NCERT Solution for Class 8 Maths Chapter 1 Rational Numbers
NCERT Solution for Class 8 Maths Chapter 1 Rational Numbers: Rational Numbers Class 8 solutions are provided here to help the students understand the concepts from the basics. Students must understand the basic concepts of class 8 Maths chapter 1 as it will continue in class 9 and class 10. Going through these solutions for Class 8 Maths Chapter 1 Rational Numbers is very helpful to score good marks in the Class 8 examination. These NCERT Class 8 Maths Chapter 1 Rational Numbers solutions help students understand the concepts in a better way.
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NCERT Solution for Class 8 Maths Chapter 1 Rational Numbers
Students can go through the Rational Numbers Class 8 solutions below to score good marks in the examination.
Exercise 1.1 Page: 14
1. Using appropriate properties, find:
(i) -2/3 × 3/5 + 5/2 – 3/5 × 1/6
Solution:
-2/3 × 3/5 + 5/2 – 3/5 × 1/6
= -2/3 × 3/5– 3/5 × 1/6+ 5/2 (by commutativity)
= 3/5 (-2/3 – 1/6)+ 5/2
= 3/5 ((- 4 – 1)/6)+ 5/2
= 3/5 ((–5)/6)+ 5/2 (by distributivity)
= – 15 /30 + 5/2
= – 1 /2 + 5/2
= 4/2
= 2
(ii) 2/5 × (- 3/7) – 1/6 × 3/2 + 1/14 × 2/5
Solution:
2/5 × (- 3/7) – 1/6 × 3/2 + 1/14 × 2/5
= 2/5 × (- 3/7) + 1/14 × 2/5 – (1/6 × 3/2) (by commutativity)
= 2/5 × (- 3/7 + 1/14) – 3/12
= 2/5 × ((- 6 + 1)/14) – 3/12
= 2/5 × ((- 5)/14)) – 1/4
= (-10/70) – 1/4
= – 1/7 – 1/4
= (– 4– 7)/28
= – 11/28
Read More: NCERT Solution for Class 8 Maths Chapter 2
2. Write the additive inverse of each of the following:
Solution:
(i) 2/8
The Additive inverse of 2/8 is – 2/8
(ii) -5/9
The additive inverse of -5/9 is 5/9
(iii) -6/-5 = 6/5
The additive inverse of 6/5 is -6/5
(iv) 2/-9 = -2/9
The additive inverse of -2/9 is 2/9
(v) 19/-16 = -19/16
The additive inverse of -19/16 is 19/16
3. Verify that: -(-x) = x for:
(i) x = 11/15
(ii) x = -13/17
Solution:
(i) x = 11/15
We have, x = 11/15
The additive inverse of x is – x (as x + (-x) = 0).
Then, the additive inverse of 11/15 is – 11/15 (as 11/15 + (-11/15) = 0).
The same equality, 11/15 + (-11/15) = 0, shows that the additive inverse of -11/15 is 11/15.
Or, – (-11/15) = 11/15
i.e., -(-x) = x
(ii) -13/17
We have, x = -13/17
The additive inverse of x is – x (as x + (-x) = 0).
Then, the additive inverse of -13/17 is 13/17 (as 13/17 + (-13/17) = 0).
The same equality (-13/17 + 13/17) = 0, shows that the additive inverse of 13/17 is -13/17.
Or, – (13/17) = -13/17,
i.e., -(-x) = x
4. Find the multiplicative inverse of the following:
(i) -13 (ii) -13/19 (iii) 1/5 (iv) -5/8 × (-3/7) (v) -1 × (-2/5) (vi) -1
Solution:
(i) -13
Multiplicative inverse of -13 is -1/13.
(ii) -13/19
Multiplicative inverse of -13/19 is -19/13.
(iii) 1/5
Multiplicative inverse of 1/5 is 5.
(iv) -5/8 × (-3/7) = 15/56
Multiplicative inverse of 15/56 is 56/15.
(v) -1 × (-2/5) = 2/5
Multiplicative inverse of 2/5 is 5/2.
(vi) -1
Multiplicative inverse of -1 is -1.
5. Name the property under multiplication used in each of the following:
(i) -4/5 × 1 = 1 × (-4/5) = -4/5
(ii) -13/17 × (-2/7) = -2/7 × (-13/17)
(iii) -19/29 × 29/-19 = 1
Solution:
(i) -4/5 × 1 = 1 × (-4/5) = -4/5
Here 1 is the multiplicative identity.
(ii) -13/17 × (-2/7) = -2/7 × (-13/17)
The property of commutativity is used in the equation.
(iii) -19/29 × 29/-19 = 1
The multiplicative inverse is the property used in this equation.
6. Multiply 6/13 by the reciprocal of -7/16.
Solution:
Reciprocal of -7/16 = 16/-7 = -16/7
According to the question,
6/13 × (Reciprocal of -7/16)
6/13 × (-16/7) = -96/91
Check out: Class 8th Combo Set of 5 Books
7. Tell what property allows you to compute 1/3 × (6 × 4/3) as (1/3 × 6) × 4/3.
Solution:
1/3 × (6 × 4/3) = (1/3 × 6) × 4/3
Here, the way in which factors are grouped in a multiplication problem supposedly does not change the product. Hence, the Associativity Property is used here.
8. Is 8/9 the multiplication inverse of –
Solution:
–
[Multiplicative inverse ⟹ product should be 1]
According to the question,
8/9 × (-9/8) = -1 ≠ 1
Therefore, 8/9 is not the multiplicative inverse of –
9. If 0.3 is the multiplicative inverse of
Solution:
0.3 = 3/10
[Multiplicative inverse ⟹ product should be 1]
According to the question,
3/10 × 10/3 = 1
Therefore, 0.3 is the multiplicative inverse of
10. Write:
(i) The rational number that does not have a reciprocal.
(ii) The rational numbers that are equal to their reciprocals.
(iii) The rational number that is equal to its negative.
Solution:
(I) The rational number that does not have a reciprocal is 0.
Reason:
0 = 0/1
Reciprocal of 0 = 1/0, which is not defined.
(ii) The rational numbers that are equal to their reciprocals are 1 and -1.
Reason:
1 = 1/1
Reciprocal of 1 = 1/1 = 1, similarly, reciprocal of -1 = – 1
(iii) The rational number that is equal to its negative is 0.
Reason:
Negative of 0=-0=0
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11. Fill in the blanks.
(i) Zero has _______reciprocal.
(ii) The numbers ______and _______are their own reciprocals
(iii) The reciprocal of – 5 is ________.
(iv) Reciprocal of 1/x, where x ≠ 0 is _________.
(v) The product of two rational numbers is always a ________.
(vi) The reciprocal of a positive rational number is _________.
Solution:
(i) Zero has no reciprocal.
(ii) The numbers -1 and 1 are their own reciprocals
(iii) The reciprocal of – 5 is -1/5.
(iv) Reciprocal of 1/x, where x ≠ 0 is x.
(v) The product of two rational numbers is always a rational number.
(vi) The reciprocal of a positive rational number is positive.
Exercise 1.2 Page: 20
1. Represent these numbers on the number line.
(i) 7/4
(ii) -5/6
Solution:
(i) 7/4
Divide the line between the whole numbers into 4 parts, i.e. divide the line between 0 and 1 to 4 parts, 1 and 2 to 4 parts, and so on.
Thus, the rational number 7/4 lies at a distance of 7 points away from 0 towards the positive number line.
(ii) -5/6
Divide the line between the integers into 4 parts, i.e. divide the line between 0 and -1 to 6 parts, -1 and -2 to 6 parts, and so on. Here, since the numerator is less than the denominator, dividing 0 to – 1 into 6 parts is sufficient.
Thus, the rational number -5/6 lies at a distance of 5 points, away from 0, towards the negative number line.
2. Represent -2/11, -5/11, -9/11 on a number line.
Solution:
Divide the line between the integers into 11 parts.
Thus, the rational numbers -2/11, -5/11, and -9/11 lie at a distance of 2, 5, and 9 points away from 0, towards the negative number line, respectively.
3. Write five rational numbers which are smaller than 2.
Solution:
The number 2 can be written as 20/10
Hence, we can say that the five rational numbers which are smaller than 2 are:
2/10, 5/10, 10/10, 15/10, 19/10
4. Find the rational numbers between -2/5 and ½.
Solution:
Let us make the denominators the same, say 50.
-2/5 = (-2 × 10)/(5 × 10) = -20/50
½ = (1 × 25)/(2 × 25) = 25/50
Ten rational numbers between -2/5 and ½ = ten rational numbers between -20/50 and 25/50.
Therefore, ten rational numbers between -20/50 and 25/50 = -18/50, -15/50, -5/50, -2/50, 4/50, 5/50, 8/50, 12/50, 15/50, 20/50.
5. Find five rational numbers between:
(i) 2/3 and 4/5
(ii) -3/2 and 5/3
(iii) ¼ and ½
Solution:
(i) 2/3 and 4/5
Let us make the denominators the same, say 60
i.e., 2/3 and 4/5 can be written as:
2/3 = (2 × 20)/(3 × 20) = 40/60
4/5 = (4 × 12)/(5 × 12) = 48/60
Five rational numbers between 2/3 and 4/5 = five rational numbers between 40/60 and 48/60.
Therefore, five rational numbers between 40/60 and 48/60 = 41/60, 42/60, 43/60, 44/60, 45/60.
378
(ii) -3/2 and 5/3
Let us make the denominators the same, say 6
i.e., -3/2 and 5/3 can be written as:
-3/2 = (-3 × 3)/(2× 3) = -9/6
5/3 = (5 × 2)/(3 × 2) = 10/6
Five rational numbers between -3/2 and 5/3 = five rational numbers between -9/6 and 10/6.
Therefore, five rational numbers between -9/6 and 10/6 = -1/6, 2/6, 3/6, 4/6, 5/6.
(iii) ¼ and ½
Let us make the denominators the same, say 24
i.e., ¼ and ½ can be written as:
¼ = (1 × 6)/(4 × 6) = 6/24
½ = (1 × 12)/(2 × 12) = 12/24
Five rational numbers between ¼ and ½ = five rational numbers between 6/24 and 12/24.
Therefore, five rational numbers between 6/24 and 12/24 = 7/24, 8/24, 9/24, 10/24, 11/24.
6. Write five rational numbers greater than -2.
Solution:
-2 can be written as – 20/10
Hence, we can say that the five rational numbers greater than -2 are
-10/10, -5/10, -1/10, 5/10, 7/10
7. Find ten rational numbers between 3/5 and ¾.
Solution:
Let us make the denominators the same, say 80.
3/5 = (3 × 16)/(5× 16) = 48/80
3/4 = (3 × 20)/(4 × 20) = 60/80
Ten rational numbers between 3/5 and ¾ = ten rational numbers between 48/80 and 60/80.
Therefore, ten rational numbers between 48/80 and 60/80 = 49/80, 50/80, 51/80, 52/80, 54/80, 55/80, 56/80, 57/80, 58/80, 59/80.
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Rational Number Class 8 Summary
A rational number is any number that can be written as a fraction, where both the top number (called the numerator) and the bottom number (called the denominator) are integers, and the denominator is not zero. For example, 3/4 and -5/2 are rational numbers. The key idea is that rational numbers include both positive and negative numbers, as well as zero.
Rational numbers can be whole numbers, like 7 or -3, since these can be written as 7/1 or -3/1. They can also be fractions like 1/2 or -4/7. Even numbers like 0.5 or 1.75 are rational because they can be expressed as fractions (1/2 and 7/4, respectively).
You can perform the usual math operations with rational numbers. You can add, subtract, multiply, and divide them, as long as you remember that dividing by zero is not allowed. For example, to add 1/4 and 2/4, you add the numerators to get 3/4.
Rational numbers can also be represented as decimal numbers. Some rational numbers have decimals that repeat in a pattern, like 1/3 which is 0.333..., while others have decimals that end after a few digits, like 1/4 which is 0.25.
In summary, rational numbers are any numbers that can be written as fractions where the denominator is not zero, and they include many types of numbers you encounter in everyday life.
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Rational Numbers Class 8 FAQs
Q1. How many exercises are there in rational numbers class 8?
Ans. Class 8 Rational Number is divided into 2 exercises. The first exercise is based on the application of property formulas while the second one sees the use of a number line to represent rational numbers.
Q2. Is 0 a rational number?
Ans. Yes, zero is also a rational number.
Q3. Can 8 be a rational number?
Ans. Yes, 8 is a rational number. Because a rational number can be represented as decimals values as well as in the form of fractions.
Q4. Which is the most easiest chapter in class 8 maths?
Ans. The most easiest chapter in class 8 maths are rational numbers, squares and square roots, exponents and powers, direct and inverse variations, etc.
Q5. Who invented rational numbers?
Ans. Rational Number is invented by Pythagoras who was an early Greek mathematician.
