[Arithmetic 6] Comparing fractions

❄ COMPARING TWO FRACTIONS

To compare two fractions, we do as follows:

Way 1: Convert two fractions to the same denominator (review here) or numerator.

Way 2: Compare two fractions with suitable numbers (integers or fractions).
Note:
1) The fraction in which numerator and denominator have the same sign integers is greater than 0. A fraction greater than 0 is called a positive fraction. Conversely, the fraction in which numerator and denominator have the different sign integers is less than 0. A fraction less than 0 is called a negative fraction.
2) For a positive fraction (numerator and denominator have the same sign), if the numerator is greater than the denominator, the fraction is greater than 1 and conversely.
3) Of two positive fractions with the same numerator, the fraction with the smaller denominator is the larger fraction and conversely.
4) Given two fractions $\dfrac{a}{b}$ and $\dfrac{c}{d}$ (a, b, c, d $\in$ $\mathbb{N}^*$, a < c, b > d) then $\dfrac{a}{b} < \dfrac{c}{b} < \dfrac{c}{d}$
5) When we just want to know if two fractions are equivalent, we apply the following property $\dfrac{a}{b} = \dfrac{c}{d} \Leftrightarrow a.d = b.c$

Example 1: Are the follwing pairs of fractions equivalent?

1) $\dfrac{2}{3}$ and $\dfrac{6}{9}$

$\dfrac{2}{3} = \dfrac{6}{9}$ since 2.9 = 3.6

2) $\dfrac{{ - 3}}{4}$ and $\dfrac{5}{{ - 7}}$

$\dfrac{{ - 3}}{4} \ne \dfrac{5}{{ - 7}}$ since $( - 3).( - 7) \ne 4.5$

Example 2: Compare the following fractions:

1) $\dfrac{14}{{21}}$ and $\dfrac{ - 60}{{ - 72}}$

$\dfrac{{14}}{{21}} = \dfrac{{2}}{{3}}$, $\dfrac{{ - 60}}{{ - 72}} = \dfrac{{5}}{{6}}$

CD = 6

$\dfrac{2}{3} = \dfrac{{2.2}}{{3.2}} = \dfrac{4}{6}$

Since $\dfrac{{4}}{{6}} < \dfrac{{ 5}}{6}$,

$\dfrac{{14}}{{21}} < \dfrac{{ - 60}}{ - 72}$

2) $\dfrac{-2}{{7}}$ and $\dfrac{24}{{-56}}$

$\dfrac{{24}}{{ - 56}} = \dfrac{{ - 3}}{{7}}$ (divide both the numerator and denominator by $-$8)

Since $\dfrac{{-2}}{{7}} > \dfrac{{ - 3}}{7}$,

$\dfrac{{-2}}{{7}} > \dfrac{{24}}{-56}$

3) $\dfrac{11033}{{15045}}$ and $\dfrac{1111}{{1515}}$

$\dfrac{{11033}}{{15045}} = \dfrac{{11}}{{15}}$ (divide both the numerator and denominator by 1003)

$\dfrac{{1111}}{{1515}} = \dfrac{{11}}{{15}}$ (divide both the numerator and denominator by 101)

So, $\dfrac{{11033}}{{15045}} = \dfrac{{1111}}{{1515}}$

4) $\dfrac{141}{{893}}$ and $\dfrac{159}{{901}}$

$\dfrac{{141}}{{893}} = \dfrac{3}{{19}}$, $\dfrac{{159}}{{901}} = \dfrac{3}{{17}}$

Since $\dfrac{{3}}{{19}} < \dfrac{{3}}{17}$,

$\dfrac{{141}}{{893}} < \dfrac{{159}}{901}$

5) $\dfrac{ - 19}{{7}}$ and $\dfrac{1}{{6}}$

$\dfrac{{ - 19}}{7} < 0 < \dfrac{1}{6}$

$ \Rightarrow $ $\dfrac{{ - 19}}{7} < \dfrac{1}{6}$

6) $\dfrac{{79}}{{97}}$ and $\dfrac{71}{{17}}$

$\dfrac{{79}}{97} < 1 < \dfrac{71}{17}$

$ \Rightarrow $ $\dfrac{{79}}{97} < \dfrac{71}{17}$

7) $\dfrac{{37}}{{195}}$ and $\dfrac{73}{{179}}$

$\dfrac{{37}}{{195}} < \dfrac{{73}}{{195}} < \dfrac{{73}}{{179}}$

$ \Rightarrow $ $\dfrac{{37}}{{195}} < \dfrac{{73}}{{179}}$

8) $\dfrac{{12}}{{29}}$ and $\dfrac{16}{{41}}$

$\dfrac{{12}}{{29}} > \dfrac{{12}}{{30}} = \dfrac{2}{5} = \dfrac{{16}}{{40}} > \dfrac{{16}}{{41}}$

$ \Rightarrow $ $\dfrac{{12}}{{29}} > \dfrac{{16}}{{41}}$

9) $\dfrac{{22}}{{7}}$ and $\dfrac{34}{{11}}$

$\dfrac{{22}}{7} = 3\dfrac{1}{7}\;\left( { = 3 + \dfrac{1}{7}} \right)$, $\dfrac{{34}}{{11}} = 3\dfrac{1}{{11}}\;\left( { = 3 + \dfrac{1}{{11}}} \right)$

Since $\dfrac{{1}}{{7}} > \dfrac{{1}}{11}$,

$\dfrac{{22}}{{7}} > \dfrac{{34}}{11}$