuring lengths when units are subdivided differently.

FAQs

Q1: Why can’t we just add denominators?
A1: Because denominators represent the size of each part. You can only add fractions that have equal-sized parts.

Q2: What if the denominators are prime numbers, like 5 and 7?
A2: Multiply them together (35) to get a common denominator, then adjust both fractions.

Q3: Can I simplify before adding?
A3: Only if both fractions have factors that can be reduced first. Otherwise, simplify after adding.

Study Tip

Always look for the smallest common denominator — it makes calculations quicker and avoids large numbers. Practise converting and adding pairs like \(\frac{2}{5}+\frac{3}{10}\) or \(\frac{1}{6}+\frac{1}{4}\) until the process becomes automatic. Strong fraction fluency supports later GCSE Maths topics such as algebraic fractions, ratios, and proportional reasoning.

By mastering addition with different denominators, you build a solid foundation for more advanced number skills, ensuring speed and confidence in your exams and real-world applications.