Fractions: Equivalence and Operations

≈ 30 min

A fraction is unchanged when you multiply or divide top and bottom by the same number — that gives equivalent fractions. To simplify, divide both by their highest common factor: $\dfrac{12}{18} = \dfrac{12 \div 6}{18 \div 6} = \dfrac{2}{3}$ .

Adding or subtracting needs a common denominator:

\frac{1}{4} + \frac{2}{3} = \frac{3}{12} + \frac{8}{12} = \frac{11}{12}.

Multiplying is direct — multiply tops and multiply bottoms, then simplify:

\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}.

Dividing means multiply by the reciprocal (flip the second fraction):

\frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times \frac{2}{1} = \frac{6}{4} = \frac{3}{2}.

Exam checkpoint

Fractions name equal parts of one whole. Before adding or subtracting, make the parts the same size by finding a common denominator. For multiplication, multiply numerators and denominators; for division, multiply by the reciprocal. Simplify only by factors that divide both numerator and denominator exactly.

Short worked example. $\frac{2}{3}+\frac{1}{6}=\frac{4}{6}+\frac{1}{6}=\frac{5}{6}$ . The denominator 6 tells us all pieces are sixths before we count them.

Simplify $\dfrac{12}{18}$ fully. Give your answer as a fraction in the form a/b.

Simplify: $\dfrac{2}{3} \times \dfrac{3}{4}$

$\dfrac{1}{2}$

$\dfrac{6}{7}$

$\dfrac{5}{7}$

$\dfrac{5}{12}$

Calculate $\dfrac{1}{4} + \dfrac{2}{3}$ . Give your answer as a fraction in the form a/b.

A rusk recipe needs $\dfrac{3}{4}$ cup of sugar. You make half the recipe. How much sugar do you need? Give your answer as a fraction in the form a/b.