Chapter 9, Rational Numbers (परिमेय संख्याएँ), extends the number system to include numbers that can be expressed as a ratio of two integers. The rationalized syllabus focuses on the definition, standard form, and basic operations.

Exercise 9.1: Introduction and Standard Form

This exercise deals with identifying rational numbers and reducing them to their standard form.

Q. No.	Type	Example (Step-by-Step Solution)	Key Concept
1	Identification	Is every integer a rational number? Yes. Any integer $z$ can be written as $\frac{z}{1}$.	Rational Number: A number that can be written in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.
2	Standard Form	Reduce $\frac{-45}{30}$ to the standard form. 1. Find HCF of 45 and 30 (which is 15). 2. Divide numerator and denominator by 15: $\frac{-45 \div 15}{30 \div 15} = \frac{-3}{2}$. Standard Form: $\mathbf{\frac{-3}{2}}$	Standard Form: The denominator is positive, and the numerator and denominator have no common factor other than 1.
3	Equivalent R.N.	Write four rational numbers equivalent to $\frac{2}{7}$. Multiply numerator and denominator by 2, 3, 4, etc.: $\mathbf{\frac{4}{14}, \frac{6}{21}, \frac{8}{28}, \frac{10}{35}}$	Equivalent Rational Numbers: Obtained by multiplying/dividing the numerator and denominator by the same non-zero integer.
4	Number Line	Locate $\frac{-3}{4}$ on the number line. It lies between $-1$ and $0$. Divide the segment into 4 equal parts and take the 3rd mark to the left of 0.

| Rational numbers can be plotted on a number line. |

Exercise 9.2: Operations on Rational Numbers

This exercise applies the four basic operations (addition, subtraction, multiplication, and division) to rational numbers.

Q. No.	Type	Example (Step-by-Step Solution)	Key Concept
1	Addition	Find the sum: $\frac{5}{4} + (\frac{-11}{4})$ Since the denominator is the same: $\frac{5 + (-11)}{4} = \frac{-6}{4}$. Simplify: $\mathbf{\frac{-3}{2}}$	Addition with Same Denominator: Add the numerators.
2	Addition (LCM)	Find the sum: $\frac{3}{5} + \frac{2}{3}$ LCM of 5 and 3 is 15. $\frac{3 \times 3}{15} + \frac{2 \times 5}{15} = \frac{9}{15} + \frac{10}{15} = \mathbf{\frac{19}{15}}$	Addition with Different Denominators: Use the LCM to make the denominators equal.
3	Subtraction	Find the difference: $\frac{5}{6} – \frac{1}{3}$ LCM of 6 and 3 is 6. $\frac{5}{6} – \frac{1 \times 2}{6} = \frac{5 – 2}{6} = \frac{3}{6}$. Simplify: $\mathbf{\frac{1}{2}}$	Subtraction follows the same LCM rule as addition.
4	Multiplication	Find the product: $\frac{9}{2} \times (\frac{-7}{4})$ $\frac{9 \times (-7)}{2 \times 4} = \mathbf{\frac{-63}{8}}$	Multiplication: Product of numerators divided by product of denominators.
5	Division	Find the value: $\frac{-4}{5} \div (-2)$ Division is multiplication by the reciprocal. $\frac{-4}{5} \times \frac{1}{-2} = \frac{-4}{-10} = \frac{4}{10}$. Simplify: $\mathbf{\frac{2}{5}}$	Division: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$.