Equivalent fractions

Foundation and Higher

The idea

Multiplying or dividing the top and bottom of a fraction by the same number keeps its value: $\tfrac{3}{4} = \tfrac{9}{12}$ . A mixed number is a whole number and a fraction: $2\tfrac{1}{3} = \tfrac{7}{3}$ .

To simplify, divide top and bottom by their HCF.
Mixed to improper: whole $\times$ bottom $+$ top, over the same bottom: $2\tfrac{1}{3} = \tfrac{2 \times 3 + 1}{3}$ .
Improper to mixed: divide, then the remainder goes over the bottom.

Watch out The whole number becomes thirds (or halves, or fifths), not just a $2$ added to the top.

See it worked

Worked exampleNon-calculator4 marks

Answer each part without a calculator.

(a) Write $\frac{18}{24}$ in its simplest form.

(b) Write $2\frac{3}{5}$ as an improper fraction.

(c) Write $\frac{17}{4}$ as a mixed number.

(d) Which is larger, $\frac{5}{8}$ or $0.6$ ?

(a) The highest common factor of 18 and 24 is 6. $\frac{18 \div 6}{24 \div 6} = \frac{3}{4}$ .
(b) $2$ wholes are $\frac{10}{5}$ . Add $\frac{3}{5}$ : $\frac{10}{5} + \frac{3}{5} = \frac{13}{5}$ .
(c) $17 \div 4 = 4$ remainder $1$ , so $\frac{17}{4} = 4\frac{1}{4}$ .
(d) $\frac{5}{8} = 5 \div 8 = 0.625$ , and $0.625 > 0.600$ .

Answer (a) $\frac{3}{4}$ ; (b) $\frac{13}{5}$ ; (c) $4\frac{1}{4}$ ; (d) $\frac{5}{8}$

Your turn

Question 1Non-calculator5 marks

Answer each part without a calculator.

(a) Write $\frac{3}{8}$ as a decimal.

(b) Write $2\frac{1}{5}$ as a decimal.

Question 2Non-calculator4 marks

A carpenter has planks of length $2\frac{3}{5}$ m.

(a) Write $2\frac{3}{5}$ as an improper fraction.

(b) Four planks are laid end to end. Find the total length. Give your answer as a mixed number.

Question 3Non-calculator3 marks

A tank is full. Exactly 2/7 of the water is removed. Adding 18 litres then makes the tank exactly 3/4 full. Find the capacity of the tank.

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