NCERT Solutions for Class 7 Maths Chapter 8: Rational Numbers — Free PDF

Chapter Overview: Rational Numbers

Rational numbers represent one of the most important expansions of the number system that students encounter in middle school. Until now, you have worked with natural numbers, whole numbers, integers, and fractions. Chapter 8 brings all of these together under one umbrella: rational numbers.

A rational number is any number that can be written as $\dfrac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$. This single definition encompasses positive fractions, negative fractions, all integers, zero, and much more. The word "rational" comes from "ratio," reflecting the fact that these numbers represent ratios of integers.

The chapter has 2 exercises covering understanding rational numbers (equivalence, comparison, number line) and operations (addition, subtraction, multiplication, division). This chapter bridges the gap between fractions and the broader number system, and is essential preparation for Class 8 where properties like closure, commutativity, associativity, and distributivity are studied in depth.

Key Concepts and Definitions

Let us establish the essential vocabulary and rules before solving problems.

What Is a Rational Number?

Rational Number: Any number that can be expressed in the form $\dfrac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.

Examples: $\dfrac{3}{5}$, $\dfrac{-7}{2}$, $0 = \dfrac{0}{1}$, $5 = \dfrac{5}{1}$, $-3 = \dfrac{-3}{1}$.

Positive rational numbers: Both numerator and denominator have the same sign. Example: $\dfrac{3}{5}$, $\dfrac{-4}{-7} = \dfrac{4}{7}$.

Negative rational numbers: Numerator and denominator have opposite signs. Example: $\dfrac{-3}{5}$, $\dfrac{4}{-7}$.

Zero is neither positive nor negative. $\dfrac{0}{q} = 0$ for any $q \neq 0$.

Equivalence and Standard Form

Equivalent rational numbers: $\dfrac{p}{q} = \dfrac{p \times k}{q \times k}$ for any non-zero integer $k$. Multiplying (or dividing) both numerator and denominator by the same non-zero number gives an equivalent rational number.

Standard form: A rational number $\dfrac{p}{q}$ is in standard form if:
1. The denominator $q$ is positive.
2. The HCF of $|p|$ and $q$ is $1$ (numerator and denominator are coprime).

Example: $\dfrac{-2}{3}$ is in standard form, but $\dfrac{4}{-6}$ is not (should be $\dfrac{-2}{3}$).

Comparison and Number Line

Comparison: To compare two rational numbers, convert them to equivalent fractions with the same positive denominator (use LCM), then compare numerators. The one with the larger numerator is greater.

Number line: Every rational number has a unique position on the number line. Positive rational numbers lie to the right of $0$, negative ones to the left. To plot $\dfrac{-3}{4}$, divide the segment from $-1$ to $0$ into $4$ equal parts and mark the third division from $0$.

Between any two rational numbers, there are infinitely many rational numbers. One method to find a rational number between two given ones is to take their average: the number between $\dfrac{a}{b}$ and $\dfrac{c}{d}$ is $\dfrac{1}{2}\left(\dfrac{a}{b} + \dfrac{c}{d}\right)$.

Operations on Rational Numbers

Addition/Subtraction: Find LCM of denominators, convert to common denominator, then add/subtract numerators.

Multiplication: Multiply numerators and multiply denominators.

Division: Multiply by the reciprocal of the divisor.

Reciprocal: The reciprocal of $\dfrac{p}{q}$ is $\dfrac{q}{p}$ (provided $p \neq 0$). Zero has no reciprocal.

Exercise 8.1 — Understanding Rational Numbers (Solved)

Exercise 8.1 covers equivalence, comparison, standard form, and number line representation.

Q1: Rational Numbers Between -1 and 0

Problem: List five rational numbers between $-1$ and $0$.

Solution:

These all satisfy $-1 < \dfrac{p}{q} < 0$. There are infinitely many such numbers — you can always find more by choosing different denominators or by taking averages.

Q2: Standard Form

Problem: Write $\dfrac{-4}{6}$ in standard form.

Solution:
Divide numerator and denominator by their HCF ($2$):

A rational number is in standard form when the denominator is positive and the HCF of $|\text{numerator}|$ and denominator is $1$.

Q3: Comparing Rational Numbers

Problem: Which is greater: $\dfrac{-3}{5}$ or $\dfrac{-2}{3}$?

Solution:
Find common denominator (LCM of $5$ and $3$ is $15$):

Since $-9 > -10$, we have $\dfrac{-3}{5} > \dfrac{-2}{3}$.

Remember: on the number line, $\dfrac{-3}{5}$ is to the right of $\dfrac{-2}{3}$, so it is greater.

Q4: Number Line Representation

Problem: Represent $\dfrac{-3}{4}$ on the number line.

Solution:
Divide the segment from $-1$ to $0$ into $4$ equal parts. The third mark from $0$ (towards $-1$) represents $\dfrac{-3}{4}$.

Alternatively, the first mark from $-1$ (towards $0$) represents $\dfrac{-3}{4}$ since $-1 + \dfrac{1}{4} = \dfrac{-3}{4}$.

Q5: Equivalent Rational Numbers

Problem: Find two equivalent rational numbers for $\dfrac{3}{7}$.

Solution:

Multiply both numerator and denominator by the same non-zero integer ($2$ and $3$ respectively).

Q6: Arranging in Ascending Order

Problem: Arrange in ascending order: $\dfrac{-3}{7}, \dfrac{-3}{2}, \dfrac{-3}{4}$.

Solution:
LCM of $7, 2, 4$ is $28$.

Since $-42 < -21 < -12$:

Exercise 8.2 — Operations on Rational Numbers (Solved)

Exercise 8.2 covers addition, subtraction, multiplication, and division of rational numbers.

Q1: Addition

Problem: Add $\dfrac{-3}{5}$ and $\dfrac{2}{3}$.

Solution:
LCM of $5$ and $3$ is $15$.

Q2: Subtraction

Problem: Subtract $\dfrac{5}{6}$ from $\dfrac{1}{3}$.

Solution:

Always reduce the answer to standard form.

Q3: Multiplication

Problem: Multiply $\dfrac{-3}{7}$ and $\dfrac{5}{4}$.

Solution:

For multiplication: multiply numerators together and denominators together.

Q4: Division

Problem: Divide $\dfrac{-2}{3}$ by $\dfrac{4}{5}$.

Solution:

For division: multiply by the reciprocal of the divisor.

Q5: Multi-Step Computation

Problem: Simplify: $\dfrac{-3}{4} + \dfrac{5}{6} - \dfrac{1}{2}$.

Solution:
LCM of $4, 6, 2$ is $12$.

Q6: Product with Negative Signs

Problem: Find the product: $\dfrac{-4}{9} \times \dfrac{3}{-16}$.

Solution:

Note: negative divided by negative gives positive.

Q7: Division of Two Negative Rationals

Problem: Divide $\dfrac{-7}{8}$ by $\dfrac{-3}{4}$.

Solution:

Worked Examples — Additional Practice

Here are additional examples covering common exam patterns.

Example 1: Finding Rational Numbers Between Two Values

Problem: Find five rational numbers between $\dfrac{1}{4}$ and $\dfrac{1}{2}$.

Solution:
Convert to equivalent fractions with a larger denominator.
$\dfrac{1}{4} = \dfrac{6}{24}$ and $\dfrac{1}{2} = \dfrac{12}{24}$.

Five rational numbers between them: $\dfrac{7}{24}, \dfrac{8}{24}, \dfrac{9}{24}, \dfrac{10}{24}, \dfrac{11}{24}$

Simplified: $\dfrac{7}{24}, \dfrac{1}{3}, \dfrac{3}{8}, \dfrac{5}{12}, \dfrac{11}{24}$.

Example 2: Finding a Missing Number

Problem: The sum of two rational numbers is $\dfrac{-1}{3}$. If one of them is $\dfrac{5}{6}$, find the other.

Solution:
Let the other number be $x$.

Verification: $\dfrac{-7}{6} + \dfrac{5}{6} = \dfrac{-2}{6} = \dfrac{-1}{3}$ ✓

Example 3: Distributive Property Verification

Problem: Verify that $\dfrac{-3}{5} \times \left(\dfrac{2}{7} + \dfrac{4}{9}\right) = \dfrac{-3}{5} \times \dfrac{2}{7} + \dfrac{-3}{5} \times \dfrac{4}{9}$.

Solution:
LHS: $\dfrac{2}{7} + \dfrac{4}{9} = \dfrac{18}{63} + \dfrac{28}{63} = \dfrac{46}{63}$
$\dfrac{-3}{5} \times \dfrac{46}{63} = \dfrac{-138}{315} = \dfrac{-46}{105}$

RHS: $\dfrac{-3}{5} \times \dfrac{2}{7} = \dfrac{-6}{35}$ and $\dfrac{-3}{5} \times \dfrac{4}{9} = \dfrac{-12}{45} = \dfrac{-4}{15}$
$\dfrac{-6}{35} + \dfrac{-4}{15} = \dfrac{-18}{105} + \dfrac{-28}{105} = \dfrac{-46}{105}$

LHS $=$ RHS ✓. The distributive property holds.

Example 4: BODMAS with Rational Numbers

Problem: Simplify $\dfrac{-5}{8} + \dfrac{3}{4} \times \dfrac{2}{3} - \dfrac{1}{2}$.

Solution:
Follow BODMAS — multiplication first:
$\dfrac{3}{4} \times \dfrac{2}{3} = \dfrac{6}{12} = \dfrac{1}{2}$

Now: $\dfrac{-5}{8} + \dfrac{1}{2} - \dfrac{1}{2} = \dfrac{-5}{8} + 0 = \dfrac{-5}{8}$

Common Mistakes to Avoid

Mistake 1: Adding numerators and denominators separately.
Wrong: $\dfrac{1}{3} + \dfrac{2}{5} = \dfrac{3}{8}$. Correct: Find LCM of denominators first, then add numerators. $\dfrac{1}{3} + \dfrac{2}{5} = \dfrac{5}{15} + \dfrac{6}{15} = \dfrac{11}{15}$.

Mistake 2: Forgetting to reduce to standard form.
After computation, always simplify. $\dfrac{-3}{6}$ should be written as $\dfrac{-1}{2}$.

Mistake 3: Sign errors with negative rational numbers.
$\dfrac{-3}{-5} = \dfrac{3}{5}$ (positive, not negative). $\dfrac{3}{-5} = \dfrac{-3}{5}$ (negative). Always move the negative sign to the numerator in standard form.

Mistake 4: Comparing without a common denominator.
You cannot compare $\dfrac{-3}{5}$ and $\dfrac{-2}{3}$ by just looking at the numerators. Convert to the same denominator first.

Mistake 5: Wrong reciprocal during division.
The reciprocal of $\dfrac{-3}{4}$ is $\dfrac{4}{-3} = \dfrac{-4}{3}$. Students sometimes flip the wrong fraction or forget to flip at all.

Practice Questions with Answers

Test yourself with these problems.

Q1: Standard Form

Question: Write $\dfrac{18}{-42}$ in standard form.

Answer: HCF of $18$ and $42$ is $6$. $\dfrac{18}{-42} = \dfrac{-18}{42} = \dfrac{-3}{7}$.

Q2: Addition and Subtraction

Question: Find $\dfrac{-2}{3} + \dfrac{5}{4} - \dfrac{1}{6}$.

Answer: LCM of $3, 4, 6$ is $12$.
$\dfrac{-8}{12} + \dfrac{15}{12} - \dfrac{2}{12} = \dfrac{-8 + 15 - 2}{12} = \dfrac{5}{12}$.

Q3: Product

Question: Find $\dfrac{-5}{9} \times \dfrac{-3}{10}$.

Answer: $\dfrac{(-5)(-3)}{(9)(10)} = \dfrac{15}{90} = \dfrac{1}{6}$.

Q4: Word Problem

Question: A rope of length $\dfrac{7}{2}$ m is cut into pieces of length $\dfrac{1}{4}$ m each. How many pieces are there?

Answer: Number of pieces $= \dfrac{7}{2} \div \dfrac{1}{4} = \dfrac{7}{2} \times \dfrac{4}{1} = \dfrac{28}{2} = 14$ pieces.

Exam Tips for Rational Numbers

Key Takeaways

- A rational number is $\dfrac{p}{q}$ where $q \neq 0$. Every integer, fraction, and terminating/repeating decimal is rational.
- Standard form: Denominator is positive, and HCF of numerator and denominator is $1$.
- To compare rational numbers: Convert to the same denominator, then compare numerators.
- Addition/Subtraction: Find LCM of denominators, convert, then add/subtract numerators.
- Multiplication: $\dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{ac}{bd}$.
- Division: $\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c}$.
- Between any two rational numbers, there are infinitely many rational numbers.
- The number $0$ is a rational number that is neither positive nor negative.
- Every natural number, whole number, and integer is a rational number (with denominator $1$).

Common Mistakes Students Make in Rational Numbers

Rational Numbers takes the fraction rules you learned in Chapter 2 and layers sign handling on top. That combination produces more student mistakes than almost any other chapter in Class 7. Here are the ten biggest traps with clear wrong vs correct examples.

Mistake 1: Leaving a negative denominator in the answer
❌ Wrong: Writing the final answer as $\tfrac{3}{-5}$.
✅ Correct: Standard form is $\tfrac{-3}{5}$.
The denominator of a rational number in standard form must be positive. Move the negative sign to the numerator.

Mistake 2: Forgetting to reduce to standard form
❌ Wrong: Writing the answer as $\tfrac{12}{18}$.
✅ Correct: $\tfrac{12}{18} = \tfrac{2}{3}$.
Always divide numerator and denominator by their HCF.

Mistake 3: Comparing rational numbers with different sign denominators
❌ Wrong: Saying $\tfrac{3}{-5} > \tfrac{-4}{7}$ because $3 > -4$.
✅ Correct: First put in standard form: $\tfrac{-3}{5}$ and $\tfrac{-4}{7}$. Get common denominators: $\tfrac{-21}{35}$ and $\tfrac{-20}{35}$. Since $-20 > -21$, $\tfrac{-4}{7} > \tfrac{-3}{5}$.

Mistake 4: Sign error when multiplying rational numbers
❌ Wrong: $\tfrac{-2}{3} \times \tfrac{-5}{7} = \tfrac{-10}{21}$.
✅ Correct: $\tfrac{-2}{3} \times \tfrac{-5}{7} = \tfrac{10}{21}$.
Negative times negative is positive.

Mistake 5: Flipping the wrong fraction when dividing
❌ Wrong: $\tfrac{-3}{4} \div \tfrac{2}{5} = \tfrac{-4}{3} \times \tfrac{2}{5}$.
✅ Correct: $\tfrac{-3}{4} \div \tfrac{2}{5} = \tfrac{-3}{4} \times \tfrac{5}{2} = \tfrac{-15}{8}$.
Always flip the second fraction, never the first.

Mistake 6: Thinking every fraction is a rational number and vice versa
❌ Wrong: All rational numbers are fractions.
✅ Correct: Every fraction is a rational number, but rational numbers include negative fractions and integers as well.

Mistake 7: Not using LCM when adding
❌ Wrong: $\tfrac{-1}{2} + \tfrac{1}{3} = \tfrac{0}{5}$.
✅ Correct: LCM of $2$ and $3$ is $6$. $\tfrac{-3}{6} + \tfrac{2}{6} = \tfrac{-1}{6}$.

Mistake 8: Cancelling across addition
❌ Wrong: $\tfrac{-3 + 5}{6} = \tfrac{-1}{2}$ (cancelling the $3$ with the $6$).
✅ Correct: $\tfrac{-3 + 5}{6} = \tfrac{2}{6} = \tfrac{1}{3}$.
Never cancel across a plus sign.

Mistake 9: Saying zero has a reciprocal
❌ Wrong: Reciprocal of $0$ is $\tfrac{1}{0}$.
✅ Correct: Zero has no reciprocal because $\tfrac{1}{0}$ is undefined.

Mistake 10: Forgetting that between any two rational numbers there are infinitely many more
❌ Wrong: There are only a few rational numbers between $\tfrac{1}{2}$ and $\tfrac{1}{3}$.
✅ Correct: There are infinitely many. This is an important CBSE textbook fact.

Previous Year Questions (PYQs) with Solutions

These are CBSE Class 7 style questions from the past few years on Rational Numbers.

Q1. Write $\tfrac{-28}{42}$ in standard form. `(CBSE 2021, 2 marks)`
Solution: HCF of $28$ and $42$ is $14$.
$\tfrac{-28}{42} = \tfrac{-2}{3}$.
Answer: $\tfrac{-2}{3}$

Q2. Find $\tfrac{-7}{8} + \tfrac{5}{6}$. `(CBSE 2022, 2 marks)`
Solution: LCM of $8$ and $6$ is $24$.
$\tfrac{-21}{24} + \tfrac{20}{24} = \tfrac{-1}{24}$.
Answer: $\tfrac{-1}{24}$

Q3. Simplify $\tfrac{-3}{5} \times \tfrac{-10}{9}$. `(CBSE 2023, 2 marks)`
Solution: $\tfrac{(-3)(-10)}{5 \times 9} = \tfrac{30}{45} = \tfrac{2}{3}$.
Answer: $\tfrac{2}{3}$

Q4. Divide $\tfrac{-4}{9}$ by $\tfrac{2}{3}$. `(CBSE 2022, 2 marks)`
Solution: $\tfrac{-4}{9} \div \tfrac{2}{3} = \tfrac{-4}{9} \times \tfrac{3}{2} = \tfrac{-12}{18} = \tfrac{-2}{3}$.
Answer: $\tfrac{-2}{3}$

Q5. Find five rational numbers between $\tfrac{2}{3}$ and $\tfrac{4}{5}$. `(CBSE 2023, 3 marks)`
Solution: LCM of $3$ and $5$ is $15$.
$\tfrac{2}{3} = \tfrac{10}{15}$ and $\tfrac{4}{5} = \tfrac{12}{15}$. Multiply numerator and denominator by $6$: $\tfrac{60}{90}$ and $\tfrac{72}{90}$.
Five rational numbers between them: $\tfrac{61}{90}, \tfrac{63}{90}, \tfrac{65}{90}, \tfrac{68}{90}, \tfrac{70}{90}$.

Q6. Which is greater, $\tfrac{-7}{8}$ or $\tfrac{5}{-6}$? `(CBSE 2021, 3 marks)`
Solution: Standard form: $\tfrac{-7}{8}$ and $\tfrac{-5}{6}$. LCM of $8$ and $6$ is $24$.
$\tfrac{-21}{24}$ and $\tfrac{-20}{24}$. Since $-20 > -21$, $\tfrac{-5}{6} > \tfrac{-7}{8}$.
Answer: $\tfrac{5}{-6}$ is greater.

Q7. A rope of length $\tfrac{15}{4}$ m is cut into pieces of length $\tfrac{3}{8}$ m each. How many pieces are obtained? `(CBSE 2024, 3 marks)`
Solution:
$\tfrac{15}{4} \div \tfrac{3}{8} = \tfrac{15}{4} \times \tfrac{8}{3} = \tfrac{120}{12} = 10$.
Answer: $10$ pieces.

Q8 (HOTS). The product of two rational numbers is $\tfrac{-8}{9}$. If one of them is $\tfrac{-4}{15}$, find the other. `(CBSE 2024, 4 marks)`
Solution: Other $= \tfrac{-8}{9} \div \tfrac{-4}{15} = \tfrac{-8}{9} \times \tfrac{15}{-4} = \tfrac{-120}{-36} = \tfrac{10}{3}$.
Answer: $\tfrac{10}{3}$

Practice This Chapter on SparkEd

Rational Numbers is the chapter where you finally combine everything you have learned about fractions and integers. It is also the chapter most students leave half mastered because the sign handling tires them out. The SparkEd Rational Numbers module fixes that by giving you targeted practice with immediate feedback.

The module has over 60 practice problems across three levels. Level 1 builds confidence with standard form conversions, simple comparisons, and basic operations with clean numbers. Level 2 moves into mixed operations with fractions that need LCMs and careful sign work. Level 3 is the CBSE HOTS zone where you tackle problems like finding multiple rational numbers between two given ones, and word problems involving rational quantities.

The instant AI feedback is particularly useful here because most errors in this chapter are small sign slips or forgotten standard form conversions. The AI shows you the exact step where things went wrong so you learn the habit, not just the answer. Gamification keeps you coming back, and the adaptive engine gives you more of the exact types of problems you struggle with.

Practice Rational Numbers on SparkEd →

Download Rational Numbers worksheet

Commit to twenty minutes a day for a week and you will be handling signed fraction arithmetic as fluently as ordinary multiplication.