Problems on Comparison Between Rational Numbers

Rational numbers are in the form of fractions. In this topic we will solve the problems based on the comparison between the fractions. Methods of comparing the fraction is based upon the types of fractions we have to compare. Here we have to compare between two types of fractions: like fractions and unlike fractions.

Like Fractions: These fractions are those which have same denominator. Since they have the same denominator we only need to compare their numerators. The one having larger numerator will be the greater of two fractions.

Unlike Fractions: These fractions are those which have different denominators and their comparison method differs with like fractions by one step only. First we have to make their denominators equal and rest of the process will be same as that to the like fraction.

Notes:

(i) Always remember that the denominators of the fractions should be positive.

(ii) Always remember that a positive integer is greater that the negative integer.

Let us solve some examples to have better understanding of the the topic:

1. Compare $\frac{3}{5}$ and $\frac{7}{5}$.

Solution:

The given fractions are like fractions as their denominators are equal. So, the one having larger numerator will be greater of the two. Since, 3 < 7 so, $\frac{3}{5}$ is less than $\frac{7}{5}$.

2. Compare $\frac{5}{9}$and $\frac{7}{3}$.

Solution:

The given fractions are unlike fractions as their denominators are unequal. To have a comparison between them first we need to convert them to like fractions by making their denominators equal. So, the L.C.M. of 9 and 3 is 9.

So, we have two fractions as:

$\frac{5}{9}$ and $\frac{7 × 3}{9}$

⟹ $\frac{5}{9}$ and $\frac{21}{9}$

Since they have become like fractions and the one having larger denominator will be greater of the two. Since, 21 > 5.

Hence, $\frac{21}{9}$ > $\frac{5}{9}$.

3. Compare and arrange the following fractions into ascending order.

$\frac{1}{17}$, $\frac{5}{17}$, $\frac{32}{17}$, $\frac{4}{17}$, $\frac{19}{17}$

Solution:

Since the given fractions are like fractions. So, we just need to compare their numerators. Since,

1 < 4 < 5 < 19 < 32

So, the ascending order arrangement is:

$\frac{1}{17}$ < $\frac{4}{17}$ < $\frac{5}{17}$ < $\frac{19}{17}$ < $\frac{32}{17}$.

4. Compare and arrange the following in descending order:

$\frac{2}{5}$, $\frac{4}{15}$, $\frac{5}{6}$, $\frac{7}{20}$

Solution:

The given fractions are unlike fractions. So, first we need to convert them to like fractions and then carry out the comparison process. So, the L.C.M. of 5, 15, 6 and 20 is 60.

Now the fractions become:

$\frac{2 × 12}{60}$, $\frac{4 × 4}{60}$, $\frac{5 × 10}{60}$, $\frac{7 × 3}{60}$,

i.e., $\frac{24}{60}$, $\frac{16}{60}$, $\frac{50}{60}$ and $\frac{21}{60}$.

Now, we need to compare the like fractions.

Since, 50 > 24 > 21 > 16. So, the required descending order of the fractions is as:

$\frac{50}{60}$ > $\frac{24}{60}$ > $\frac{21}{60}$ > $\frac{16}{60}$

i.e., $\frac{5}{6}$ > $\frac{2}{5}$ > $\frac{7}{20}$ > $\frac{4}{15}$

Rational Numbers

Decimal Representation of Rational Numbers

Rational Numbers in Terminating and Non-Terminating Decimals

Recurring Decimals as Rational Numbers

Laws of Algebra for Rational Numbers