Rational Numbers — Questions and Answers
Welcome to HSLC Guru. In this lesson we present the complete summary, worked examples, and fully solved Exercise 1.1 and Exercise 1.2 of Chapter 1 Rational Numbers from ASSEB (Assam State School Education Board) Class 8 General Mathematics, along with extra questions for exam practice.
Summary
The numbers that can be written in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \ne 0$, are called rational numbers. The denominator $q$ can never be zero because division by zero is undefined. Every integer can be written as $\frac{p}{q}$, so all integers are rational numbers, and terminating and recurring decimals are rational too.
If $p$ and $q$ are both positive or both negative, then $\frac{p}{q}$ is a positive rational number; if one is positive and the other negative, it is a negative rational number. By convention the denominator of a rational number is written as positive, e.g. $\frac{8}{-9} = \frac{-8}{9}$.
Rational numbers are closed under addition, subtraction and multiplication, but not under division (they are closed under division only when zero is excluded). Addition and multiplication are commutative and associative, but subtraction and division are neither. Multiplication is distributive over addition and subtraction. Zero (0) is the additive identity and one (1) is the multiplicative identity. If the sum of two numbers is zero, each is the additive inverse of the other; if the product is one, each is the multiplicative inverse (reciprocal) of the other.
Rational numbers can be represented as points on the number line. Between any two rational numbers there are infinitely many rational numbers, which can be found by equalising denominators or by the mean (average) method.
Summary: This ASSEB Class 8 General Mathematics Chapter 1 Rational Numbers guide explains numbers of the form p/q (q ≠ 0), positive and negative rational numbers, and their closure, commutative, associative and distributive properties, plus the additive identity 0, multiplicative identity 1, additive and multiplicative inverses (reciprocals), representation on the number line, and finding infinitely many rational numbers between two rationals. It gives fully worked solutions to Exercise 1.1 and Exercise 1.2 with extra MCQs, fill-in-the-blanks, true/false and short-answer practice.
Textbook Questions and Answers
Solved Examples (Properties of Rational Numbers)
Example 1. Using properties of numbers, find the value of $\frac{2}{5}\times\frac{-2}{7}-\frac{1}{12}-\frac{3}{7}\times\frac{2}{5}$.
Answer: Using the commutative and distributive properties, take out the common factor $\frac{2}{5}$.
$$\frac{2}{5}\times\frac{-2}{7}-\frac{3}{7}\times\frac{2}{5}-\frac{1}{12}=\frac{2}{5}\left(\frac{-2}{7}-\frac{3}{7}\right)-\frac{1}{12}=\frac{2}{5}\times\frac{-5}{7}-\frac{1}{12}=\frac{-2}{7}-\frac{1}{12}=\frac{-24-7}{84}=\frac{-31}{84}$$
[LCM of 7 and 12 $= 84$]
Example 2. Write the additive inverse of $-2\times\frac{5}{7}$.
Answer: $-2\times\frac{5}{7}=\frac{-10}{7}$; its additive inverse $=\frac{10}{7}$.
Example 3. Write the multiplicative inverse of $2\times\frac{-6}{7}$.
Answer: $2\times\frac{-6}{7}=\frac{-12}{7}$; its multiplicative inverse $=\frac{-7}{12}$.
Solved Examples (Representation on the Number Line)
Example 1. Represent $\frac{7}{9}$ on the number line.
Answer: In $\frac{7}{9}$ the denominator is $9$, so the segment between 0 and 1 is divided into 9 equal parts. Counting 7 parts to the right of 0 gives the point $\frac{7}{9}$.
Example 2. Represent $\frac{-5}{4}$ on the number line.
Answer: $\frac{-5}{4}=-1\frac{1}{4}$, so it lies between $-1$ and $-2$. Divide the segment between $-1$ and $-2$ into 4 equal parts; the first part to the left of $-1$ is the point $\frac{-5}{4}$.
Solved Examples (Rational Numbers Between Two Rationals)
Example 1. Find some rational numbers between $\frac{-3}{5}$ and $\frac{2}{7}$.
Answer: Convert both to a common denominator [LCM of 5 and 7 $= 35$].
$$\frac{-3}{5}=\frac{-21}{35},\qquad \frac{2}{7}=\frac{10}{35}$$
So some rational numbers between $\frac{-3}{5}$ and $\frac{2}{7}$ are $\frac{-20}{35},\ \frac{-19}{35},\ \ldots,\ \frac{9}{35}$.
Example 2. Find rational numbers between $-3$ and $-1$.
Answer: $-3=\frac{-30}{10}$ and $-1=\frac{-10}{10}$. So $\frac{-29}{10},\ \frac{-28}{10},\ \ldots,\ \frac{-11}{10}$ lie between $-3$ and $-1$. In this way we can find infinitely many rational numbers.
Example 3. Using the mean (average) method, find two rational numbers between $\frac{1}{3}$ and $\frac{1}{5}$.
Answer: Mean of $\frac{1}{3}$ and $\frac{1}{5}$ $=\left(\frac{1}{3}+\frac{1}{5}\right)\div 2=\frac{8}{15}\div 2=\frac{4}{15}$; so $\frac{4}{15}$ lies between them. Again, mean of $\frac{4}{15}$ and $\frac{1}{3}$ $=\left(\frac{4}{15}+\frac{5}{15}\right)\div 2=\frac{9}{15}\div 2=\frac{3}{10}$. Hence $\frac{4}{15}$ and $\frac{3}{10}$ are two rational numbers between $\frac{1}{5}$ and $\frac{1}{3}$.
Exercise 1.1
1. State whether the following are true or false:
(i) Rational numbers are commutative under addition and multiplication.
Answer: True. For addition and multiplication, $a+b=b+a$ and $a\times b=b\times a$.
(ii) Rational numbers are associative under subtraction and division.
Answer: False. Subtraction and division are not associative, i.e. $(a-b)-c \ne a-(b-c)$ and $(a\div b)\div c \ne a\div(b\div c)$.
(iii) Rational numbers are not closed under subtraction.
Answer: False. The difference of two rational numbers is again a rational number, so rational numbers are closed under subtraction.
(iv) Zero does not have any reciprocal.
Answer: True. Since division by zero is undefined, zero has no reciprocal (multiplicative inverse).
(v) $0$ is a rational number.
Answer: True. $0=\frac{0}{1}$ can be written in the form $\frac{p}{q}$, so it is a rational number.
(vi) $\frac{3}{2}\times\frac{5}{6}\times\frac{4}{7}=\frac{4}{7}\times\frac{5}{6}\times\frac{3}{2}$
Answer: True. By the commutative property of multiplication both sides equal $\frac{5}{7}$.
(vii) $\frac{5}{9}\times\left(\frac{1}{3}+\frac{7}{3}\right)=\frac{5}{9}\times\frac{1}{3}+\frac{7}{3}$
Answer: False. LHS $=\frac{5}{9}\times\frac{8}{3}=\frac{40}{27}$, but RHS $=\frac{5}{27}+\frac{7}{3}=\frac{68}{27}$. The correct distributive form is $\frac{5}{9}\times\frac{1}{3}+\frac{5}{9}\times\frac{7}{3}$.
(viii) $\frac{7}{12}-\frac{3}{7}+\frac{11}{12}=\frac{7}{12}-\left(\frac{3}{7}+\frac{11}{12}\right)$
Answer: False. LHS $=\frac{15}{14}$ but RHS $=\frac{-16}{21}$; subtraction is not associative, so the two sides are unequal.
(ix) $\frac{-3}{11}\times\frac{2}{9}=\frac{-3}{11}$
Answer: False. $\frac{-3}{11}\times\frac{2}{9}=\frac{-6}{99}=\frac{-2}{33} \ne \frac{-3}{11}$.
(x) $\frac{4}{9}\times\left(\frac{11}{12}+\frac{-5}{6}\right)=\frac{4}{9}\times\frac{11}{12}+\frac{4}{9}\times\frac{-5}{6}$
Answer: True. This is the distributive property of multiplication over addition; both sides equal $\frac{1}{27}$.
2. Write the additive inverse of the following:
Answer: The additive inverse of a number $a$ is $-a$ (the number that adds to give 0).
- (i) Additive inverse of $\frac{-7}{9}$ $=\frac{7}{9}$
- (ii) Additive inverse of $3$ $=-3$
- (iii) Additive inverse of $\frac{-2}{8}$ $=\frac{2}{8}=\frac{1}{4}$
- (iv) $\frac{19}{-7}=\frac{-19}{7}$; its additive inverse $=\frac{19}{7}$
- (v) Additive inverse of $\frac{-a}{c}$ $=\frac{a}{c}$
3. Write the additive inverse of $\frac{-20}{11}$ and $\frac{5}{6}$.
Answer: Additive inverse of $\frac{-20}{11}$ $=\frac{20}{11}$ and additive inverse of $\frac{5}{6}$ $=\frac{-5}{6}$.
4. Write the multiplicative inverse of the following:
Answer: The multiplicative inverse of a non-zero number $a$ is $\frac{1}{a}$ (the number that multiplies to give 1).
- (i) Multiplicative inverse of $-13$ $=\frac{-1}{13}$
- (ii) $\frac{-4}{9}\times\frac{-2}{7}=\frac{8}{63}$; its multiplicative inverse $=\frac{63}{8}$
- (iii) $-2\times\frac{2}{5}=\frac{-4}{5}$; its multiplicative inverse $=\frac{-5}{4}$
- (iv) Multiplicative inverse of $-1$ $=-1$ (since $-1\times-1=1$)
- (v) Multiplicative inverse of $\frac{2n}{5}$ $=\frac{5}{2n}$
5. Is $\frac{5}{3}$ a multiplicative inverse of $-1\frac{2}{3}$? Give reason.
Answer: No. $-1\frac{2}{3}=\frac{-5}{3}$. Its multiplicative inverse must give a product of $1$, but $\frac{-5}{3}\times\frac{5}{3}=\frac{-25}{9}\ne 1$. In fact $\frac{-5}{3}\times\frac{-3}{5}=1$, so the multiplicative inverse of $\frac{-5}{3}$ is $\frac{-3}{5}$, not $\frac{5}{3}$.
6. What are the multiplicative and additive inverses of 1?
Answer: Multiplicative inverse of $1$ $=1$ (since $1\times1=1$) and additive inverse of $1$ $=-1$ (since $1+(-1)=0$).
7. What are the multiplicative inverses of $\frac{-4}{9}$ and $\frac{11}{16}$?
Answer: Multiplicative inverse of $\frac{-4}{9}$ $=\frac{-9}{4}$ and of $\frac{11}{16}$ $=\frac{16}{11}$.
8. Write the reciprocals of the following: (i) $\frac{2}{3}$ (ii) $\frac{-5}{12}$
Answer: (i) Reciprocal of $\frac{2}{3}$ $=\frac{3}{2}$. (ii) Reciprocal of $\frac{-5}{12}$ $=\frac{-12}{5}$.
9. Write the rational numbers whose reciprocals are the numbers themselves.
Answer: $1$ and $-1$, because reciprocal of $1$ is $\frac{1}{1}=1$ and reciprocal of $-1$ is $\frac{1}{-1}=-1$.
10. Multiply $\frac{-25}{26}$ with the reciprocal of $\frac{5}{13}$.
Answer: Reciprocal of $\frac{5}{13}$ $=\frac{13}{5}$.
$$\frac{-25}{26}\times\frac{13}{5}=\frac{-25\times13}{26\times5}=\frac{-5}{2}$$
11. Write the properties used in the following:
- (i) $\frac{-3}{5}\times\frac{-2}{7}=\frac{-2}{7}\times\frac{-3}{5}$ — Commutative property of multiplication.
- (ii) $\frac{-4}{5}+0=\frac{-4}{5}=0+\frac{-4}{5}$ — Additive identity (0 is the additive identity).
- (iii) $\frac{2}{9}\times1=1\times\frac{2}{9}=\frac{2}{9}$ — Multiplicative identity (1 is the multiplicative identity).
12. Find the value of the following using relevant properties:
(i) $\frac{6}{7}+\frac{2}{5}+\frac{2}{7}+\frac{1}{5}$
Answer: $$\frac{6}{7}+\frac{2}{5}+\frac{2}{7}+\frac{1}{5}=\left(\frac{6}{7}+\frac{2}{7}\right)+\left(\frac{2}{5}+\frac{1}{5}\right)=\frac{8}{7}+\frac{3}{5}=\frac{40+21}{35}=\frac{61}{35}$$
(ii) $670\times\frac{7}{11}+670\times\frac{1}{3}$
Answer: $$670\times\frac{7}{11}+670\times\frac{1}{3}=670\left(\frac{7}{11}+\frac{1}{3}\right)=670\times\frac{21+11}{33}=670\times\frac{32}{33}=\frac{21440}{33}$$
(iii) $\frac{7}{5}\times\frac{-3}{8}+\frac{3}{4}\times\frac{7}{5}$
Answer: $$\frac{7}{5}\times\frac{-3}{8}+\frac{7}{5}\times\frac{3}{4}=\frac{7}{5}\left(\frac{-3}{8}+\frac{3}{4}\right)=\frac{7}{5}\times\frac{-3+6}{8}=\frac{7}{5}\times\frac{3}{8}=\frac{21}{40}$$
(iv) $\frac{-5}{9}\left(\frac{-27}{15}+\frac{36}{25}\right)$
Answer: $$\frac{-5}{9}\left(\frac{-27}{15}+\frac{36}{25}\right)=\frac{-5}{9}\times\frac{-135+108}{75}=\frac{-5}{9}\times\frac{-27}{75}=\frac{135}{675}=\frac{1}{5}$$
(v) $\frac{2}{3}+\frac{3}{3}-\frac{1}{4}\times\frac{2}{3}+\frac{2}{3}\times\frac{1}{2}$
Answer: $$\frac{2}{3}+1+\frac{2}{3}\left(\frac{-1}{4}+\frac{1}{2}\right)=\frac{2}{3}+1+\frac{2}{3}\times\frac{1}{4}=\frac{2}{3}+1+\frac{1}{6}=\frac{4+6+1}{6}=\frac{11}{6}$$
(vi) $\frac{4}{7}\times\frac{-5}{9}+\frac{4}{7}\times\frac{1}{5}$
Answer: $$\frac{4}{7}\left(\frac{-5}{9}+\frac{1}{5}\right)=\frac{4}{7}\times\frac{-25+9}{45}=\frac{4}{7}\times\frac{-16}{45}=\frac{-64}{315}$$
(vii) $\frac{2}{3}\times\frac{-5}{7}-\frac{1}{6}\times\frac{3}{7}+\frac{1}{14}\times\frac{2}{3}$
Answer: $$\frac{2}{3}\times\frac{-5}{7}+\frac{2}{3}\times\frac{1}{14}-\frac{1}{6}\times\frac{3}{7}=\frac{2}{3}\left(\frac{-5}{7}+\frac{1}{14}\right)-\frac{1}{14}=\frac{2}{3}\times\frac{-9}{14}-\frac{1}{14}=\frac{-3}{7}-\frac{1}{14}=\frac{-7}{14}=\frac{-1}{2}$$
13. Check whether the numbers $\frac{-2}{3},\ \frac{3}{8}$ and $\frac{4}{5}$ are associative under addition and under multiplication.
Answer: Let $a=\frac{-2}{3},\ b=\frac{3}{8},\ c=\frac{4}{5}$.
Addition: $$(a+b)+c=\frac{-7}{24}+\frac{4}{5}=\frac{61}{120}\quad\text{and}\quad a+(b+c)=\frac{-2}{3}+\frac{47}{40}=\frac{61}{120}$$
Multiplication: $$(a\times b)\times c=\frac{-1}{4}\times\frac{4}{5}=\frac{-1}{5}\quad\text{and}\quad a\times(b\times c)=\frac{-2}{3}\times\frac{3}{10}=\frac{-1}{5}$$
In both cases $(a+b)+c=a+(b+c)$ and $(a\times b)\times c=a\times(b\times c)$; hence these numbers are associative under both addition and multiplication.
14. Give an example to show that rational numbers are not closed under division.
Answer: $6$ and $0$ are both rational numbers, but $6\div0$ is undefined (meaningless), so it is not a rational number. Thus the quotient of two rational numbers is not always rational, i.e. rational numbers are not closed under division.
Exercise 1.2
1. Represent the following numbers on the number line: (i) $\frac{9}{5}$ (ii) $\frac{-2}{13}$ (iii) $\frac{-5}{7}$ (iv) $\frac{-1}{3}$ (v) $\frac{3}{2}$
Answer: The denominator of each number tells the number of equal parts into which one unit segment is divided.
- (i) $\frac{9}{5}=1\frac{4}{5}$: divide the segment from 1 to 2 into 5 parts; take 4 parts right of 1 ($=1.8$).
- (ii) $\frac{-2}{13}$: divide the segment from 0 to $-1$ into 13 parts; take 2 parts left of 0 ($\approx -0.15$).
- (iii) $\frac{-5}{7}$: divide the segment from 0 to $-1$ into 7 parts; take 5 parts left of 0 ($\approx -0.71$).
- (iv) $\frac{-1}{3}$: divide the segment from 0 to $-1$ into 3 parts; take 1 part left of 0 ($\approx -0.33$).
- (v) $\frac{3}{2}=1\frac{1}{2}$: the midpoint of 1 and 2 ($=1.5$).
2. Write 5 rational numbers between the following numbers:
(i) $\frac{-3}{5}$ and $\frac{1}{3}$
Answer: LCM $15$; $\frac{-3}{5}=\frac{-9}{15}$, $\frac{1}{3}=\frac{5}{15}$. So five rational numbers are $\frac{-8}{15},\ \frac{-7}{15},\ \frac{-6}{15},\ \frac{-5}{15},\ \frac{-4}{15}$.
(ii) $\frac{-2}{3}$ and $\frac{5}{3}$
Answer: Same denominator $3$; five rational numbers between $\frac{-2}{3}$ and $\frac{5}{3}$ are $\frac{-1}{3},\ 0,\ \frac{1}{3},\ \frac{2}{3},\ \frac{4}{3}$.
(iii) $-5$ and $-4$
Answer: $-5=\frac{-50}{10}$, $-4=\frac{-40}{10}$. So five rational numbers are $\frac{-49}{10},\ \frac{-48}{10},\ \frac{-47}{10},\ \frac{-46}{10},\ \frac{-45}{10}$.
(iv) $\frac{-2}{-3}$ and $\frac{3}{7}$
Answer: $\frac{-2}{-3}=\frac{2}{3}$; with LCM $42$, $\frac{2}{3}=\frac{28}{42}$ and $\frac{3}{7}=\frac{18}{42}$. So five rational numbers between $\frac{3}{7}$ and $\frac{2}{3}$ are $\frac{19}{42},\ \frac{20}{42},\ \frac{21}{42},\ \frac{22}{42},\ \frac{23}{42}$.
3. Find 5 rational numbers greater than $\frac{-2}{5}$ and less than $\frac{1}{2}$.
Answer: LCM $10$; $\frac{-2}{5}=\frac{-4}{10}$, $\frac{1}{2}=\frac{5}{10}$. So five such rational numbers are $\frac{-3}{10},\ \frac{-2}{10},\ \frac{-1}{10},\ 0,\ \frac{1}{10}$.
Additional Questions and Answers
Multiple Choice Questions (MCQ)
1. A number of the form $\frac{p}{q}$ is called a rational number if— (a) $q=0$ (b) $q \ne 0$ (c) $p=0$ (d) $p \ne 0$
Answer: (b) $q \ne 0$.
2. The additive identity is— (a) $1$ (b) $-1$ (c) $0$ (d) $2$
Answer: (c) $0$.
3. The multiplicative identity is— (a) $0$ (b) $1$ (c) $-1$ (d) $10$
Answer: (b) $1$.
4. The multiplicative inverse of $\frac{-4}{9}$ is— (a) $\frac{9}{4}$ (b) $\frac{-9}{4}$ (c) $\frac{4}{9}$ (d) $\frac{-4}{9}$
Answer: (b) $\frac{-9}{4}$.
5. Rational numbers are not closed under which operation? (a) Addition (b) Subtraction (c) Multiplication (d) Division
Answer: (d) Division.
6. How many rational numbers lie between two rational numbers? (a) One (b) Two (c) Finite (d) Infinite
Answer: (d) Infinite.
7. The reciprocal of $\frac{2}{3}$ is— (a) $\frac{3}{2}$ (b) $\frac{-3}{2}$ (c) $\frac{2}{3}$ (d) $\frac{-2}{3}$
Answer: (a) $\frac{3}{2}$.
8. Which number has no multiplicative inverse? (a) $1$ (b) $-1$ (c) $0$ (d) $2$
Answer: (c) $0$.
9. Which properties do NOT hold for subtraction and division? (a) Only commutative (b) Only associative (c) Both commutative and associative (d) Distributive
Answer: (c) Both commutative and associative.
10. The multiplicative inverse of $-1\frac{2}{3}$ is— (a) $\frac{5}{3}$ (b) $\frac{-3}{5}$ (c) $\frac{3}{5}$ (d) $\frac{-5}{3}$
Answer: (b) $\frac{-3}{5}$.
Fill in the Blanks
- 1. In a rational number $\frac{p}{q}$, the denominator $q$ is never ______. (zero / 0)
- 2. Zero (0) is the ______ identity. (additive)
- 3. One (1) is the ______ identity. (multiplicative)
- 4. For any rational number $a$, $a+(-a)=$ ______. (0)
- 5. ______ has no reciprocal. (Zero / 0)
True or False
- 1. Every integer is a rational number. (True)
- 2. Rational numbers are associative under addition and multiplication. (True)
- 3. Rational numbers are commutative under division. (False)
- 4. There is only one rational number between $\frac{1}{2}$ and $\frac{1}{3}$. (False)
- 5. The reciprocal of $-1$ is $-1$. (True)
Short Answer Questions
1. What is a rational number? Give the definition.
Answer: A number that can be written in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \ne 0$, is called a rational number.
2. Find the sum of $\frac{3}{4}$ and $\frac{5}{6}$ and show that it is a rational number.
Answer: $\frac{3}{4}+\frac{5}{6}=\frac{9}{12}+\frac{10}{12}=\frac{19}{12}$. Since $\frac{19}{12}$ is of the form $\frac{p}{q}$ with $q \ne 0$, it is a rational number — this illustrates the closure property under addition.
3. State the distributive law and give an example.
Answer: For any three rational numbers $a, b, c$, $a\times(b+c)=a\times b+a\times c$; this is the distributive law of multiplication over addition. Example: $\frac{4}{9}\times\left(\frac{11}{12}+\frac{-5}{6}\right)=\frac{4}{9}\times\frac{11}{12}+\frac{4}{9}\times\frac{-5}{6}=\frac{1}{27}$.
4. Find two rational numbers between $\frac{-5}{6}$ and $\frac{2}{3}$.
Answer: $\frac{2}{3}=\frac{4}{6}$; so two rational numbers between $\frac{-5}{6}$ and $\frac{4}{6}$ are $\frac{-1}{2}\left(=\frac{-3}{6}\right)$ and $\frac{1}{6}$.
Key Terms
| Term | Meaning |
|---|---|
| Rational number | A number of the form $\frac{p}{q}$ where $p, q$ are integers and $q \ne 0$ |
| Integer | Numbers $\ldots,-2,-1,0,1,2,\ldots$ |
| Numerator / Denominator | The top / bottom number of a fraction |
| Closure property | The result of the operation is a number of the same kind |
| Commutative property | $a+b=b+a$, $a\times b=b\times a$ |
| Associative property | $(a+b)+c=a+(b+c)$ |
| Distributive law | $a\times(b+c)=a\times b+a\times c$ |
| Additive identity | Zero (0); adding it leaves the number unchanged |
| Multiplicative identity | One (1); multiplying by it leaves the number unchanged |
| Additive inverse | The number that gives sum 0 ($a+(-a)=0$) |
| Multiplicative inverse / Reciprocal | The number that gives product 1 ($a\times\frac{1}{a}=1$) |
| Number line | A line on which rational numbers are shown as points |