WBBSE Solutions For Class 8 Maths Algebra Chapter 2 Rational Number
Natural Numbers
The numbers, which were known to us in the beginning, were used to count something. These numbers are natural numbers. Thus, 1, 2, 3, 4… to infinity are called natural numbers.
Whole Numbers
When 0 is included with natural numbers, then they form whole numbers or integers. Thus, 0, 1, 2, 3, 4… to infinity are called
Positive and Negative Integers
whole numbers or integers.
The numbers 0, 1, 2, 3, 4… etc. are called positive integers, and -1, -2, -3, -4, … are negative integers.
WBBSE Class 8 Rational Numbers Notes
Rational Number
Any number of the form p/q where p and q are both integers and q * 0 is called a rational number. While writing a rational number usually take the denominator as a positive integer p and express p/q in the lowest form.
Example : 3/8, 8/11, -7/12 etc.
are the rational numbers. All natural numbers, integers, and fractions are included in rational numbers. We can imagine the natural number 5 as a rational number. Because 5 =5/1 therefore, it can be expressed in the form p/q where p =5 and q = 1, p and q are both integers and q ≠ 0.
But √2 is not a rational number because it cannot be expressed in the form p/q where p and q are integers. In fact, √2 = 1.414213…
Any decimal fraction and recurring decimal fraction may be expressed in the form p/q where p and q are integers.
Therefore, they are also rational numbers.
Read And Learn More WBBSE Solutions For Class 8 Maths
Example: 0.5 = 5/10 = 1/2;
0.9 = 9/9 = 1/1
The numbers which are not rational are called irrational numbers. Examples of some irrational numbers are
Some Decisions
1. Sum of two integers is an integer: 7 + 8=15.
2. Product of two integers is an integer: 7 x 8 = 56.
3. The difference between two integers in a positive or negative integer: 8-4 = 4,
4-8
= -4.
4. The quotient of two integers is not always an integer: 7÷8 = 7/8,
8 ÷ 7 = 8/7.
(Only when two numbers are the same their quotient is an integer. For example, 5÷5 = 1.)
Some more decisions
If a and b are two rational numbers :
1. a + b is a rational number.
For example 1/2 + 2/3
Given That, a + b is a rational number.
Adding The Terms 1/2 And 2/3
= 3+4 / 6
= 7/6.
Here, 7/6 is a rational number.
2. a – b is a rational number.
For example 1/2 – 2/3
Given That, a – b is a rational number
So Subtracting The Terms 1/2 And 2/3
= 3-4 / 6
= -1/6.
Here, is a rational number.
Understanding Rational Numbers in Algebra
3. a x b is a rational number.
For example 1/2 x 2/3
Given a x b is a rational number.
So Multiply The Terms 1/2 And 2/3
1/2 x 2/3
= 1/3.
Here, 1/3 is a rational number.
4. a÷b is a rational number.
For example 1/2 ÷ 2/3
Given a ÷ b is a rational number.
= 1/2 ÷ 2/3
= 3/4
Here, 3/4 is a rational number.
Some Properties of rational numbers
1. If a and b are two rational numbers, then a + b = b + a.
That means rational numbers obey the commutative law of addition.
For example 1/2 + 2/3
= 3+4 / 6
= 7/6
2/3 + 1/2
= 4+3/6
= 7/6
∴ 1/2 + 2/3 = 2/3 + 1/2.
2. If a and b are two rational numbers, then a – b ≠ b – a.
That means the rational numbers do not obey the commutative law of subtraction.
That means the rational numbers do not obey the commutative law of subtraction.
1/2 – 2/3
= 3-4 / 6
= -1/6
2/3 – 1/2
= 4-3/6
= 1/6
∴ 1/2 – 2/3 ≠ 2/3 – 1/2.
3. If a and b are two rational numbers, then a x b = b x a.
That means rational numbers obey the commutative law of multiplication.
1/2 x 2/3 = 1/3
2/3 x 1/2 = 1/3
∴ 1/2 x 2/3 = 2/3 x 1/2.
4. If a and b are two rational numbers, then a + b x b + a.
That means the rational numbers do not obey the commutative law of division.
1/2 ÷ 2/3 = 1/2 x 3/2 = 3/4
2/3 ÷ 1/2 = 2/3 x 2/1 = 4/3
∴ 1/2 ÷ 2/3 ≠ 2/3 ÷ 1/2.
Properties of Rational Numbers Explained
Some more Properties of rational
1. If a, 6, and c are three rational numbers then, a + (b + c) = (a + b) + c.
That means the rational numbers obey the associative law of addition
1/2 + (2/3 + 3/4 ) = 1/2 + 8+9 / 12
= 1/2 + 17/12+= 6+17 / 12+= 23/12
(1/2 + 2/3 ) + 3/4 = 3+4 / 6 + 3/4
= 7/6 + 3/4
= 14+9 / 12
= 23/12
∴ 1/2 + (2/3 + 3/4) = (1/2 + 2/3) + 3/4.
2. If a, b, and c are three rational numbers then, a – (b – c) * (a – b) – c.
That means rational numbers do not obey the associative law of subtraction.
1/2 – (2/3 – 3/4) = 1/2 – (8-9 / 12)
= 1/2 + 1/12
= 6+1 / 12
= 7/ 12
(1/2 – 2/3 – 3/4 = 3-4 / 6 – 3/4
= -1/6 – 3/4
= -2-9 / 12
= -11/12
∴ 1/2 – (2/3 – 3/4) ≠ (1/2 – 2/3) – 3/4.
3. If a, b, and c are three rational numbers then, z x ( b x c) = (a x b) x c.
That means rational numbers obey the associative law of multiplication.
1/2 x (2/3 x 3/4) = 1/2 x 1/2
= 1/4
(1/2 x 2/3 ) x 3/4 = 1/3 x 3/4
= 1/4
∴ 1/2 x (2/3 x 3/4) = (1/2 x 2/3) x 3/4.
Practice Problems on Rational Numbers
4. If a, b, and c are three rational numbers then, a ÷ (6 ÷ c) ≠ (a ÷ b) ÷ c.
That means rational numbers do not obey the associative law of divisions.
\(\frac{1}{2} \div\left(\frac{2}{3} \div \frac{3}{4}\right)=\frac{1}{2} \div\left(\frac{2}{3} \times \frac{4}{3}\right)\) \(=\frac{1}{2} \div \frac{8}{9}=\frac{1}{2} \times \frac{9}{8}=\frac{9}{16}\) \(\left(\frac{1}{2} \div \frac{2}{3}\right) \div \frac{3}{4}=\left(\frac{1}{2} \times \frac{3}{2}\right) \div \frac{3}{4}=\frac{3}{4} \times \frac{4}{3}=1\) \(\frac{1}{2} \div\left(\frac{2}{3} \div \frac{3}{4}\right) \neq\left(\frac{1}{2} \div \frac{2}{3}\right) \div \frac{3}{4} \text {. }\)Distributive law
If a, b, and c are three rational numbers then,
a(b + c) = ab + ac
a(b – c) = ab – ac
\(\frac{1}{2}\left(\frac{2}{3}+\frac{3}{4}\right)=\frac{1}{2}\left(\frac{8+9}{12}\right)=\frac{1}{2} \times \frac{17}{12}=\frac{17}{24}\) \(\frac{1}{2} \times \frac{2}{3}=\frac{1}{3}, \frac{1}{2} \times \frac{3}{4}=\frac{3}{8}\) \(\frac{1}{2} \times \frac{2}{3}+\frac{1}{2} \times \frac{3}{4}=\frac{1}{3}+\frac{3}{8}=\frac{8+9}{24}=\frac{17}{24}\) \(\frac{1}{2}\left(\frac{2}{3}+\frac{3}{4}\right)=\frac{1}{2} \times \frac{2}{3}+\frac{1}{2} \times \frac{3}{4}\) \(\text { Similarly, } \frac{1}{2}\left(\frac{2}{3}-\frac{3}{4}\right)=\frac{1}{2} \times \frac{2}{3}-\frac{1}{2} \times \frac{3}{4} \text {. }\)Algebra Chapter 12 Equations Some Examples
Example 1
Find the opposite number of addition.
1. – 3/7
2. 15/17.
Solution:
1. The opposite number of addition of – 3/7 is 3/7 because,
= – 3/7 + 3/7
= -3+3 / 7
= 0/7
= 0.
– 3/7 the opposite number of addition = 0.
2. The opposite number of addition of 15/17 is – 15/17 because,
= 15/17 – 15/17
= 15-15 / 17
= 0/17
= 0.
15/17 the opposite number of addition = 0.
Examples of Adding and Subtracting Rational Numbers
Example 2
What do you mean by the reciprocal of a rational number?
Solution :
The number by which, if a rational number is multiplied to get 1 as a product is called the reciprocal or opposite number with respect to the multiplication of that rational number. It is also known as multiplicative inverse.
For Example, the reciprocal of 5/7 is 7/5 because
5/7 x 7/5 = 1
Example 3
Solution :
Let, 3/7 be any rational number.
The opposite number with respect to the addition of 3/7 is – 3/7
Again, the opposite number with respect to the addition of – 3/7 is 3/7
∴ – (-x) = x.
Example 4
Find the product: – 4/5 x 3/7 x 15/16 x – 14/9
Solution:
The given expression
= ( – 4/5 x 15/16 ) x (3/7 x – 14/9)
= – 3/4 x -2/3
= 1/2.
( – 4/5 x 15/16 ) x (3/7 x – 14/9) = 1/2.
Example 5
Find the value of 3/7 + (-6/11) + (-8/21) + 5/22.
Solution :
The given expression
= { 3/7 + (-8 / 21)} + (-6/11 + 5/22}
= 9-8 / 21 + (-12 + 5) / 22
= 1/21 – 7/22
= 22-147 / 462
= -125 / 462.
{ 3/7 + (-8 / 21)} + (-6/11 + 5/22} { 3/7 + (-8 / 21)} + (-6/11 + 5/22}
Finding Rational Numbers Between Two Values
Example 6
Write 4 rational numbers between 1 and 2.
Solution :
1. If x and y be two rational numbers such that x < y, then 1/2 ( x + y)is a rational number between x and y.
Observe how this rule is applied here
1 rational number between 1 and 2
= 1/2 (1+2)
= 3/2
4 rational numbers between 1 and 2 = 3/2
2. Again, 1 rational number between 1 and 3/2
= 1/2(1+2)
= 1/2(1 + 3/2)
= 5/4
1 rational number between 1 and 3/2 = 5/4
3. 1 rational number between 1 and 5/4
= 1/2 (1 + 5/4)
= 9/8
1 rational number between 1 and 5/4 = 9/8
4. 1 rational number between 1 and 9/8
= 1/2 (1 + 9/8)
= 17/16
1 rational number between 1 and 9/8 = 17/16
∴ 4 rational numbers between 1 and 2 are,
17/16, 9/8, 5/4, 3/2.
Example 7
Write 4 rational numbers equivalent to 2/3.
Solution :
\(\frac{2}{3}=\frac{2 \times 2}{3 \times 2}=\frac{4}{6}, \quad \frac{2}{3}=\frac{2 \times 3}{3 \times 3}=\frac{6}{99},\) \(\frac{2}{3}=\frac{2 \times 4}{3 \times 4}=\frac{8}{12}, \quad \frac{2}{3}=\frac{2 \times 5}{3 \times 5}=\frac{10}{15}\)∴ 4 rational numbers equivalent to 2/3 are,
4/6, 6/9, 8/12, 10/15.
Conceptual Questions on Operations with Rational Numbers
Example 8
Write the rational number 4 as the
1. sum and
2. difference between two irrational numbers.
Solution :
1. 4 = (2 + √3) + (2 – √3).
2. 4 = (√3 + 2) – (√3 – 2).
Example 9
What should be added to 7/12 to get – 4/12?
Solution :
Let the required number be x.
Then, 7/12 + x = – 4/15
or, x = – 4/15 – 7/12
= -(4/15 + 7/12)
= – (16+35 / 60)
= – 51/60
= – 17/20.
Example 10
The sum of two rational numbers is -3. If one of them is 1/3, then find the other one.
Solution :
Given That,
The sum of two rational numbers is -3.
one of them is 1/3.
Let the other number be x.
Then, x + 1/3 = -3
or, x = – 3 – 1/3
or, x = -(3+1/3)
= -(9+1 / 3)
= -10/3
Example 11
Find the reciprocal of -5/2 x 32/15.
Solution:
Given -5/2 x 32/15.
We have, \(-\frac{5}{8} \times \frac{32}{15}=-\frac{5 \times 32^4}{3 \times 15_3}=-\frac{4}{3}\)
∴ Reciprocal of \(-\frac{5}{8} \times \frac{32}{15}\)
= Reciprocal of \(-\frac{4}{3}\)
= \(\frac{3}{4}\)
Example 12
If x = 3 and y = 2, then find the value of (3x-4y)y-x ÷ (4x-3y)2y-x
Solution:
Given
x = 3 And y = 2.
We have, (3x -4y) = (3 x 3 – 4 x 2)
= (9 – 8)
= 1
(3x-4y)y-x ÷ (4x-3y)2y-x = 1
(4x – 3y) = (4 x 3 – 3 x 2) = (12 – 6) = 6
∴ \((3 x-4 y)^{y-x} \div(4 x-3 y)^{2 y-x}\)
= \(\frac{(3 x-4 y)^{y-x}}{(4 x-3 y)^{2 y-x}}=\frac{(1)^{2-3}}{(6)^{2 \times 2-3}}=\frac{(1)^{-1}}{(6)^{4-3}}\)
= \(\frac{(1)^{-1}}{6}=\frac{1}{1 \times 6}=\frac{1}{6}\)