Anand Classes provides the best SSC exam preparation study material, covering important mathematics topics like Rational Numbers, Properties and Decimal Representation of Rational Numbers with detailed explanations, examples, and solved questions. These concepts form the foundation for SSC CGL, SSC CHSL, SSC MTS, SSC GD, and other competitive exams where number system questions are frequently asked. Our SSC Maths notes are designed to help aspirants understand the basics clearly and practice effectively with exam-oriented questions. Click the print button to download study material and notes.

What are Rational Numbers ?

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

Examples: $\frac{7}{3}, -2, \frac{-2}{7}, 0$ etc. are all rational numbers.

What are Properties of Rational Numbers ?

• Every integer (positive, negative, or zero) is a rational number.
  Examples: $5, -12, 0$, etc.
• Every fraction is a rational number.
  Examples: $\frac{3}{9}, \frac{8}{5}, \frac{1}{8}$, etc.
• The rational number $-\frac{p}{q}$ can also be written as $\frac{-p}{q}$.
  Thus, $-\frac{2}{3}$ and $\frac{-2}{3}$ are equal.
• If $p$ and $q$ are both positive integers, then $\frac{p}{q}$ is called a positive rational number, and $\frac{-p}{q}$ is called a negative rational number.
• Every terminating decimal number is a rational number.
  Examples:
  $0.5 = \frac{1}{2}, \quad 0.28 = \frac{7}{25}$
• Every non-terminating repeating decimal number is a rational number.
  Examples:
  $\frac{421}{99} = 4.25\overline{25}, \quad 6.666… = 6.\overline{6}, \quad -3 = -3.000…$
• Non-terminating, non-repeating decimal numbers are called irrational numbers.
  Examples:
  $1.41421… = \sqrt{2}, \quad 1.73205… = \sqrt{3}$

Solved Examples on Rational Numbers

Q. Express $0.75$ as a rational number.

Solution:
$0.75 = \frac{75}{100} = \frac{3}{4}$

Thus, $0.75$ is a rational number.

Q. Convert $0.333…$ into the form $\frac{p}{q}$.

Solution:
Let $x = 0.\overline{3}$

Multiply both sides by $10$:
$10x = 3.\overline{3}$

Subtracting:
$10x – x = 3.\overline{3} – 0.\overline{3}$
$9x = 3$

So, $x = \frac{1}{3}$

Hence, $0.333… = \frac{1}{3}$.

Q. Find two rational numbers between $\frac{1}{2}$ and $\frac{2}{3}$.

Solution:
We know:
$\frac{1}{2} = \frac{3}{6}$ and $\frac{2}{3} = \frac{4}{6}$

Between them lies: $\frac{7}{12}, \frac{5}{9}$, etc.

Thus, two rational numbers between $\frac{1}{2}$ and $\frac{2}{3}$ are $\frac{7}{12}$ and $\frac{5}{9}$.

Q. Check whether $\frac{7}{20}$ is a terminating decimal.

Solution:
Denominator = $20 = 2^2 \times 5$

Since denominator has only the prime factors $2$ and $5$, $\frac{7}{20}$ is a terminating decimal.

Dividing:
$\frac{7}{20} = 0.35$

Q. Express $0.142857142857…$ as a rational number.

Solution:
This is a repeating decimal $0.\overline{142857}$

We know:
$\frac{1}{7} = 0.\overline{142857}$

Thus, $0.\overline{142857} = \frac{1}{7}$

FAQs on Rational Numbers

1. What is a rational number?

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

2. Are all integers rational numbers?

Yes. Every integer $n$ can be expressed as $\frac{n}{1}$, which is a rational number.

3. Are all fractions rational numbers?

Yes. Every fraction $\frac{a}{b}$ (with $b \neq 0$) is a rational number.

4. What is the difference between terminating and non-terminating decimals?

• A terminating decimal has a finite number of digits after the decimal point.
  Example: $\frac{1}{4} = 0.25$
• A non-terminating recurring decimal repeats its digits infinitely.
  Example: $\frac{1}{3} = 0.\overline{3}$

5. How do we identify whether a rational number is terminating?

A rational number $\frac{p}{q}$ (in lowest terms) is terminating if the prime factors of $q$ are only $2$ and/or $5$.

6. Can every decimal be written as a rational number?

• Yes, if it is terminating or repeating.
• No, if it is non-terminating and non-repeating (e.g., $\sqrt{2}, \pi$).

7. Are repeating decimals always rational?

Yes. Every repeating decimal can be expressed as a fraction.
Example: $0.\overline{7} = \frac{7}{9}$

8. What is the density property of rational numbers?

Between any two rational numbers, there exist infinitely many rational numbers.

Example: Between $\frac{1}{2}$ and $\frac{2}{3}$, numbers like $\frac{3}{5}, \frac{7}{12}, \frac{5}{9}$ lie.

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

The additive inverse of a rational number $a$ is $-a$.
Example: The additive inverse of $\frac{3}{7}$ is $-\frac{3}{7}$.

10. What is the multiplicative inverse of a rational number?

The multiplicative inverse of $\frac{p}{q}$ is $\frac{q}{p}$ (where $p \neq 0$).
Example: The multiplicative inverse of $\frac{5}{8}$ is $\frac{8}{5}$.

Multiple Choice Questions (MCQs) on Rational Numbers

1. Which of the following is a rational number?

A) $\sqrt{2}$
B) $\pi$
C) $\dfrac{7}{3}$
D) $0.1010010001…$

✅ Answer: C) $\dfrac{7}{3}$

2. Which of the following decimals is a rational number?

A) $0.121221222…$
B) $0.25$
C) $\pi$
D) $\sqrt{3}$

✅ Answer: B) $0.25$

3. The decimal expansion of $\dfrac{1}{6}$ is:

A) Terminating
B) Non-terminating, repeating
C) Non-terminating, non-repeating
D) None of these

✅ Answer: B) Non-terminating, repeating

4. Which of the following fractions will have a terminating decimal expansion?

A) $\dfrac{7}{12}$
B) $\dfrac{13}{50}$
C) $\dfrac{29}{90}$
D) $\dfrac{41}{45}$

✅ Answer: B) $\dfrac{13}{50}$

5. Between $\dfrac{2}{3}$ and $\dfrac{3}{4}$, which of the following lies?

A) $\dfrac{11}{16}$
B) $\dfrac{5}{8}$
C) $\dfrac{7}{9}$
D) $\dfrac{4}{5}$

✅ Answer: B) $\dfrac{5}{8}$

6. The multiplicative inverse of $-\dfrac{7}{11}$ is:

A) $-\dfrac{11}{7}$
B) $\dfrac{7}{11}$
C) $\dfrac{11}{7}$
D) $-7$

✅ Answer: A) $-\dfrac{11}{7}$

7. The sum of a rational number and an irrational number is:

A) Always rational
B) Always irrational
C) Sometimes rational
D) None of these

✅ Answer: B) Always irrational

8. Which of the following is NOT a rational number?

A) $0.\overline{7}$
B) $\dfrac{4}{9}$
C) $-5$
D) $\sqrt{5}$

✅ Answer: D) $\sqrt{5}$

Worksheet – Rational Numbers

Concept Recap

• Any number of the form $\dfrac{p}{q}$, where $p, q \in \mathbb{Z}$ and $q \neq 0$, is a rational number.
• Decimal expansions of rational numbers are either terminating or non-terminating recurring.
• Between any two rational numbers, infinitely many rational numbers exist.

Solved Example

Q1. Express $0.6\overline{3}$ as a rational number.

Let $x = 0.6\overline{3}$
Then $10x = 6.\overline{3}$
Subtracting, $10x – x = 6.\overline{3} – 0.6\overline{3}$
$\;\;\; 9x = 5.7$
$\;\;\; x = \dfrac{57}{90} = \dfrac{19}{30}$

✅ So, $0.6\overline{3} = \dfrac{19}{30}$

Practice Problems

Part A – Very Short Answer

1. Write the decimal form of $\dfrac{7}{8}$.
2. Is $0.125$ a rational number? Justify.
3. Find two rational numbers between $\dfrac{1}{2}$ and $\dfrac{2}{3}$.

Part B – Short Answer

4. Express $0.\overline{27}$ as a rational number.
5. Write the decimal expansion of $\dfrac{41}{2^3 \cdot 5^2}$. State whether it is terminating or non-terminating.

Part C – Long Answer

6. Show that between any two rational numbers $\dfrac{p}{q}$ and $\dfrac{r}{s}$, there exists another rational number.
7. Express $1.272727…$ as a rational number.

MCQs

1. Which of the following has a terminating decimal expansion?
   A) $\dfrac{7}{12}$
   B) $\dfrac{5}{16}$
   C) $\dfrac{3}{7}$
   D) $\dfrac{2}{11}$ ✅ Answer: B) $\dfrac{5}{16}$
2. The multiplicative inverse of $-\dfrac{9}{4}$ is:
   A) $-\dfrac{4}{9}$
   B) $\dfrac{4}{9}$
   C) $-\dfrac{9}{4}$
   D) $\dfrac{9}{4}$ ✅ Answer: A) $-\dfrac{4}{9}$

Answer Key (Practice Problems)

1. $0.875$
2. Yes, because $0.125 = \dfrac{125}{1000} = \dfrac{1}{8}$
3. Example: $\dfrac{9}{16}, \dfrac{5}{8}$
4. $\dfrac{27}{99} = \dfrac{3}{11}$
5. $0.41$ (terminating)
6. By using $\dfrac{p}{q}$ and $\dfrac{r}{s}$ → $\dfrac{ps+rq}{2qs}$ lies between them.
7. $1.272727… = \dfrac{127}{99}$

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