NCERT Solutions for Class 8 Maths Chapter 1- Rational Numbers is a resourceful article which was developed by GFG experts to aid students in answering questions they may have as they go through problems from the NCERT textbook.
This chapter contains the following topics:
Rational Numbers: Exercise 1.1
Question 1: Using appropriate properties find.
(i) -2/3 × 3/5 + 5/2 – 3/5 × 1/6
(ii) 2/5 × (- 3/7) – 1/6 × 3/2 + 1/14 × 2/5
Solution:
(i) -2/3 × 3/5 + 5/2 – 3/5 × 1/6
Given equation: -2/3 × 3/5 + 5/2 – 3/5 × 1/6
By regrouping we get,
= -2/3 × 3/5 – 3/5 × 1/6 + 5/2
= 3/5 (-2/3 – 1/6)+ 5/2 [taking 3/5 as common]
= 3/5 ((-2×2/3×2 -1×1/6×1 )+ 5/2 [by using distributive property]
= 3/5 ((-4-1)/6)+ 5/2
= 3/5 ((–5)/6)+ 5/2
= – 15/30 + 5/2 [Dividing -15 and 30 by 2 we get -1/2]
= – 1/2 + 5/2
= 4/2
= 2
Therefore,
-2/3 × 3/5 + 5/2 – 3/5 × 1/6 = 2
(ii) 2/5 × (- 3/7) – 1/6 × 3/2 + 1/14 × 2/5
Given equation: 2/5 × (- 3/7) – 1/6 × 3/2 + 1/14 × 2/5
By regrouping we get,
= 2/5 × (-3/7) + 1/14 × 2/5 – (1/6 × 3/2)
= 2/5 × (-3/7 + 1/14) – 3/12
= 2/5 × ((-6 + 1)/14) – 3/12 [by using distributive property]
= 2/5 × ((-5)/14)) – 1/4
= (-10/70) – 1/4 [Dividing -10 and 70 by 10 we get -1/7]
= -1/7 – 1/4
= (-4 -7)/28
= -11/28
Therefore,
2/5 × (- 3/7) – 1/6 × 3/2 + 1/14 × 2/5 = -11/28
Question 2: Write the additive inverse of each of the following
(i) 2/8
(ii) -5/9
(iii) -6/-5
(iv) 2/-9
(v) 19/-16
Solution:
We know that the additive inverse of x will be -x,
(i) 2/8
Given: 2/8
Additive inverse of 2/8 will be -2/8
(ii) -5/9
Given: -5/9
Additive inverse of -5/9 will be 5/9
(iii) -6/-5
Given: -6/-5
-6/-5 = 6/5 [Dividing both by -1 ]
Additive inverse of 6/5 will be -6/5
(iv) 2/-9
Given: 2/-9
2/-9 = -2/9
Additive inverse of -2/9 will be 2/9
(v) 19/-16
Given: 19/-16
19/-16 = -19/16
Additive inverse of -19/16 will be 19/16
Question 3: Verify that: -(-x) = x for.
(i) x = 11/15
(ii) x = -13/17
Solution:
(i) x = 11/15
Given, x = 11/15
Since, additive inverse of x will be -x
Therefore, the additive inverse of 11/15 will be -11/15 (as 11/15 + (-11/15) = 0)
We can also represent the following as 11/15 = -(-11/15)
Thus, -x = -11/15
-(-x) = -(-11/15) = (11/15) = x
Hence, verified: -(-x) = x
(ii) -13/17
Given, x = -13/17
Since, additive inverse of x will be -x as x + (-x) = 0
Therefore, the additive inverse of -13/17 will be 13/17 as 13/17 + (-13/17) = 0
We can also represent the following as 13/17 = -(-13/17)
Thus, -x = -13/17
-(-x) = -(-13/17) = (13/17) = x
Hence, verified: -(-x) = x
Question 4: Find the multiplicative inverse of the
(i) -13
(ii) -13/19
(iii) 1/5
(iv) -5/8 × (-3/7)
(v) -1 × (-2/5)
(vi) -1
Solution:
We know that the multiplicative inverse of x will be 1/x as a × 1/a = 1
(i) -13
Given: -13
The multiplicative inverse of -13 will be -1/13
(ii) -13/19
Given: -13/19
The multiplicative inverse of -13/19 will be -19/13
(iii) 1/5
Given: 1/5
The multiplicative inverse of 1/5 will be 5
(iv) -5/8 × (-3/7)
Given: -5/8 × (-3/7)
-5/8 × (-3/7) = 15/56
The multiplicative inverse of 15/56 will be 56/15
(v) -1 × (-2/5)
Given: -1 × (-2/5)
-1 × (-2/5) = 2/5
The multiplicative inverse of 2/5 will be 5/2
(vi) -1
Given: -1
The multiplicative inverse of -1 will be -1
Question 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
Given: -4/5 × 1 = 1 × (-4/5) = -4/5
It is representing the property of multiplicative identity.
(ii) -13/17 × (-2/7) = -2/7 × (-13/17)
Given: -13/17 × (-2/7) = -2/7 × (-13/17)
It is representing the property of commutativity.
(iii) -19/29 × 29/-19 = 1
Given: -19/29 × 29/-19 = 1
It is representing the property of multiplicative inverse
Question 6: Multiply 6/13 by the reciprocal of -7/16
Solution:
Given: 6/13 × (Reciprocal of -7/16)
Since, reciprocal of -7/16 = 16/-7 = -16/7
Therefore,
6/13 × (-16/7) = -96/91
Question 7: Tell what property allows you to compute 1/3 × (6 × 4/3) as (1/3 × 6) × 4/3
Solution:
Given: 1/3 × (6 × 4/3) = (1/3 × 6) × 4/3
Here, the product of their multiplication does not change. Hence, Associativity Property is used in the given equation.
Question 8: Is 8/9 the multiplication inverse of -1 1/8? Why or why not?
Solution:
Given: -1 1/8 which is equal to -9/8
Since it is the multiplication inverse, therefore the product should be 1.
8/9 × (-9/8) = -1 ≠1
Hence, 8/9 is not the multiplication inverse of -1 1/8
Question 9: If 0.3 the multiplicative inverse of 3 1/3? Why or why not?
Solution:
Give: 3 1/3 = 10/3
Since it is the multiplication inverse, therefore the product should be 1.
0.3 × 10/3 = 3/3 = 1
Hence, 0.3 is the multiplicative inverse of 3 1/3.
Question 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.
Since, 0 = 0/1
Therefore, the reciprocal of 0 = 1/0, which is not defined.
Hence, the rational number that does not have a reciprocal is 0.
(ii) The rational numbers that are equal to their reciprocals.
Since, 1 = 1/1
Therefore, the reciprocal of 1 = 1/1 = 1
Similarly,
-1 = -1/1
Therefore, the reciprocal of -1 = -1/1 = -1
Hence, the rational numbers that are equal to their reciprocals are 1 and -1
(iii) The rational number that is equal to its negative.
Since negative of 0 = -0 = 0
Therefore, the rational number that is equal to its negative is 0.
Question 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.
Rational Numbers: Exercise 1.2
Question 1. Represent these numbers on the number line?
(i) 7/4 (ii) -5/6
Solution:
(i) In number line we have to cover zero to positive integer 1 which signifies the whole no 1, after that we have to divide 1 and 2 into 4 parts and we have to cover 3 places away from 0, which denotes 3/4. And the total of seven places away from 0 represents 7/4. P represents 7/4.
(ii) For representing – 5/6 we have to divide 0 to – 1 integer into 6 parts and we have to go 5 places away from 0 for – 5/6.
Question 2. Represent – 2/11, -5/11, -9/11 on the number line?
Solution:
We have to divide 0 to – 1 integer into 11 parts and the distance of 2, 5, 9 from 0 towards the left of it represents – 2/11, -5/11, -9/11 marked A, B, C, respectively.
Question 3. Write five rational numbers that are smaller than 2?
Solution:
We can write the number 2 as 6 / 3
Hence, we can write, the five rational numbers which are smaller than 2 are:
1 / 3 , 2 / 3 , 3 / 3 , 4/ 3 , 5 / 3
Question 4. Find ten rational numbers between – 2/5 and 1/2?
Solution:
For finding rational numbers between fractions we have to take L. C. M. of their denominators or its multiples. Here L. C. M. Of 5 and 2 is 10 and for finding fractions between them we have to take multiple of 10. Let us take 20 as denominator.
So,
-2 / 5 = (- 2 / 5) × (4 / 4) = -8 / 20
Also,
1 / 2 = (1 / 2) × (10 / 10) = 10 / 20
Hence ten rational numbers between – 2 / 5 to 1 / 2 are same as rational numbers between – 8 / 20 and 10 / 20. And those are as follows
-7 / 20, -6 / 20, -5 / 20, -4 / 20, -3 / 20, -2 / 20, -1 / 20, 0, 1 / 20, 2 / 20
Question 5. Find five rational numbers between
(i) 2/3 and 4/5 (ii) – 3/2 and 5/3 (iii)1/4 and 1/2
Solution:
(i) 2 / 3 and 4 / 5
For finding rational numbers between fractions we have to take L. C. M. of their denominators or its multiples.
Here L. C. M. Of 3 and 5 is 15
And we take the denominators as multiple of 15, as 60
Hence
2 / 3 = ( 2 / 3 ) × ( 20 / 20 ) = 40 / 60
4 / 5 = ( 4 / 5 ) × ( 12 / 12 ) = 48 / 60
Five rational numbers between 2 / 3 and 4 / 5 same as five rational numbers between
40 / 60 and 48 / 60
Therefore, Five rational numbers between 40 / 60 and 48 / 60 are as follows
41 / 60, 42 / 60, 43 / 60, 44 / 60, 45 / 60
(ii) -3 / 2 and 5 / 3
Similarly,
L. C. M. of 2 and 3 is 6.
Here we take denominators same as 6.
-3 / 2 = ( -3 / 2 ) × ( 3 / 3 ) = -9 / 6
5 / 3 = ( 5 / 3 ) × ( 2 / 2 ) = 10 / 6
Hence five rational numbers between -3 / 2 and 5 / 3 are same as five rational numbers between -9 / 6 and 10 / 6 and those are as follows
-8 / 6, -7 / 6, -1, -5 / 6, -4 / 6
(iii) 1 / 4 and 1 / 2
Here L. C. M. of 4 and 2 is 8.
Here we take denominator as multiple of 8 say 32.
Hence
1 / 4 = ( 1 / 4 ) × (8 / 8) = 8 / 32
1 / 2 = ( 1 / 2 ) × ( 16 / 16 ) = 16 / 32
Hence five rational numbers between 1 / 4 and 1 / 2 are same as five rational numbers between 8/32 and 16/32 and those are as follows
9 / 32, 10 / 32, 11 / 32, 12 / 32, 13 / 32
Question 6. Write five rational numbers greater than –2?
Solution:
We can write -2 as -10 / 5
Hence five rational numbers greater than -2 are as follows
-1 / 5, -2 / 5, -3 / 5, -4 / 5 ,-1
Question 7. Find ten rational numbers between 3/5 and 3/4?
Solution:
L .C. M. of 4 and 5 is 20. For finding rational number between them we should make denominator same or multiple of L .C.M.
Here we take 80.
3 / 5 = ( 3 / 5) × ( 16 / 16 ) = 48 / 80
3 / 4 = ( 3 / 4 ) × ( 20 / 20 ) = 60 / 80
Ten rational numbers between 3 / 5 and 3 / 4 are same as ten rational numbers between 48 / 80 and 60 / 80
Ten rational numbers between 48 / 80 and 60 / 80 are as follows
49 / 80, 50 / 80, 51 / 80, 52 / 80, 54 / 80, 55 / 80, 56 / 80, 57 / 80, 58 / 80, 59 / 80
Important Points to Remember:
- These NCERT solutions are developed by the GfG team, with a focus on students’ benefit.
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- Each solution is presented in a step-by-step format with comprehensive explanations of the intermediate steps.
FAQs on NCERT Solutions for Class 8 Maths Chapter 1- Rational Numbers
1. Why is it important to learn Rational Numbers?
Learning about rational numbers is important for class 8 students because it helps them simplify fractions and understand real-life applications involving measurements and division.
2. What topics are covered in NCERT Solutions for Chapter 1– Rational Numbers?
NCERT Solutions for Class 8 Maths Chapter 1- Rational Numbers covers topics such as basic introduction of rational numbers, their representation on number line, comparing rational numbers and finding rational numbers between two rational numbers.
3. How can NCERT Solutions for Class 8 Maths Chapter 1- Rational Numbers help me?
NCERT Solutions for Class can help you solve the NCERT exercise without any limitations. If you are stuck on a problem, you can find its solution in these solutions and free yourself from the frustration of being stuck on some question.
4. How many exercises are there in Class 8 Maths Chapter 1- Rational Numbers ?
There are 2 exercises in the Class 8 Maths Chapter 1- Rational Numbers which covers all the important topics and sub-topics.
5. Where can I find NCERT Solutions for Class 8 Maths Chapter 1- Rational Numbers?
You can find these NCERT Solutions in this article created by our team of experts at GeeksforGeeks.
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