Comparing Fractions

When comparing decimals, we start by comparing the integer parts of these numbers; if the integer parts are equal, we compare the tenths; if the tenths are equal, we compare the hundredths, and so on.


However, it is not so obvious when comparing fractions. What are the techniques?

I. Comparing fractions with a common denominator

A. Rule

To compare positive fractions with a common denominator we simply compare the numerators.


More precisely: Where a, b, and c are three positive numbers and c is a number other than 0, if a < b, then $\frac{a}{c}<\frac{b}{c}$ (and, of course, if a > b, then $\frac{a}{c}>\frac{b}{c}$ ).

B. Examples

We want to compare $\frac{3}{7}$ and $\frac{5}{7}$.


We know that 3 < 5, so $\frac{3}{7}<\frac{5}{7}$.


We want to compare $\frac{2.5}{0.03}$ and $\frac{2.3}{00.5}$.


We know that 2.5 > 2.3, so $\frac{2.5}{0.03}>\frac{2.3}{00.5}$.

II. Comparing fractions with different denominators

At this point we are only looking at cases where one denominator is a multiple of the other (or where it can easily be reduced to this). We will study the general case later.

A. By reducing the fractions to a common denominator

To compare fractions with different denominators we start by reducing them to a common denominator. This means that we replace them with equivalent fractions that have a common denominator. So we can apply the rule from paragraph I.A.


To replace one fraction with another we use the following rule:  $\frac{a}{b}=\frac{a\times k}{b\times k}$
(where b and k are not 0).


Example 1: Compare $\frac{5}{8}$ and $\frac{3}{4}$.


Note that: 8 = 4 × 2, therefore $\frac{3}{4}=\frac{3\times 2}{4\times 2}=\frac{6}{8}$.


We have: $\frac{5}{8}<\frac{6}{8}$ (according to the rule from paragraph I.A), therefore $\frac{5}{8}<\frac{3}{4}$.


Example 2: Compare 5 and $\frac{34}{7}$.

$5=\frac{5\times 7}{7}=\frac{35}{7}$


We have: $5=\frac{5\times 7}{7}=\frac{35}{7}$, therefore $5>\frac{34}{7}$.

B. By calculating decimal approximations

To compare two fractions we can look for decimal approximations by using a calculator:
If we find exact values, comparing these values will give us the result;
otherwise we have to use values that are rounded to sufficient accuracy in order to reach a conclusion.


Example 1: We want to compare $\frac{25}{8}$ and $\frac{77}{25}$.


Using a calculator (or by hand), we find that: $\frac{25}{8}=3.125$ and $\frac{77}{25}=3.08$.


As 3.125 > 3.08, we deduce from this that $\frac{25}{8}>\frac{77}{25}$.


In this example we have used exact values.


Example 2: We want to compare $\frac{62}{37}$ and $\frac{32}{19}$.


Using a calculator (or by hand), we find that: $\frac{62}{37}\approx 1.676$ and $\frac{32}{19}\approx 1.684$ (values rounded to the nearest thousandth).


As 1.676 < 1.684, we deduce that $\frac{62}{37}<\frac{32}{19}$.

In this example, we have used values rounded to the nearest thousandth. Values rounded to the nearest hundredth would have been inconclusive because $\frac{62}{37}\approx 1.68$ and $\frac{32}{19}\approx 1.68$.