Operations with Fractions

This lesson covers the four operations applied to fractions and mixed numbers — adding, subtracting, multiplying and dividing — together with finding a fraction of an amount and the idea of a reciprocal. Fluency with fractions is essential across the whole of OCR GCSE Mathematics (J560), especially on the non-calculator Paper 1 where calculator shortcuts are not available, and it feeds directly into ratio, proportion and algebra later.

This lesson mainly builds AO1 fluency in fraction arithmetic, with AO2 reasoning when you decide whether an answer is sensible, and AO3 problem-solving in the multi-step worded examples.

Key Vocabulary

Term	Meaning
Numerator	The top number of a fraction (how many parts)
Denominator	The bottom number (how many equal parts in the whole)
Mixed number	A whole number together with a proper fraction, e.g. $2\tfrac{3}{4}$
Improper fraction	A fraction whose numerator $\ge$ denominator, e.g. $\dfrac{11}{4}$
Equivalent fractions	Fractions of equal value, e.g. $\dfrac{2}{3} = \dfrac{8}{12}$
Reciprocal	The fraction "turned upside down"; $\dfrac{a}{b}$ has reciprocal $\dfrac{b}{a}$
Lowest common denominator (LCD)	The LCM of the denominators, used to add or subtract

Equivalent Fractions and Simplifying

Everything in this lesson rests on equivalent fractions — different-looking fractions with the same value. Multiplying (or dividing) the numerator and denominator by the same number does not change a fraction's value, because you are really multiplying by a disguised $1$ . For instance $\dfrac{2}{3} = \dfrac{2 \times 4}{3 \times 4} = \dfrac{8}{12}$ , and $\dfrac{8}{12} = \dfrac{8 \div 4}{12 \div 4} = \dfrac{2}{3}$ again.

Worked Example 0

Write $\dfrac{18}{24}$ in its simplest form.

The HCF of $18$ and $24$ is $6$ , so divide both by $6$ : $\dfrac{18}{24} = \dfrac{3}{4}$ .

Answer: $\dfrac{3}{4}$ .

This skill matters because every fraction answer should be given in its simplest form unless the question says otherwise, and because rewriting fractions with a common denominator (next section) is just making equivalent fractions on purpose.

Adding and Subtracting Fractions

To add or subtract fractions you need a common denominator. Find the LCM of the denominators, rewrite each fraction with that denominator (using equivalent fractions), then add or subtract the numerators only. The denominator tells you the size of the parts, so the parts must be the same size before you can combine them — you cannot add fifths to quarters any more than you can add metres to kilograms without converting first.

Worked Example 1

Work out $\dfrac{2}{5} + \dfrac{1}{4}$ .

The LCM of $5$ and $4$ is $20$ . Rewrite: $\dfrac{2}{5} = \dfrac{8}{20}$ and $\dfrac{1}{4} = \dfrac{5}{20}$ .

$\dfrac{8}{20} + \dfrac{5}{20} = \dfrac{13}{20}.$

Answer: $\dfrac{13}{20}$ (already in simplest form).

Common error: Adding the denominators too, writing $\dfrac{3}{9}$ . Once the denominators match, you add the numerators only.

Worked Example 2

Work out $\dfrac{5}{6} - \dfrac{3}{8}$ .

LCM of $6$ and $8$ is $24$ . Rewrite: $\dfrac{5}{6} = \dfrac{20}{24}$ and $\dfrac{3}{8} = \dfrac{9}{24}$ .

$\dfrac{20}{24} - \dfrac{9}{24} = \dfrac{11}{24}.$

Answer: $\dfrac{11}{24}$ .

Worked Example 2b

Work out $\dfrac{1}{2} + \dfrac{1}{3} + \dfrac{1}{4}$ .

The LCM of $2$ , $3$ and $4$ is $12$ . Rewrite each: $\dfrac{1}{2} = \dfrac{6}{12}$ , $\dfrac{1}{3} = \dfrac{4}{12}$ , $\dfrac{1}{4} = \dfrac{3}{12}$ .

$\dfrac{6}{12} + \dfrac{4}{12} + \dfrac{3}{12} = \dfrac{13}{12} = 1\dfrac{1}{12}.$

Answer: $1\dfrac{1}{12}$ . With three or more fractions, find a single common denominator for all of them at once rather than combining two at a time.

Mixed Numbers

To add or subtract mixed numbers, the safest method is to convert each to an improper fraction first, work as above, then convert back.

Worked Example 3

Work out $3\dfrac{1}{2} + 1\dfrac{2}{3}$ .

Convert: $3\dfrac{1}{2} = \dfrac{7}{2}$ and $1\dfrac{2}{3} = \dfrac{5}{3}$ . LCM of $2$ and $3$ is $6$ .

$\dfrac{7}{2} + \dfrac{5}{3} = \dfrac{21}{6} + \dfrac{10}{6} = \dfrac{31}{6} = 5\dfrac{1}{6}.$

Answer: $5\dfrac{1}{6}$ .

Worked Example 4

Work out $4\dfrac{1}{4} - 1\dfrac{2}{3}$ .

Convert: $4\dfrac{1}{4} = \dfrac{17}{4}$ , $1\dfrac{2}{3} = \dfrac{5}{3}$ . LCM of $4$ and $3$ is $12$ .

$\dfrac{17}{4} - \dfrac{5}{3} = \dfrac{51}{12} - \dfrac{20}{12} = \dfrac{31}{12} = 2\dfrac{7}{12}.$

Answer: $2\dfrac{7}{12}$ .

Common error: Subtracting the whole numbers and fractions separately and getting stuck when the second fraction is bigger ( $\tfrac{1}{4} - \tfrac{2}{3}$ is negative). Converting to improper fractions avoids this entirely.

Multiplying and Dividing Fractions

Multiplying fractions is actually simpler than adding them — there is no need for a common denominator. To multiply, multiply the numerators together and the denominators together; cancel common factors first to keep numbers small. This works because " $\dfrac{3}{8}$ of $\dfrac{4}{9}$ " means taking three-eighths of four-ninths, and multiplying tops and bottoms captures exactly that. To divide, multiply by the reciprocal of the second fraction ("keep, change, flip"). Dividing by $\dfrac{2}{3}$ is the same as multiplying by $\dfrac{3}{2}$ , because asking "how many two-thirds fit into a number" is the inverse of multiplying by two-thirds.

Worked Example 5

Work out $\dfrac{3}{8} \times \dfrac{4}{9}$ .

Cancel before multiplying: $4$ and $8$ share a factor of $4$ ( $\to 1$ and $2$ ); $3$ and $9$ share a factor of $3$ ( $\to 1$ and $3$ ).

$\dfrac{3}{8} \times \dfrac{4}{9} = \dfrac{1}{2} \times \dfrac{1}{3} = \dfrac{1}{6}.$

Answer: $\dfrac{1}{6}$ .

Worked Example 6

Work out $\dfrac{5}{6} \div \dfrac{10}{9}$ .

Multiply by the reciprocal of $\dfrac{10}{9}$ :

$\dfrac{5}{6} \div \dfrac{10}{9} = \dfrac{5}{6} \times \dfrac{9}{10} = \dfrac{45}{60} = \dfrac{3}{4}.$

Answer: $\dfrac{3}{4}$ .

Common error: Flipping the first fraction instead of the second. Keep the first, change $\div$ to $\times$ , flip the second.

Reciprocals

The reciprocal of a number is $1$ divided by that number; for a fraction you simply swap numerator and denominator. The reciprocal of a whole number $n$ is $\dfrac{1}{n}$ , and any number multiplied by its reciprocal gives $1$ .

Worked Example 7

Write down the reciprocal of (a) $\dfrac{3}{7}$ , (b) $5$ , (c) $2\dfrac{1}{2}$ .

(a) Swap: $\dfrac{7}{3}$ . (b) $\dfrac{1}{5}$ . (c) First write $2\dfrac{1}{2} = \dfrac{5}{2}$ , then the reciprocal is $\dfrac{2}{5}$ .

Answers: (a) $\dfrac{7}{3}$ , (b) $\dfrac{1}{5}$ , (c) $\dfrac{2}{5}$ .

Fractions of an Amount

To find a fraction of an amount, divide by the denominator and multiply by the numerator.

Worked Example 8

Work out $\dfrac{3}{8}$ of £56.

$56 \div 8 = 7$ (so £7), then $7 \times 3 = 21$ (so £21).

Answer: £21.

OCR Exam Tip: " $\dfrac{3}{8}$ of" means " $\dfrac{3}{8} \times$ ". Divide by the bottom, multiply by the top — that order keeps the numbers small.

Extended Worked Examples

Worked Example 9: Multiplying mixed numbers

Work out $2\dfrac{2}{3} \times 1\dfrac{1}{2}$ .

Convert both to improper fractions: $2\dfrac{2}{3} = \dfrac{8}{3}$ , $1\dfrac{1}{2} = \dfrac{3}{2}$ .

$\dfrac{8}{3} \times \dfrac{3}{2} = \dfrac{8 \times 3}{3 \times 2} = \dfrac{24}{6} = 4.$

Answer: $4$ . (Cancelling the $3$ s first gives $\dfrac{8}{1} \times \dfrac{1}{2} = 4$ more quickly.)

Common error: Multiplying the whole numbers and the fractions separately ( $2 \times 1$ and $\tfrac{2}{3} \times \tfrac{1}{2}$ ). That does not work for multiplication — you must convert to improper fractions first.

Worked Example 10: Dividing mixed numbers in context

A plank is $4\dfrac{1}{2}$ m long. How many $\dfrac{3}{4}$ m pieces can be cut from it?

This is $4\dfrac{1}{2} \div \dfrac{3}{4}$ . Convert: $4\dfrac{1}{2} = \dfrac{9}{2}$ .

$\dfrac{9}{2} \div \dfrac{3}{4} = \dfrac{9}{2} \times \dfrac{4}{3} = \dfrac{36}{6} = 6.$

Answer: $6$ pieces (with none left over).

Worked Example 11: Combining operations (BIDMAS with fractions)

Work out $\dfrac{1}{2} + \dfrac{1}{3} \times \dfrac{3}{4}$ .

By BIDMAS, multiply first: $\dfrac{1}{3} \times \dfrac{3}{4} = \dfrac{3}{12} = \dfrac{1}{4}$ .

Then add: $\dfrac{1}{2} + \dfrac{1}{4} = \dfrac{2}{4} + \dfrac{1}{4} = \dfrac{3}{4}$ .

Answer: $\dfrac{3}{4}$ .

Worked Example 12: Fraction of an amount, two stages

In a class of $32$ students, $\dfrac{3}{8}$ walk to school and $\dfrac{1}{4}$ of the remainder cycle. Work out how many students cycle.

Walkers: $\dfrac{3}{8} \times 32 = 12$ . Remainder $= 32 - 12 = 20$ . Cyclists: $\dfrac{1}{4} \times 20 = 5$ .

Answer: $5$ students cycle.

Worked Example 13: Comparing two fractions

Which is larger, $\dfrac{5}{8}$ or $\dfrac{7}{12}$ ?

Put both over a common denominator (the LCM of $8$ and $12$ is $24$ ): $\dfrac{5}{8} = \dfrac{15}{24}$ and $\dfrac{7}{12} = \dfrac{14}{24}$ . Since $15 > 14$ , $\dfrac{5}{8}$ is the larger.

Answer: $\dfrac{5}{8} > \dfrac{7}{12}$ .

Common error: Assuming $\dfrac{7}{12}$ is bigger "because the numbers are bigger". The size of a fraction depends on the relationship between numerator and denominator, so always compare over a common denominator (or convert to decimals).

Operations with Fractions

Operations with Fractions

Key Vocabulary

Equivalent Fractions and Simplifying

Worked Example 0

Adding and Subtracting Fractions

Worked Example 1

Worked Example 2

Worked Example 2b

Mixed Numbers

Worked Example 3

Worked Example 4

Multiplying and Dividing Fractions

Worked Example 5

Worked Example 6

Reciprocals

Worked Example 7

Fractions of an Amount

Worked Example 8

Extended Worked Examples

Worked Example 9: Multiplying mixed numbers

Worked Example 10: Dividing mixed numbers in context

Worked Example 11: Combining operations (BIDMAS with fractions)

Worked Example 12: Fraction of an amount, two stages

Worked Example 13: Comparing two fractions

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