GCSE Maths Practice: fractions

Question 8 of 11

This Higher GCSE question combines addition and subtraction of fractions with mixed-number conversion. You must handle brackets first, align denominators, and simplify into a mixed number to give a clear final result.

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\( \begin{array}{l}\text{Evaluate: }\frac{5}{8}+\left(\frac{3}{4}-\frac{1}{8}\right).\\\text{Express your answer as a mixed number.}\end{array} \)

Choose one option:

Follow BIDMAS: do brackets first, then addition. Find a common denominator for any subtraction or addition, and simplify the final result into a mixed number.

At Higher GCSE level, you will often need to deal with expressions containing both mixed numbers and multiple fraction operations. These test fluency in converting between mixed and improper forms, applying BIDMAS correctly, and simplifying results accurately.

Key Principles

  1. Convert mixed numbers into improper fractions before operating on them.
  2. Use a common denominator to add or subtract fractions.
  3. After the calculation, if the fraction is improper (numerator > denominator), convert back into a mixed number to show understanding.
  4. Remember that “of” in word problems translates into multiplication, and brackets must always be handled first.

Worked Example 1 – Mixed number and subtraction in brackets

\( \tfrac{5}{8}+\left(\tfrac{3}{4}-\tfrac{1}{8}\right) \)

  1. Work inside brackets: \( \tfrac{3}{4}=\tfrac{6}{8} \Rightarrow \tfrac{6}{8}-\tfrac{1}{8}=\tfrac{5}{8}. \)
  2. Add: \( \tfrac{5}{8}+\tfrac{5}{8}=\tfrac{10}{8}=1\tfrac{1}{4}. \)

Example 2 – Mixed number + improper fraction

\( 1\tfrac{2}{3}+\tfrac{5}{6} \)

  1. Convert \( 1\tfrac{2}{3}=\tfrac{5}{3}. \)
  2. Find LCM of 3 and 6 → 6.
  3. \( \tfrac{10}{6}+\tfrac{5}{6}=\tfrac{15}{6}=2\tfrac{1}{2}. \)

Example 3 – Compound with multiplication first

\( \tfrac{1}{2}+\tfrac{3}{4}\times\tfrac{2}{3} \)

  1. Multiply first: \( \tfrac{3}{4}\times\tfrac{2}{3}=\tfrac{6}{12}=\tfrac{1}{2}. \)
  2. Add: \( \tfrac{1}{2}+\tfrac{1}{2}=1. \)

Common Mistakes

  • Changing both numerator and denominator incorrectly when converting to an equivalent fraction.
  • Adding denominators directly instead of finding a common one.
  • Forgetting to convert mixed numbers to improper form before operating.
  • Not simplifying at the end — always check for common factors or convert to mixed number.

Why This Matters

Complex fraction problems appear in geometry, ratio, and proportion contexts — for example, combining parts of an area or length measurements. Expressing answers clearly as mixed numbers shows full comprehension of the part–whole relationship.

Quick FAQs

  • Q: Do I always need to convert to a mixed number?
    A: Only if the question requests it — otherwise, an improper fraction is acceptable.
  • Q: Can I simplify before converting?
    A: Yes. Simplify first to reduce errors when dividing into whole and fractional parts.
  • Q: How can I check my result?
    A: Convert the mixed number back to an improper fraction and see if it matches your earlier step.

Study Tip

Use the “common-denominator ladder”: find the smallest multiple of both denominators, write equivalent fractions, combine, simplify, and then convert. It keeps your layout tidy and logical in the exam.

Try These Yourself

  • \( \tfrac{3}{5}+\left(\tfrac{2}{3}-\tfrac{1}{6}\right) \)
  • \( 1\tfrac{1}{4}+\tfrac{2}{3} \)
  • \( \tfrac{7}{8}+\left(\tfrac{5}{12}-\tfrac{1}{6}\right) \)