Mixed Number Arithmetic (IGCSE Fractions)

Mixed number arithmetic is where Edexcel IGCSE fractions get serious: you turn whole-and-part numbers like two and a third into improper fractions, then add, subtract, multiply and divide them with confidence. This page shows the convert-first method step by step, with clear worked examples and a bank of randomly generated, auto-marked questions below so you can practise every type until it sticks.

Prior Knowledge

Be confident with fraction arithmetic basics (adding, subtracting, multiplying and dividing proper fractions) and with finding the HCF and LCM for common denominators. This page builds straight on both.

⇩ Jump to Practice Questions ⇩

Core Ideas

Mixed number arithmetic follows one reliable pattern: convert to improper fractions, do the operation, then write the answer back as a mixed number. Here is each operation at a glance, with full steps and worked examples below.

Add

Make the denominators the same, add the numerators, then convert back to a mixed number.

Subtract

Same as adding, but subtract the numerators. Converting first avoids borrowing across the whole number.

Multiply

Convert to improper fractions, cancel common factors, then multiply tops and bottoms.

Divide

Convert to improper fractions, then Keep, Change, Flip: multiply by the reciprocal.

Converting Between Mixed and Improper Fractions

Mixed to improper: multiply the whole number by the denominator, add the numerator, and keep the same denominator.

Improper to mixed: divide the numerator by the denominator. The quotient is the whole part, the remainder is the new numerator, and the denominator stays the same. Simplify to lowest terms.

💡 Example 1: Mixed to improper, \(4\dfrac{2}{5}\)

\[ \begin{array}{rcl} 4\dfrac{2}{5} &=& \dfrac{4\times 5 + 2}{5} \\ &=& \dfrac{22}{5} \end{array} \]
What is happening?
  • Whole times denominator: \(4\times 5 = 20\).
  • Add the numerator: \(20+2 = 22\).
  • The denominator stays \(5\).

💡 Example 2: Improper to mixed, \(\dfrac{29}{6}\)

\[ \begin{array}{rcl} 29 \div 6 &=& 4 \text{ r } 5 \\ \dfrac{29}{6} &=& 4\dfrac{5}{6} \end{array} \]
What is happening?
  • How many whole \(6\)s in \(29\)? Four, since \(4\times 6 = 24\).
  • The remainder \(29-24 = 5\) is the new numerator.
  • The denominator stays \(6\).

Good to know: you do not have to convert to add or subtract. You can add or subtract the whole numbers and the fraction parts on their own, carrying or borrowing across the whole number when the fraction part needs it. Converting everything to improper fractions is the safe route though: it always works and skips the carrying and borrowing, so if you think you might forget, just convert every time. (Multiplying and dividing always need the conversion.)

How to Add and Subtract Mixed Numbers

  1. Convert each mixed number to an improper fraction.
  2. Rewrite them over a common denominator (use the lowest common denominator).
  3. Add or subtract the numerators and keep the denominator.
  4. Simplify, and write a top-heavy answer back as a mixed number.

💡 Example 3: Add, \(2\dfrac{3}{4} + 1\dfrac{5}{6}\)

\[ \begin{array}{rcl} 2\dfrac{3}{4} + 1\dfrac{5}{6} &=& \dfrac{11}{4} + \dfrac{11}{6} \\ &=& \dfrac{33}{12} + \dfrac{22}{12} \\ &=& \dfrac{55}{12} \\ &=& 4\dfrac{7}{12} \end{array} \]
What is happening?
  • Convert both to improper fractions.
  • Lowest common denominator of \(4\) and \(6\) is \(12\).
  • Add the numerators, keep the denominator.
  • Convert the top-heavy answer back to a mixed number.

💡 Example 4: Subtract, \(5\dfrac{1}{4} - 2\dfrac{5}{6}\)

\[ \begin{array}{rcl} 5\dfrac{1}{4} - 2\dfrac{5}{6} &=& \dfrac{21}{4} - \dfrac{17}{6} \\ &=& \dfrac{63}{12} - \dfrac{34}{12} \\ &=& \dfrac{29}{12} \\ &=& 2\dfrac{5}{12} \end{array} \]
What is happening?
  • Converting first avoids any borrowing.
  • Common denominator of \(4\) and \(6\) is \(12\).
  • Subtract the numerators, keep the denominator.
  • Write the answer back as a mixed number.

How to Multiply and Divide Mixed Numbers

  1. Convert each mixed number to an improper fraction.
  2. To divide, flip the second fraction (multiply by its reciprocal). To multiply, leave the fractions as they are.
  3. Cancel any common factors, then multiply the tops together and the bottoms together.
  4. Simplify, and write a top-heavy answer back as a mixed number.

💡 Example 5: Multiply, \(2\dfrac{1}{2} \times 1\dfrac{3}{5}\)

\[ \begin{array}{rcl} 2\dfrac{1}{2} \times 1\dfrac{3}{5} &=& \dfrac{5}{2} \times \dfrac{8}{5} \\ &=& \dfrac{5 \times 8}{2 \times 5} \\ &=& \dfrac{40}{10} \\ &=& 4 \end{array} \]
What is happening?
  • Convert to improper fractions.
  • The \(5\)s cancel and a factor of \(2\) divides out.
  • Multiply tops and bottoms.
  • Here the answer is a whole number, \(4\).

💡 Example 6: Divide, \(3\dfrac{1}{3} \div 1\dfrac{1}{9}\)

\[ \begin{array}{rcl} 3\dfrac{1}{3} \div 1\dfrac{1}{9} &=& \dfrac{10}{3} \div \dfrac{10}{9} \\ &=& \dfrac{10}{3} \times \dfrac{9}{10} \\ &=& \dfrac{10 \times 9}{3 \times 10} \\ &=& 3 \end{array} \]
What is happening?
  • Convert to improper fractions.
  • Keep, Change, Flip: multiply by the reciprocal of the second fraction.
  • Cancel the \(10\)s and divide by \(3\).
  • Here the answer is a whole number, \(3\).

🔑 Key Points

  • Convert mixed numbers to improper fractions before multiplying or dividing.
  • Add and subtract need a common denominator; multiply and divide do not.
  • Cancel common factors before multiplying to keep the numbers small.
  • For division, Keep, Change, Flip, then multiply.
  • Always simplify, and write a top-heavy answer back as a mixed number.

⚠️ Common Pitfalls

  • Adding the denominators instead of finding a common denominator.
  • Multiplying or dividing the whole parts and fraction parts separately.
  • Flipping the wrong fraction in division: always flip the second one.
  • Leaving a top-heavy answer when a mixed number is expected.
  • Forgetting to reduce the final fraction to its lowest terms.
⇩ Jump to Practice Questions ⇩

Ready to practise? Work through the auto-marked questions below. For the proper-fraction basics see Fraction Arithmetic, or move on to order of operations next.

Next: BIDMAS →

Mixed Number Arithmetic: Practice Rooms

Randomly generated, auto-marked practice in mixed number arithmetic for Edexcel IGCSE Maths. Room 0 converts between mixed and improper fractions; Rooms 1 to 4 add, subtract, multiply, divide and then mix every type. Every grid gives 16 fresh questions. Type / for a fraction, and put a space before the fraction for a mixed number, for example 2 1/3; it renders live as real maths as you type. Give every answer in its simplest form: whole numbers as a plain number, and any answer bigger than 1 as a mixed number. Each correct answer adds to your global streak.

✓ Correct 0
✗ Re-attempts 0
🔥 Streak 0
🏆 Best 0

Acceptable answers, for example: 4 (a whole number), 7/20 (a proper fraction) or 2 5/12 (a mixed number). An equivalent but top-heavy or un-reduced answer is marked wrong. Spacing does not matter.