Comparison of Like Fractions

Any two like fractions can be compared by comparing their numerators. The fraction with larger numerator is greater than the fraction with smaller numerator, for example $\frac{7}{13}$ > $\frac{2}{13}$ because 7 > 2.

1. In comparison of like fractions here are some rectangular figures.

(i)

$Comparison of Like Fractions$

In (i) shaded portion represents $\frac{2}{7}$

(ii)

$Comparison of Like Fractions$

In (ii) shaded portion represents $\frac{3}{7}$

(iii)

$Comparison of Like Fractions$

In (iii) shaded portion represents $\frac{5}{7}$

It is clear that $\frac{2}{7}$ < $\frac{3}{7}$ < $\frac{5}{7}$

or $\frac{5}{7}$ > $\frac{3}{7}$ > $\frac{2}{7}$

Thus, in like fractions or fractions having same denominator, that fraction is greater which has the greater numerator.

Accordingly, $\frac{3}{5}$ > $\frac{2}{5}$; $\frac{2}{5}$ < $\frac{3}{5}$

$\frac{15}{17}$ > $\frac{10}{17}$; $\frac{10}{17}$ < $\frac{15}{17}$

2. Again, let us consider $\frac{2}{5}$ and $\frac{3}{5}$.

$2 by 5$

$\frac{2}{5}$ represents 2 parts out of 5 equal parts on the strip.

$3 by 5$

$\frac{3}{5}$ represents 3 parts out of 5 equal parts on the strip.

3 > 2

Hence, for any two like fractions, the fraction with the larger numerator is greater than the fraction with smaller numerator.

$Comparison of Like Fractions$

If there are three or more like fractions, they may be arranged in ascending (increasing) and descending (decreasing) order. The order will be according to the order of the numerators.

(a) Ascending order: $\frac{1}{9}$, $\frac{2}{9}$, $\frac{3}{9}$, $\frac{4}{9}$, $\frac{5}{9}$, $\frac{6}{9}$, $\frac{7}{9}$, $\frac{8}{9}$:

as, 1 < 2 < 3 < 4 < 5 < 6 < 7 < 8

(b) Descending order: $\frac{8}{9}$, $\frac{7}{9}$, $\frac{5}{9}$, $\frac{4}{9}$, $\frac{3}{9}$, $\frac{2}{9}$, $\frac{1}{9}$:

as, 8 > 7 > 5 > 4 > 3 > 2 > 1

Similarly again;

(a) Ascending order: $\frac{5}{17}$, $\frac{7}{17}$, $\frac{8}{17}$, $\frac{11}{17}$, $\frac{13}{17}$, $\frac{14}{17}$, $\frac{16}{17}$:

as, 5 < 7 < 8 < 11 < 13 < 14 < 16

(b) Descending order: $\frac{16}{17}$, $\frac{14}{17}$, $\frac{13}{17}$, $\frac{11}{17}$, $\frac{8}{17}$, $\frac{7}{17}$, $\frac{5}{17}$:

as, 16 > 14 > 13 > 11 > 8 > 7 > 5

Comparison of Fractions:

Case I: Fractions with same Denominator:

If two fractions have the same denominator, the fraction with greater numerator denotes the greater fraction.

For example,

$\frac{6}{9}$ > $\frac{4}{9}$

$\frac{2}{3}$ < $\frac{5}{3}$

Comparison of fractions with the same denominator

Observe the following figures.

$Comparison of Fractions$

In the first figure, 2 parts out of 6 equal parts are shaded.

In the second figure, 3 parts out of 6 equal parts are shaded.

Clearly, shaded parts in the second circle are more than those in the first circle.

Thus, $\frac{3}{6}$ > $\frac{2}{6}$ or $\frac{2}{6}$ < $\frac{3}{6}$

Hence, among two fractions having the same denominator, the fraction with the greater numerator is greater than the other.

1. Compare $\frac{2}{5}$ and $\frac{4}{5}$

Solution:

Consider $\frac{2}{5}$, $\frac{4}{5}$

Since, 4 > 2, hence $\frac{4}{5}$ > $\frac{2}{5}$ or $\frac{2}{5}$ < $\frac{4}{5}$

2. Compare $\frac{7}{15}$ and $\frac{8}{15}$

Solution:

Consider $\frac{7}{15}$, $\frac{8}{15}$

Since, 8 > 7, hence $\frac{8}{15}$ > $\frac{7}{15}$ or $\frac{7}{15}$ < $\frac{8}{15}$

Three or more like fractions can be arranged in ascending or descending order by arranging their numerators in ascending or descending order.

For example, the fractions $\frac{8}{11}$, $\frac{7}{11}$, $\frac{10}{11}$, $\frac{3}{11}$ can be arranged in the ascending order as $\frac{3}{11}$ < $\frac{7}{11}$ < $\frac{8}{11}$ < $\frac{10}{11}$

Case II: Fraction with Like Numerators:

The fraction with smaller denominator is greater.

Examples:

$\frac{4}{7}$ > $\frac{4}{10}$;

$\frac{8}{13}$ > $\frac{8}{17}$;

$\frac{18}{33}$ > $\frac{18}{41}$ etc.

Similarly, $\frac{3}{5}$ < $\frac{3}{4}$;

$\frac{11}{13}$ < $\frac{11}{10}$;

$\frac{25}{42}$ < $\frac{25}{37}$ etc.