Technique

Comparing two fractions

Arithmetic • Number System • Rational Numbers

Statement

For fractions

\frac{a}{b}

and

\frac{c}{d}

with

b > 0

and

d > 0

\frac{a}{b} > \frac{c}{d}

exactly when

a d > b c

; equivalently, rewrite both over a common denominator, or divide each out to a decimal and compare the decimals.

How to use it

1For a single pair, cross-multiply: compare $a d$ against $b c$ and keep the denominators positive.
2For a list of four or five, divide each out to three or four decimal places, which puts them all on one scale at once.
3To sort a list quickly, compare every entry against one convenient benchmark such as $\frac{1}{2}$ or $\frac{3}{4}$ , then cross-multiply only among the survivors.

Worked example

Which is larger,

\frac{5}{8}

\frac{7}{11}

Step-by-step

1
$5 \times 11 = 55$ against $8 \times 7 = 56$
Why: Cross-multiply, keeping each numerator with the other fraction's denominator.
2
$55 < 56 \Rightarrow \frac{5}{8} < \frac{7}{11}$

Answer

\frac{7}{11}

Watch out

The cross-multiplication test reverses if a denominator is negative, so normalise the signs first. The rule of thumb 'the smaller gap between numerator and denominator wins' is false unless the denominators agree:

\frac{2}{3}

has gap

1

and

\frac{7}{10}

has gap

3

, yet

\frac{7}{10}

is the larger.

Also known as

compare fractions, cross-multiplication, ordering rational numbers, common denominator, which fraction is larger

Appears in past papers

Statement

Worked example

Appears in past papers

2020 (II)

2020 (I)

2019 (II)

2018 (II)

2018 (I)

2017 (II)

2017 (I)