📝 Worksheet: Rational Numbers – Class 9 CBSE

✳️ Section A: Definitions (1 mark each)

1. Define a rational number.

2. What is meant by co-prime numbers?

3. Why is the denominator of a rational number never zero?

4. What is the origin of the word “rational”?

5. Write the symbol used to represent the set of rational numbers.

✳️ Section B: Fill in the Blanks (1 mark each)

1. A rational number is a number of the form __________, where p and q are integers and q ≠ 0.

2. All whole numbers are __________ numbers.

3. The rational numbers $\frac{3}{5}$ and $\frac{6}{10}$ are __________ rational numbers.

4. The decimal expansion of a rational number is either __________ or __________.

5. The number 0 can be written as $\frac{0}{1}, \frac{0}{2}, \frac{0}{-5}$ where __________.

✳️ Section C: True or False (Write T/F and give reasons) (2 marks each)

1. Every whole number is a natural number.

2. Every rational number is an integer.

3. Zero is a rational number.

4. Every integer is a rational number.

5. Every natural number is a rational number.

✳️ Section D: Reasoning Questions (2–3 marks each)

1. Is 0 a rational number? Explain with three different ways of writing it as a rational number.

2. Is $\frac{3}{0}$ a rational number? Why or why not?

3. How are equivalent rational numbers formed? Give an example.

4. Is $\frac{-8}{12}$ in simplest form? If not, simplify it.

✳️ Section E: Find Rational Numbers Between (2–3 marks each)

1. Find five rational numbers between 1 and 2.

2. Find six rational numbers between 3 and 4.

3. Find five rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$.

4. Find three rational numbers between $\frac{-1}{2}$ and $\frac{1}{2}$.

5. Find two rational numbers between -1 and 0.

✳️ Section F: Challenge Question (4 marks)

Q: If a rational number lies between $\frac{2}{7}$ and $\frac{5}{7}$, how many such numbers can be found? Show at least six examples and explain your method.

✅ Answer Key: Rational Numbers – Class 9 CBSE

✳️ Section A: Definitions

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

2. Co-prime numbers are two numbers that have no common factor other than 1.

3. The denominator of a rational number is never zero because division by zero is undefined.

4. The word “rational” comes from "ratio", because rational numbers are expressed as a ratio of two integers.

5. The set of rational numbers is denoted by Q.

✳️ Section B: Fill in the Blanks

1. $\frac{p}{q}$

2. Rational

3. Equivalent

4. Terminating or Non-terminating Repeating

5. $q \ne 0$
   

✳️ Section C: True or False with Reasons

1. False – Every whole number is not a natural number. For example, 0 is a whole number but not a natural number.

2. False – Not every rational number is an integer. For example, $\frac{1}{2}$ is a rational number but not an integer.

3. True – 0 is a rational number because it can be written as $\frac{0}{1}, \frac{0}{2}, \frac{0}{-5}$, etc., where the denominator is not zero.

4. True – Every integer is a rational number because any integer $a$ can be written as $\frac{a}{1}$.

5. True – Every natural number can be written in the form $\frac{p}{q}$, like $\frac{3}{1}$, so it is a rational number.

✳️ Section D: Reasoning Questions

1. Yes, 0 is a rational number because:

   $$0 = \frac{0}{1} = \frac{0}{2} = \frac{0}{-5} \quad (q \ne 0)$$
   

2. No, $\frac{3}{0}$ is not a rational number because division by zero is undefined.

3. Equivalent rational numbers are formed by multiplying or dividing both numerator and denominator by the same non-zero number.
   Example: $\frac{2}{3} = \frac{4}{6} = \frac{6}{9}$

4. $\frac{-8}{12}$ is not in simplest form.
   Divide numerator and denominator by 4: $\frac{-2}{3}$ is the simplest form.

✳️ Section E: Find Rational Numbers Between

1. Between 1 and 2:

Multiply both by 10 to get like denominators:

$$\frac{10}{10}, \frac{11}{10}, \frac{12}{10}, \frac{13}{10}, \frac{14}{10}, \frac{15}{10}, \frac{16}{10}, \frac{17}{10}, \frac{18}{10}, \frac{19}{10}, \frac{20}{10}$$

So, five rational numbers are:

$$\frac{11}{10}, \frac{12}{10}, \frac{13}{10}, \frac{14}{10}, \frac{15}{10}$$

2. Between 3 and 4:

$$\frac{31}{10}, \frac{32}{10}, \frac{33}{10}, \frac{34}{10}, \frac{35}{10}, \frac{36}{10}$$

(Equivalent to 3.1 to 3.6)

3. Between $\frac{3}{5}$ and $\frac{4}{5}$:

Convert to denominator 25:

$$\frac{15}{25}, \frac{16}{25}, \frac{17}{25}, \frac{18}{25}, \frac{19}{25}, \frac{20}{25}$$

So, five numbers are:

$$\frac{16}{25}, \frac{17}{25}, \frac{18}{25}, \frac{19}{25}, \frac{20}{25}$$

4. Between $\frac{-1}{2}$ and $\frac{1}{2}$:

$$\frac{-2}{5}, \frac{-1}{5}, 0, \frac{1}{5}$$

(Any three from these: $\frac{-1}{5}, 0, \frac{1}{5}$)

5. Between -1 and 0:

$$\frac{-9}{10}, \frac{-4}{5}$$

✳️ Section F: Challenge Question

Between $\frac{2}{7}$ and $\frac{5}{7}$:

Choose common denominator (e.g., 70):

$$\frac{20}{70}, \frac{21}{70}, \frac{22}{70}, \frac{23}{70}, \frac{24}{70}, \frac{25}{70}, \ldots, \frac{49}{70}, \frac{50}{70}$$

So we can find many rational numbers, such as:

$$\frac{21}{70}, \frac{22}{70}, \frac{23}{70}, \frac{24}{70}, \frac{25}{70}, \frac{26}{70}$$

(At least 6 shown above.)