Comparing 2 numbers

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Comparing whole numbers is easy, but fractions need a fair comparison.
The cleanest way is to rewrite both fractions with the same denominator, then just compare the numerators.

For example, compare

\frac{2}{5}

and

\frac{11}{25}

.
Rewrite

\frac{2}{5}

with denominator

25

by multiplying top and bottom by

5

\frac{2}{5} = \frac{10}{25}

.
Since

\frac{11}{25} > \frac{10}{25}

, we get

\frac{11}{25} > \frac{2}{5}

A faster shortcut is cross-multiplication.
To compare

\frac{a}{b}

and

\frac{c}{d}

, compare

a \times d

with

b \times c

.
Whichever product is bigger sits over the bigger fraction.

If you would rather not deal with fractions at all, convert each to a decimal.
For instance,

\frac{2}{5} = 0.40

and

\frac{11}{25} = 0.44

.
Comparing

0.40

and

0.44

shows

\frac{11}{25}

is larger.

Worked examples

Which is greater,

\frac{3}{4}

\frac{5}{7}

?
Cross-multiply: for

\frac{3}{4}

and

\frac{5}{7}

, compare

3 \times 7 = 21

with

4 \times 5 = 20

.
Since

21 > 20

, the fraction

\frac{3}{4}

is greater.

Which is greater,

\frac{9}{20}

\frac{11}{25}

?
Use a common denominator of

100

\frac{9}{20} = \frac{45}{100}

and

\frac{11}{25} = \frac{44}{100}

.
Since

45 > 44

, the fraction

\frac{9}{20}

is greater.

Which is greater,

\frac{5}{8}

\frac{7}{12}

?
Convert to decimals:

\frac{5}{8} = 0.625

and

\frac{7}{12} \approx 0.583

.
Since

0.625 > 0.583

, the fraction

\frac{5}{8}

is greater.

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