Operations on Rational Numbers and Their Properties for SSC CGL CHSL | Solved Examples, MCQs, pdf

Anand Classes provides complete study material on Rational Numbers, their properties, and decimal representation for SSC CGL CHSL exam. In this article, you will learn step-by-step methods to add rational numbers with the same and different denominators, and understand closure, commutative, associative properties along with additive identity and additive inverse. All solved examples are explained in detail for better understanding. Click the print button to download study material and notes.

Operations on Rational Numbers

Addition of Rational Numbers

To add two rational numbers, first express each rational number with a positive denominator (if needed).

Case I: Rational Numbers Having Same Denominator

If $\frac{p}{q}$ and $\frac{r}{q}$ ($q>0$) are two rational numbers, then

$$
\frac{p}{q} + \frac{r}{q} = \frac{p+r}{q}
$$

Example 1. Add the following pairs of rational numbers:

(i) $\frac{5}{7} + \frac{-3}{7}$ (ii) $\frac{-7}{-11} + \frac{1}{11}$

Solution:

(i)
$$
\frac{5}{7} + \frac{-3}{7} = \frac{5+(-3)}{7} = \frac{2}{7}
$$

(ii) First express $\frac{-7}{-11}$ as a rational number with positive denominator:

$$
\frac{-7}{-11} = \frac{(-7)\times(-1)}{(-11)\times(-1)} = \frac{7}{11}
$$

Now,
$$
\frac{7}{11} + \frac{1}{11} = \frac{7+1}{11} = \frac{8}{11}
$$

Case II: Rational Numbers Having Different Denominators

Find LCM of their denominators and express each rational number with this LCM as denominator, then add them as in Case I.

Example 2. Add the following pairs of rational numbers:

(i) $\frac{-3}{5} + \frac{2}{3}$ (ii) $\frac{-23}{18} + \frac{17}{24}$

Solution:

(i)
LCM of $5$ and $3 = 15$

$$
\frac{-3}{5} = \frac{(-3)\times 3}{5\times 3} = \frac{-9}{15}, \quad
\frac{2}{3} = \frac{2\times 5}{3\times 5} = \frac{10}{15}
$$

Now,
$$
\frac{-3}{5} + \frac{2}{3} = \frac{-9}{15} + \frac{10}{15} = \frac{1}{15}
$$

(ii)
LCM of $18$ and $24 = 72$

$$
\frac{-23}{18} = \frac{(-23)\times 4}{18\times 4} = \frac{-92}{72}, \quad
\frac{17}{24} = \frac{17\times 3}{24\times 3} = \frac{51}{72}
$$

Now,
$$
\frac{-23}{18} + \frac{17}{24} = \frac{-92}{72} + \frac{51}{72} = \frac{-41}{72}
$$

Closure Property

Let us add any two rational numbers and check whether the sum is a rational number.

Thus, we find that the sum of two rational numbers is a rational number.

In other words:

If $\frac{p}{q}$ and $\frac{r}{s}$ ($q,s \neq 0$) are two rational numbers, then

$$
\frac{p}{q} + \frac{r}{s} \quad \text{is also a rational number.}
$$

This is called the Closure Property of addition of rational numbers.

Commutative Property

Observe the following:

(i)
$$
\frac{-2}{3} + \frac{2}{7} = \frac{-14+15}{21} = \frac{1}{21}
$$

and

$$
\frac{2}{7} + \frac{-2}{3} = \frac{15+(-14)}{21} = \frac{1}{21}
$$

So,
$$
\frac{-2}{3} + \frac{2}{7} = \frac{2}{7} + \frac{-2}{3}
$$

Thus, in general:

If $\frac{p}{q}$ and $\frac{r}{s}$ ($q,s \neq 0$) are two rational numbers, then

$$
\frac{p}{q} + \frac{r}{s} = \frac{r}{s} + \frac{p}{q}
$$

This is called the Commutative Property of addition of rational numbers.

Associative Property

Observe the following:

(i)
$$
\frac{-2}{3} + \left(\frac{3}{7} + \frac{-5}{6}\right) = \frac{-2}{3} + \left(\frac{18+(-35)}{42}\right) $$

$$ = \frac{-2}{3} + \frac{-17}{42}
= \frac{-28+(-17)}{42} = \frac{-45}{42} $$

Now,
$$
\left(\frac{-2}{3} + \frac{3}{7}\right) + \frac{-5}{6}
= \left(\frac{-14+9}{21}\right) + \frac{-5}{6} $$

$$ = \frac{-5}{21} + \frac{-5}{6}
= \frac{-30+(-35)}{126} = \frac{-65}{126} $$

Hence both are equal, showing associativity.

Thus, in general, if $\frac{p}{q}, \frac{r}{s}, \frac{t}{u}$ are rational numbers ($q,s,u \neq 0$), then

$$
\frac{p}{q} + \left(\frac{r}{s} + \frac{t}{u}\right) = \left(\frac{p}{q} + \frac{r}{s}\right) + \frac{t}{u}
$$

This is called the Associative Property of addition of rational numbers.

Existence of Additive Identity

Observe the following:

(i)
$$
\frac{5}{7} + 0 = \frac{5}{7} = 0 + \frac{5}{7}
$$

(ii)
$$
\frac{-5}{3} + 0 = \frac{-5}{3} = 0 + \frac{-5}{3}
$$

Thus, in general:

For every rational number $\frac{p}{q}$ ($q\neq 0$), there exists $0$ such that

$$
\frac{p}{q} + 0 = \frac{p}{q} = 0 + \frac{p}{q}
$$

The number $0$ is called the Additive Identity of rational numbers.

Existence of Additive Inverse

Observe the following:

(i)
$$
\frac{2}{9} + \left(-\frac{2}{9}\right) = 0, \quad \left(-\frac{2}{9}\right) + \frac{2}{9} = 0
$$

(ii)
$$
\frac{5}{3} + \left(-\frac{5}{3}\right) = 0, \quad \left(-\frac{5}{3}\right) + \frac{5}{3} = 0
$$

Thus, in general:

For every rational number $\frac{p}{q}$ ($q\neq 0$), there exists $\frac{-p}{q}$ such that

$$
\frac{p}{q} + \frac{-p}{q} = 0 = \frac{-p}{q} + \frac{p}{q}
$$

Hence, $\frac{-p}{q}$ is called the Additive Inverse of $\frac{p}{q}$.

Note : To find the additive inverse of any rational number change its sign.

Example 3. Verify the commutative property of addition for:

(i) $\frac{-9}{5}$ and $\frac{3}{4}$ (ii) $\frac{1}{3}$ and $\frac{1}{4}$

Solution:

(i)
LHS:
$$
\frac{-9}{5} + \frac{3}{4} = \frac{-36+15}{20} = \frac{-21}{20}
$$

RHS:
$$
\frac{3}{4} + \frac{-9}{5} = \frac{15+(-36)}{20} = \frac{-21}{20}
$$

Hence verified.

(ii)
LHS:
$$
\frac{1}{3} + \frac{1}{4} = \frac{4+3}{12} = \frac{7}{12}
$$

RHS:
$$
\frac{1}{4} + \frac{1}{3} = \frac{3+4}{12} = \frac{7}{12}
$$

Hence verified.

Example 4. Show that

$$ \left(\frac{-2}{21}\right) + \left(\frac{1}{6} + \frac{-3}{14}\right) = \left(\frac{-2}{21} + \frac{1}{6}\right) + \frac{-3}{14} $$

Solution:

LHS:
$$
\frac{-2}{21} + \left(\frac{1}{6} + \frac{-3}{14}\right)
= \frac{-2}{21} + \left(\frac{7+(-9)}{42}\right)$$

$$= \frac{-2}{21} + \frac{-2}{42}
= \frac{-4+(-2)}{42} = \frac{-6}{42} = \frac{-1}{7}
$$

RHS:
$$
\left(\frac{-2}{21} + \frac{1}{6}\right) + \frac{-3}{14}
= \left(\frac{-4+7}{42}\right) + \frac{-9}{42}
= \frac{3}{42} + \frac{-9}{42} = \frac{-6}{42} = \frac{-1}{7}
$$

Hence verified.
This is the Associative Property.

Example 5. Write the additive inverse of each of the following:

(i) $\frac{-5}{7}$ (ii) $\frac{-3}{4}$

Solution:

(i)
Additive inverse of $\frac{-5}{7}$ is $\frac{5}{7}$.

(ii)
Additive inverse of $\frac{-3}{4}$ is $\frac{3}{4}$.

Example 6. Evaluate $\frac{2}{7} + \frac{-6}{11} + \frac{-8}{21} + \frac{2}{22}$ using properties of addition.

Solution:

Take LCM of denominators $7, 11, 21, 22 = 462$.

Now,
$$
\frac{2}{7} + \frac{-6}{11} + \frac{-8}{21} + \frac{2}{22}
= \frac{132}{462} + \frac{-252}{462} + \frac{-176}{462} + \frac{42}{462}
$$

$$
= \frac{132-252-176+42}{462} = \frac{-254}{462}
$$

Simplify:
$$
= \frac{-127}{231}
$$

Thus, the value is $\frac{-127}{231}$.

FAQs on Rational Numbers for SSC Exams

Q1. What is the definition of a rational number?

A rational number is any number that can be expressed in the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$.

Q2. How do you add rational numbers with the same denominator?

If $\frac{p}{q}$ and $\frac{r}{q}$ are rational numbers, then
$$
\frac{p}{q} + \frac{r}{q} = \frac{p+r}{q}
$$

Q3. How do you add rational numbers with different denominators?

Find the LCM of the denominators, make them equal, and then add.
Example:
$$
\frac{-3}{5} + \frac{2}{3} = \frac{-9}{15} + \frac{10}{15} = \frac{1}{15}
$$

Q4. What is the closure property of rational numbers?

The sum of any two rational numbers is always a rational number.

Q5. What is the additive identity of rational numbers?

Zero (0) is the additive identity because $\frac{p}{q} + 0 = \frac{p}{q}$.

Q6. What is the additive inverse of a rational number?

For every $\frac{p}{q}$, the additive inverse is $\frac{-p}{q}$ since their sum is $0$.

Q7. Are rational numbers commutative under addition?

Yes. For any two rational numbers,
$$
\frac{p}{q} + \frac{r}{s} = \frac{r}{s} + \frac{p}{q}
$$

Q8. Are rational numbers associative under addition?

Yes. For any three rational numbers,
$$
\frac{p}{q} + \left(\frac{r}{s} + \frac{t}{u}\right) = \left(\frac{p}{q} + \frac{r}{s}\right) + \frac{t}{u}
$$

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