Fractions — Simplifying & Operations

From basics to adding, subtracting, multiplying & dividing with confidence

GCSE Number
Fractions — Simplifying & Operations – cover
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What is a fraction?

A fraction represents part of a whole. It has a numerator (top) and a denominator (bottom): \(\dfrac{\text{part}}{\text{whole}}\). For example, \(\tfrac{3}{4}\) means 3 parts out of 4 equal parts.

Key vocabulary

  • Proper fraction: numerator < denominator, e.g. \(\tfrac{3}{5}\).
  • Improper fraction: numerator ≥ denominator, e.g. \(\tfrac{9}{7}\).
  • Mixed number: a whole number and a fraction together, e.g. \(1\tfrac{2}{5}\).
  • Equivalent fractions: different-looking fractions with the same value, e.g. \(\tfrac{1}{2}=\tfrac{2}{4}=\tfrac{50}{100}\).

Equivalent fractions & simplifying

Two fractions are equivalent if you multiply or divide top and bottom by the same non-zero number. To simplify a fraction, divide numerator and denominator by their greatest common divisor (GCD).

Method (simplifying)

  1. Find a common factor of the numerator and denominator.
  2. Divide both by that factor; repeat until no common factor remains.
  3. The result is in lowest terms.
Worked Example 1: Simplify \(\tfrac{24}{36}\)

Common factors: 24 and 36 share 2, 3, 4, 6, 12. The greatest is 12. \[ \frac{24}{36}=\frac{24\div12}{36\div12}=\frac{2}{3}. \] Answer: \(\tfrac{2}{3}\).

Worked Example 2: Put \(\tfrac{56}{84}\) in lowest terms

GCD\( (56,84)=28\). \[ \frac{56}{84}=\frac{56\div28}{84\div28}=\frac{2}{3}. \] Answer: \(\tfrac{2}{3}\).

Mixed numbers ↔ improper fractions

  • \(a\tfrac{b}{c}\) to improper: \(\dfrac{ac+b}{c}\).
  • Improper \(\tfrac{n}{c}\) to mixed: divide \(n\) by \(c\): \(n=qc+r\) with \(0\le r<c\). Then \(q\tfrac{r}{c}\).
Worked Example 3: Convert \(3\tfrac{1}{4}\) to an improper fraction

Multiply the whole by the denominator and add the numerator: \[ 3\times 4+1=13,\quad \Rightarrow \quad 3\tfrac{1}{4}=\frac{13}{4}. \]

Worked Example 4: Convert \(\tfrac{29}{6}\) to a mixed number

\(29\div 6=4\) remainder \(5\). \[ \frac{29}{6}=4\tfrac{5}{6}. \]

Adding and subtracting fractions

To add or subtract fractions, we need a common denominator. The safest is the LCM (lowest common multiple) of the denominators.

Same denominator

Keep the denominator, add/subtract the numerators: \[ \frac{a}{d}\pm \frac{b}{d}=\frac{a\pm b}{d}. \]

Worked Example 5: \(\tfrac{3}{8}+\tfrac{5}{8}\)

Denominator is the same (8): \(3+5=8\). \[ \frac{3}{8}+\frac{5}{8}=\frac{8}{8}=1. \]

Different denominators

  1. Find the LCM of the denominators.
  2. Convert each fraction to an equivalent with that denominator.
  3. Add/subtract numerators; simplify.
Worked Example 6: \(\tfrac{2}{3}+\tfrac{5}{6}\)

Denominators 3 and 6 → LCM = 6. \[ \frac{2}{3}=\frac{4}{6},\quad \frac{5}{6}=\frac{5}{6}. \] Add: \(\tfrac{4}{6}+\tfrac{5}{6}=\tfrac{9}{6}=\tfrac{3}{2}=1\tfrac{1}{2}.\)

Worked Example 7: \(2\tfrac{1}{5}-\tfrac{3}{10}\)

Convert to improper if easier: \(2\tfrac{1}{5}=\tfrac{11}{5}\). LCM(5,10)=10. \[ \frac{11}{5}=\frac{22}{10},\quad \frac{3}{10}=\frac{3}{10}. \] Subtract: \(\tfrac{22}{10}-\tfrac{3}{10}=\tfrac{19}{10}=1\tfrac{9}{10}.\)

Negative fractions

Carry the sign in the numerator or in front: \(-\tfrac{3}{4}=\tfrac{-3}{4}\). Use the same common-denominator method and add/subtract signed numbers carefully.

Worked Example 8: \(-\tfrac{5}{12}+\tfrac{1}{3}\)

LCM(12,3)=12. \[ -\frac{5}{12}+\frac{1}{3}=-\frac{5}{12}+\frac{4}{12}=-\frac{1}{12}. \]

Multiplying fractions

Multiply tops; multiply bottoms. Then simplify. \[ \frac{a}{b}\times \frac{c}{d}=\frac{ac}{bd}. \] Often you can cancel a common factor across a numerator and a denominator before multiplying.

Worked Example 9: \(\tfrac{3}{10}\times \tfrac{5}{6}\)

Cancel common factors: 5 in top with 10 in bottom → becomes \(\tfrac{3}{2}\times \tfrac{1}{6}\). Also cancel 3 in top with 6 in bottom → \(\tfrac{1}{2}\times \tfrac{1}{2}\). \[ =\frac{1}{4}. \]

Worked Example 10: \(2\tfrac{1}{2}\times 1\tfrac{1}{3}\)

Convert to improper: \[ 2\tfrac{1}{2}=\frac{5}{2},\quad 1\tfrac{1}{3}=\frac{4}{3}. \] Multiply: \(\tfrac{5}{2}\cdot \tfrac{4}{3}=\tfrac{20}{6}=\tfrac{10}{3}=3\tfrac{1}{3}.\)

Dividing fractions

To divide by a fraction, multiply by its reciprocal (“flip and multiply”). \[ \frac{a}{b}\div \frac{c}{d}=\frac{a}{b}\times \frac{d}{c}. \] Then simplify.

Worked Example 11: \(\tfrac{7}{9}\div \tfrac{14}{27}\)

Multiply by reciprocal: \[ \frac{7}{9}\times \frac{27}{14}. \] Cancel: \(7\) with \(14\to 1\) and \(2\); \(9\) with \(27\to 1\) and \(3\): \[ \frac{1}{1}\times \frac{3}{2}=\frac{3}{2}=1\tfrac{1}{2}. \]

Worked Example 12: \(4\div \tfrac{5}{8}\)

Write \(4\) as \(\tfrac{4}{1}\) and multiply by reciprocal \(\tfrac{8}{5}\): \[ \frac{4}{1}\times \frac{8}{5}=\frac{32}{5}=6\tfrac{2}{5}. \]

Fractions, decimals and percentages

Converting is useful for checking and for real-world problems.

  • \(\textbf{Fraction → decimal}\): divide numerator by denominator, e.g. \(\tfrac{3}{8}=0.375\).
  • \(\textbf{Decimal → fraction}\): write over 10, 100, 1000… and simplify, e.g. \(0.125=\tfrac{125}{1000}=\tfrac{1}{8}\).
  • \(\textbf{Fraction → percentage}\): multiply by 100%, e.g. \(\tfrac{3}{20}=15\%\).
Worked Example 13: Which is larger, \(\tfrac{7}{12}\) or \(58\%\)?

Compare as decimals. \(58\%=0.58\). \(\tfrac{7}{12}\approx 0.5833\ldots\) so \(\tfrac{7}{12}\) is larger.

Order of operations with fractions

Remember BIDMAS/BODMAS. Treat fraction arithmetic the same as integers: do brackets, then indices, then multiply/divide, then add/subtract.

Worked Example 14: \(\tfrac{3}{4}+\left(\tfrac{2}{3}\times \tfrac{9}{8}\right)\)

Do the multiplication first: \[ \frac{2}{3}\times \frac{9}{8}=\frac{18}{24}=\frac{3}{4}. \] Now \(\tfrac{3}{4}+\tfrac{3}{4}= \tfrac{6}{4}=1\tfrac{1}{2}\).

Checking your answers

  • Simplest form? Could a common factor reduce it further?
  • Sensible size? Estimate with decimals/percentages to see if your result is reasonable.
  • Mixed vs improper: give the form the question asks for.
  • Units/context in word problems.

Common mistakes to avoid

  • Adding/subtracting denominators directly (wrong). You must use a common denominator.
  • Forgetting to simplify at the end.
  • Not converting mixed numbers to improper fractions before multiplying/dividing.
  • Dropping negative signs when finding common denominators.
  • Flipping the wrong fraction when dividing (only the divisor flips).

Exam tips

  • Write intermediate steps clearly; examiners award method marks.
  • Cancel factors early in multiplication/division to keep numbers small.
  • When time is tight, estimate (e.g., \(\tfrac{5}{6}\approx 0.83\)) to check you’re close.
  • Label your final answer as mixed/improper exactly as requested.

Practice set A (core skills)

  1. Simplify: \(\tfrac{18}{30}\).
  2. Write as a mixed number: \(\tfrac{41}{8}\).
  3. \(\tfrac{7}{12}+\tfrac{1}{6}\).
  4. \(3\tfrac{3}{5}-\tfrac{4}{5}\).
  5. \(\tfrac{9}{10}\times \tfrac{5}{6}\).
  6. \(\tfrac{11}{12}\div \tfrac{11}{18}\).
  7. Which is larger: \(\tfrac{4}{9}\) or \(45\%\)?
Answers — Set A
  1. \(\tfrac{3}{5}\)
  2. \(5\tfrac{1}{8}\)
  3. LCM 12: \(\tfrac{7}{12}+\tfrac{2}{12}=\tfrac{9}{12}=\tfrac{3}{4}\)
  4. \(3\tfrac{3}{5}-\tfrac{4}{5}=3-\tfrac{1}{5}=2\tfrac{4}{5}\)
  5. \(\tfrac{9}{10}\times \tfrac{5}{6}=\tfrac{45}{60}=\tfrac{3}{4}\)
  6. \(\tfrac{11}{12}\div \tfrac{11}{18}=\tfrac{11}{12}\times \tfrac{18}{11}=\tfrac{18}{12}=\tfrac{3}{2}=1\tfrac{1}{2}\)
  7. \(45\%=0.45\), \(\tfrac{4}{9}\approx 0.444\ldots \Rightarrow 45\%\) is larger

Practice set B (exam-style)

  1. A recipe uses \(\tfrac{3}{4}\) cup of sugar for 1 batch. You make \(2\tfrac{1}{2}\) batches. How much sugar is needed?
  2. A tank is \(\tfrac{5}{8}\) full. After removing \(\tfrac{1}{5}\) of the water in the tank, what fraction of the tank is full?
  3. Order these from smallest to largest: \(\tfrac{2}{3}\), \(0.62\), \(65\%\), \(\tfrac{5}{8}\).
  4. Evaluate: \(\left(\tfrac{7}{10} - \tfrac{1}{4}\right)\div \tfrac{3}{5}\).
  5. The fraction of a class choosing basketball is \(\tfrac{9}{20}\). The rest prefer football or tennis in the ratio \(5:3\). What fraction choose football?
Answers — Set B
  1. \(2\tfrac{1}{2}=\tfrac{5}{2}\). Multiply: \(\tfrac{3}{4}\times \tfrac{5}{2}=\tfrac{15}{8}=1\tfrac{7}{8}\) cups.
  2. Remove \(\tfrac{1}{5}\) of current amount: remaining \(=\tfrac{4}{5}\times \tfrac{5}{8}=\tfrac{4}{8}=\tfrac{1}{2}\) of the tank.
  3. Convert to decimals: \(\tfrac{5}{8}=0.625\), \(\tfrac{2}{3}\approx 0.666\ldots\), \(0.62=0.62\), \(65\%=0.65\). Order: \(0.62\), \(\tfrac{5}{8}(0.625)\), \(65\%(0.65)\), \(\tfrac{2}{3}(0.66\ldots)\).
  4. \(\tfrac{7}{10}-\tfrac{1}{4}=\tfrac{14}{20}-\tfrac{5}{20}=\tfrac{9}{20}\). Divide by \(\tfrac{3}{5}\): \(\tfrac{9}{20}\times \tfrac{5}{3}=\tfrac{45}{60}=\tfrac{3}{4}\).
  5. Remaining fraction after basketball: \(1-\tfrac{9}{20}=\tfrac{11}{20}\). Split in ratio \(5:3\). Total parts \(=8\). Football \(=\tfrac{5}{8}\) of \(\tfrac{11}{20}=\tfrac{55}{160}=\tfrac{11}{32}\).

Further challenge

  1. Simplify \(\dfrac{6x}{9y}\times \dfrac{12y}{15x}\).
  2. If \(\tfrac{a}{b}=\tfrac{3}{5}\) with \(a,b\in \mathbb{Z}^+\) and \(a+b\le 50\), list three possible pairs \((a,b)\).
Solutions — Challenge
  1. \(\dfrac{6x}{9y}\cdot \dfrac{12y}{15x}=\dfrac{72xy}{135xy}=\dfrac{72}{135}=\dfrac{8}{15}\).
  2. Multiples of \((3,5)\): \((6,10)\), \((9,15)\), \((12,20)\), \((15,25)\), \((18,30)\), \((21,35)\), \((24,40)\), \((27,45)\).

Great work! You can now simplify, add, subtract, multiply and divide fractions, convert between forms, and tackle exam-style problems with confidence.

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