### Theory:

Unlike fractions are fractions with different denominators.
Let us see how we can order the unlike fractions with the same numerator and compare the unlike fractions with different denominator.

Ordering of unlike fractions with the same numerator:

$\frac{6}{13},\phantom{\rule{0.147em}{0ex}}\frac{6}{14},\phantom{\rule{0.147em}{0ex}}\frac{6}{15}$ are unlike fractions with the same numerator.

The fractions can be ordered as, $\frac{6}{15}>\frac{6}{13}>\frac{6}{14}$ because the unlike fraction with the highest denominator will be the greatest among unlike fractions with the same numerator.

Comparison of unlike fractions with different numerator:

$\frac{4}{5},\frac{2}{3},\frac{6}{7}$. To order unlike fractions with different numerators, we have to convert all the unlike fractions to like fractions. To do that follow the steps below:

i) Take LCM (Least Common Multiple) of all the denominators.

ii) Make the denominators of all the fractions into LCM.

iii) Compare the numerators of the like fractions, the fraction with the greatest numerator will be the greatest fraction.
Example:
Find the greatest fraction among $\frac{4}{5},\frac{2}{3},\frac{6}{7}$
i) LCM of all the denominators $$5, 3, 7$$ is $$105$$.

ii) $\frac{4×21}{5×21},\frac{2×35}{3×35},\frac{6×15}{7×15}$ $$=$$ $\frac{84}{105},\frac{70}{105},\frac{90}{105}$

• To convert $$5$$ into $$105$$, $$105 ÷ 5 = 21$$, multiply $$4/5$$ with $$21$$ in both numerator and denominator to get $$105$$ as the denominator (LCM) of the fraction.
• To convert $$3$$ into $$105$$, $$105 ÷ 3 = 35$$, multiply $$2/3$$ with $$35$$ in both numerator and denominator to get $$105$$ as the denominator (LCM) of the fraction.
• To convert $$7$$ into $$105$$, $$105 ÷ 7 = 15$$, multiply $$6/7$$ with $$15$$ in both numerator and denominator to get $$105$$ as the denominator (LCM) of the fraction.
iii) Compare the numerators of like fractions, $$84$$, $$70$$ and $$90$$. Where $$90 > 84 > 70$$ so $\frac{90}{105}>\frac{84}{105}>\frac{70}{105}$.

Write the corresponding fractions, $\frac{6}{7}>\frac{4}{5}>\frac{2}{3}$.

Hence, the greatest fraction among $\frac{4}{5},\frac{2}{3},\frac{6}{7}$ is $\frac{6}{7}$.