Learning Objectives
  • Recognise proper fractions, improper fractions and mixed numbers.
  • Simplify fractions and find equivalent fractions.
  • Add, subtract, multiply and divide fractions accurately.
  • Solve multi-step problems involving fractions of quantities.
Key Terms
Numerator
The number above the fraction line.
Denominator
The number below the fraction line.
Proper fraction
A fraction whose numerator is smaller than its denominator.
Improper fraction
A fraction whose numerator is greater than or equal to its denominator.
Mixed number
A whole number combined with a proper fraction.
Reciprocal
The number obtained by interchanging numerator and denominator; the product of a non-zero number and its reciprocal is 1.
Equivalent Fractions And Simplest Form

Equivalent fractions represent the same value. Multiplying or dividing the numerator and denominator by the same non-zero number does not change the fraction. A fraction is in simplest form when the numerator and denominator have no common factor greater than 1.

\frac{a}{b}=\frac{ka}{kb}\quad (b\ne0,\;k\ne0)
Worked Example: Simplifying

Question: Simplify \frac{84}{126}.

  1. The HCF of 84 and 126 is 42.
  2. Divide both numerator and denominator by 42: \frac{84\div42}{126\div42}.

Answer: \frac{2}{3}

Converting Improper Fractions And Mixed Numbers

To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator. To convert an improper fraction to a mixed number, divide the numerator by the denominator; the quotient is the whole-number part and the remainder becomes the numerator.

a\frac{b}{c}=\frac{ac+b}{c}
Worked Example: Conversion

Question: Convert 3\frac{5}{8} to an improper fraction and \frac{47}{6} to a mixed number.

  1. 3\frac{5}{8}=\frac{3\times8+5}{8}=\frac{29}{8}.
  2. 47\div6=7 remainder 5, so \frac{47}{6}=7\frac{5}{6}.

Answer: \frac{29}{8} and 7\frac{5}{6}.

Adding And Subtracting Fractions

Fractions can be added or subtracted only after they have a common denominator. The most efficient common denominator is usually the LCM of the denominators. Convert mixed numbers to improper fractions when the calculation is complicated or when subtraction would require borrowing.

Worked Example: Addition

Question: Work out 2\frac{1}{6}+1\frac{3}{4}.

  1. Convert to improper fractions: \frac{13}{6}+\frac{7}{4}.
  2. The LCM of 6 and 4 is 12.
  3. Rewrite: \frac{26}{12}+\frac{21}{12}=\frac{47}{12}.
  4. Convert to a mixed number: 3\frac{11}{12}.

Answer: 3\frac{11}{12}

Worked Example: Subtraction

Question: Work out 5\frac{1}{3}-2\frac{7}{8}.

  1. Convert to improper fractions: \frac{16}{3}-\frac{23}{8}.
  2. Use denominator 24: \frac{128}{24}-\frac{69}{24}=\frac{59}{24}.
  3. Convert to a mixed number.

Answer: 2\frac{11}{24}

Multiplying Fractions

Multiply numerators together and denominators together. Cancel common factors before multiplying to keep numbers small. Mixed numbers must first be changed to improper fractions.

Worked Example: Multiplication

Question: Work out 2\frac{2}{5}\times1\frac{3}{4}.

  1. Convert: \frac{12}{5}\times\frac{7}{4}.
  2. Cancel 12 with 4 to give 3 and 1.
  3. Multiply: \frac{3\times7}{5}=\frac{21}{5}.
  4. Convert to a mixed number.

Answer: 4\frac{1}{5}

Dividing Fractions

Dividing by a non-zero fraction is equivalent to multiplying by its reciprocal. Keep the first fraction, change division to multiplication and invert the second fraction.

\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\times\frac{d}{c}
Worked Example: Division

Question: Work out 3\frac{1}{2}\div\frac{7}{12}.

  1. Convert 3\frac{1}{2} to \frac{7}{2}.
  2. Multiply by the reciprocal: \frac{7}{2}\times\frac{12}{7}.
  3. Cancel 7 and simplify \frac{12}{2}=6.

Answer: 6

Fractions Of Quantities

“Of” usually means multiplication. To find \frac{3}{8} of 240, divide by 8 and multiply by 3. In reverse problems, if \frac{3}{8} of a quantity is 90, divide 90 by 3 and multiply by 8.

Worked Example: Reverse Fraction

Question: \frac{5}{12} of a number is 70. Find the number.

  1. One twelfth is 70\div5=14.
  2. Twelve twelfths are 14\times12=168.

Answer: The number is 168.

8th Edition Chapter Map
  • Proper fractions, improper fractions and mixed numbers
  • Addition and subtraction of fractions and mixed numbers
  • Multiplication and division of fractions and mixed numbers
Ordering Fractions Efficiently

Fractions can be ordered by finding a common denominator, converting to decimals, or comparing cross-products. Cross-multiplication is often fastest for two positive fractions: compare \frac{a}{b} and \frac{c}{d} by comparing ad and bc. This comparison does not require forming the decimal values.

Worked Example: Ordering Four Fractions

Order \frac{5}{8},\frac{7}{12},\frac{2}{3},\frac{11}{18} from least to greatest.

  1. The LCM of 8, 12, 3 and 18 is 72.
  2. Convert: \frac{5}{8}=\frac{45}{72}, \frac{7}{12}=\frac{42}{72}, \frac{2}{3}=\frac{48}{72}, \frac{11}{18}=\frac{44}{72}.
  3. Therefore \frac{7}{12}<\frac{11}{18}<\frac{5}{8}<\frac{2}{3}.
Combined Operations With Fractions

The order of operations still applies. Work inside brackets first, then powers, multiplication and division, followed by addition and subtraction. In a long fraction calculation, write one clear line for each stage and cancel only factors in multiplication, never terms joined by addition.

Worked Example: Brackets And Division

Question: Evaluate \left(2\frac{1}{4}-\frac{5}{6}\right)\div\frac{17}{24}.

  1. 2\frac14=\frac94.
  2. \frac94-\frac56=\frac{27}{12}-\frac{10}{12}=\frac{17}{12}.
  3. \frac{17}{12}\div\frac{17}{24}=\frac{17}{12}\times\frac{24}{17}=2.
Fractions Of Quantities And Reverse Fraction Problems

To find a fraction of a quantity, multiply by the fraction. To find the whole when a fraction is known, divide by the fraction. A diagram or unitary method can prevent a common error in reverse problems.

Worked Example: Finding The Whole

\frac{7}{12} of a tank contains 315 litres. The full capacity is 315\div\frac{7}{12}=315\times\frac{12}{7}=540 litres.

Complex Fractions

A fraction whose numerator or denominator is itself a fraction can be simplified by treating the main fraction line as division. For example, \dfrac{\frac34}{\frac56}=\frac34\div\frac56=\frac{9}{10}. Another method is to multiply the top and bottom by a common multiple of the small denominators.

Operation Reliable method Check
Add or subtract Use a common denominator, then combine numerators. The denominator must not be added or subtracted.
Multiply Multiply numerators and denominators; cancel common factors first. The result should be sensible in size.
Divide Multiply by the reciprocal of the divisor. Dividing by a proper fraction should increase a positive number.
Extended Practice

A. Evaluate 3\frac25+1\frac{7}{10}-2\frac14.

\frac{17}{5}+\frac{17}{10}-\frac94=\frac{68+34-45}{20}=\frac{57}{20}=2\frac{17}{20}.

B. A class spends \frac38 of a fund on books and \frac{5}{12} on equipment. What fraction remains?

Spent =\frac{9}{24}+\frac{10}{24}=\frac{19}{24}, so \frac{5}{24} remains.

C. Find \dfrac{1-\frac25}{1+\frac12}.

\dfrac{\frac35}{\frac32}=\frac35\times\frac23=\frac25.

Examination Guidance
  • Give fractions in simplest form unless the context requires another form.
  • Use exact fractions rather than rounded decimals when an exact answer is possible.
  • When dividing fractions, invert only the divisor, not both fractions.
  • In word problems, write a statement explaining what each fraction represents.
Common Mistakes
  • Adding denominators when adding fractions.
  • Forgetting to convert mixed numbers before multiplication or division.
  • Cancelling across addition or subtraction; cancellation is valid only for common factors in a product.
  • Leaving an improper fraction when the context clearly expects a mixed number or measurement.
Knowledge Check

1. Simplify \frac{132}{198}.

Answer

\frac{2}{3}.

2. Work out \frac{5}{6}-\frac{7}{15}.

Answer

\frac{11}{30}.

3. Work out 1\frac{3}{5}\times2\frac{1}{4}.

Answer

3\frac{3}{5}.

4. Work out \frac{9}{10}\div\frac{3}{5}.

Answer

\frac{3}{2}=1\frac{1}{2}.

5. \frac{7}{9} of a number is 56. Find the number.

Answer

72.

Mixed Review With Full Solutions

1. Evaluate \frac56-\left(\frac34-\frac23\right).

The bracket is \frac1{12}. Therefore \frac56-\frac1{12}=\frac{10-1}{12}=\frac34.

2. A recipe uses 2\frac14 cups of flour for 6 portions. How much flour is needed for 14 portions?

Per portion =\frac94\div6=\frac38 cup. For 14 portions, 14\times\frac38=\frac{42}{8}=5\frac14 cups.

3. Simplify \dfrac{\frac57-\frac3{14}}{\frac9{10}}.

Numerator =\frac{10-3}{14}=\frac12. Then \frac12\div\frac9{10}=\frac12\times\frac{10}{9}=\frac59.

4. \frac35 of a number exceeds \frac14 of the same number by 42. Find the number.

The difference is \frac35-\frac14=\frac7{20}. Hence \frac7{20}N=42, so N=42\times\frac{20}{7}=120.

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