## ML Aggarwal Class 8 Solutions for ICSE Maths Chapter 1 Rational Numbers Objective Type Questions

**Mental Maths**

Question 1.

Fill in the blanks:

(i) The product of two rational numbers is a ……….

(ii) Subtraction of rational numbers is …….. commutative.

(iii) The rational number \(\frac { -7 }{ 4 }\) lies ……. of zero on the number line.

(iv) Division of rational numbers is ……… associative.

(v) \(\frac { p }{ q }\) ÷ 0 is ……..

(vi) Negative of a rational number is called its ………

(vii) Multiplicative identity of rational numbers is ……….

(viii) Multiplication of rational numbers is ……. over addition.

(ix) Division of a rational number by a non-zero rational number is a ………..

(x) The rational number which is additive inverse of itself is …………

Solution:

(i) The product of two rational numbers is a rational number.

(ii) Subtraction of rational numbers is not commutative.

(iii) The rational number \(\frac { -7 }{ 4 }\) lies left side of zero on the number line.

(iv) Division of rational numbers is not associative.

(v) \(\frac { p }{ q }\) ÷ 0 is not defined.

(vi) Negative of a rational number is called its additive inverse.

(vii) Multiplicative identity of rational numbers is 1.

(viii) Multiplication of rational numbers is distributive over addition.

(ix) Division of a rational number by a non-zero rational number is a rational number.

(x) The rational number which is additive inverse of itself is 0.

Question 2.

State whether the following statements are true (T) or false (F):

(i) \(\frac { -5 }{ 9 }\) is the additive inverse of \(\frac { 5 }{ 9 }\).

(ii) Every integer is a rational number.

(iii) Zero has its multiplicative inverse.

(iv) Every rational number is an integer.

(v) Division of two rational numbers is alwasy closed.

(vi) Non-terminating, non-recurring decimal numbers are rational numbers.

(vii) 0 is the multiplicative identity of rational numbers.

(viii) Non-terminating recurring decimal numbers are not rational numbers.

(ix) Subtraction of two rational numbers is not associative.

(x) Reciprocal of 1 is 1.

(xi) The multiplicative inverse is also called a reciprocal.

(xii) Between two different rational numbers, there are infinitely many number of rational numbers.

Solution:

(i) \(\frac { -5 }{ 9 }\) is the additive inverse of \(\frac { 5 }{ 9 }\). (True)

(ii) Every integer is a rational number. (True)

(iii) Zero has its multiplicative inverse. (False)

Correct:

Zero has no multiplicative inverse.

(iv) Every rational number is an integer. (False)

Correct:

Rational numbers are not integers.

(v) Division of two rational numbers is alwasy closed. (False)

Correct:

Division by zero is not defined.

(vi) Non-terminating, non-recurring decimal numbers are rational numbers. (False)

Correct:

Non-terminating, recurring numbers are rationals.

(vii) 0 is the multiplicative identity of rational numbers. (False)

Correct:

1 is the multiplicative identity.

(viii) Non-terminating recurring decimal numbers are not rational numbers. (False)

Correct:

These are rational numbers.

(ix) Subtraction of two rational numbers is no associative. (True)

(x) Reciprocal of 1 is 1. (True)

(xi) The multiplicative inverse is also called a reciprocal. (True)

(xii) Between two different rational numbers,

there is infinitely many numbers of rational numbers. (True)

**Multiple Choice Questions**

Choose the correct answer from the given four options (3 to 18):

Question 3.

Additive inverse of \(\frac { -2 }{ -5 }\) is

(a) \(\frac { 2 }{ 5 }\)

(b) \(\frac { 5 }{ 2 }\)

(c) \(\frac { 2 }{ -5 }\)

(d) \(\frac { 5 }{ -2 }\)

Solution:

Additive inverse of \(\frac { -2 }{ -5 }\) is \(\frac { 2 }{ -5 }\). (c)

Question 4.

Multiplicative inverse of \(\frac { -3 }{ 7 }\) is

(a) \(\frac { 7 }{ 3 }\)

(b) \(\frac { -7 }{ 3 }\)

(c) \(\frac { 3 }{ 7 }\)

(d) None of these

Solution:

Multiplicative inverse of \(\frac { -3 }{ 7 }\) is \(\frac { -7 }{ 3 }\). (b)

Question 5.

Sum of a rational number and its additive inverse is

(a) 1

(b) 0

(c) -1

(d) None of these

Solution:

Sum of a rational number and its additive inverse is 0. (b)

Question 6.

Rational numbers are not closed under

(a) addition

(b) subtraction

(c) multiplication

(d) division

Solution:

Rational numbers are not closed under division. (d)

Question 7.

0 ÷ \(\frac { 2 }{ 3 }\) is equal to

(a) \(\frac { 2 }{ 3 }\)

(b) \(\frac { 3 }{ 2 }\)

(c) 0

(d) not defined

Solution:

0 ÷ \(\frac { 2 }{ 3 }\) is equal to 0. (c)

Question 8.

\(\frac { 2 }{ 3 }\) ÷ 0 is equal to

(a) \(\frac { 2 }{ 3 }\)

(b) \(\frac { 3 }{ 2 }\)

(c) 0

(d) not defined

Solution:

\(\frac { 2 }{ 3 }\) ÷ 0 is equal to 0. (c)

Question 9.

\(\frac { p }{ q } +\left( \frac { r }{ s } +\frac { t }{ u } \right) =\left( \frac { p }{ q } +\frac { r }{ s } \right) +\frac { t }{ u }\) is called

(a) commutative property

(b) associative property

(c) distributive property

(d) None of these

Solution:

\(\frac { p }{ q } +\left( \frac { r }{ s } +\frac { t }{ u } \right) =\left( \frac { p }{ q } +\frac { r }{ s } \right) +\frac { t }{ u }\) is called associative property. (b)

Question 10.

Multiplication of a non-zero rational number and its reciprocal is

(a) 0

(b) 1

(c) -1

(d) None of these

Solution:

Multiplication of a non-zero rational number and its reciprocal is 1. (b)

Question 11.

Product of rational number \(\frac { -2 }{ 5 }\) and its additive inverse is

(a) 0

(b) 1

(c) \(\frac { -4 }{ 25 }\)

(d) \(\frac { -5 }{ 2 }\)

Solution:

Product of a rational number\(\frac { -2 }{ 5 }\) and its additive inverse is \(\frac { -4 }{ 25 }\). (c)

Question 12.

Sum of rational number \(\frac { 4 }{ 7 }\) and its reciprocal is

(a) \(\frac { 28 }{ 65 }\)

(b) \(\frac { 65 }{ 28 }\)

(c) \(\frac { -28 }{ 65 }\)

(d) \(\frac { -65 }{ 28 }\)

Solution:

Sum of rational number \(\frac { 4 }{ 7 }\) and its reciprocal is \(\frac { 65 }{ 28 }\). (b)

Question 13.

Sum of two rational numbers is 0, if one ofthem is \(\frac { -4 }{ 5 }\), then other is

(a) \(\frac { 5 }{ 4 }\)

(b) \(\frac { 4}{ 5 }\)

(c) \(\frac { -5 }{ 4 }\)

(d) \(\frac { -4 }{ 5 }\)

Solution:

Sum of two rational numbers is 0, if one of them is \(\frac { -4 }{ 5 }\),

then other will be \(\frac { 4 }{ 5 }\). (b)

Question 14.

Product of two rational numbers is 1, if one of them is \(\frac { 10 }{ 3 }\), then other is

(a) \(\frac { 3 }{ 10 }\)

(b) \(\frac { -3}{ 10 }\)

(c) \(\frac { 10 }{ 3 }\)

(d) None of these

Solution:

Product of two rational numbers is 1.

If one of them is \(\frac { 10 }{ 3 }\), then the other is \(\frac { 3 }{ 10 }\). (a)

Question 15.

Rational number represented by the point P on the number line is

(a) \(\frac { -5 }{ 7 }\)

(b) \(\frac { -3}{ 7 }\)

(c) \(\frac { -5 }{ 8 }\)

(d) \(\frac { -4 }{ 8 }\)

Solution:

Rational number represented by the point P

on the given number line is \(\frac { -3 }{ 7 }\) (b)

Question 16.

What should be subtracted from \(\frac { -5 }{ 3 }\) to get \(\frac { -2 }{ 7 }\)?

(a) \(\frac { 29 }{ 21 }\)

(b) \(\frac { -21}{ 29 }\)

(c) \(\frac { -29 }{ 21 }\)

(d) \(\frac { 21 }{ 29 }\)

Solution:

Question 17.

Reciprocal of a negative number is

(a) positive

(b) negative

(c) can not say

(d) does not exist

Solution:

Reciprocal of a negative number is negative. (b)

Question 18.

Which of the following statement is true?

Solution:

**Value Based Questions**

Question 1.

Ram donated \(\frac { 1 }{ 10 }\) of his salary to an orphanage, \(\frac { 1 }{ 3 }\) of his salary spent on food, \(\frac { 1 }{ 4 }\) of salary on rent and electricity and \(\frac { 1 }{ 20 }\) of his salary on telephone. This month he donated ₹ 5000 in Prime Minister relief fund for Uttarakhand victims. He was left with ₹ 3000 with him, find his monthly salary. Should we donate the money for needy people? What values are being promoted?

Solution:

Let Ram’s salary = x

Then donation to orphanage = x × \(\frac { 1 }{ 10 }\) = \(\frac { x }{ 10 }\)

Amount spent on food = \(\frac { x }{ 3 }\)

Spend on rent and electricity = \(\frac { x }{ 4 }\)

Spent on telephone = \(\frac { x }{ 20 }\)

Donation to a needy institution is good,

also donation to PM relief fund is an act of kindness.

Question 2.

In an Examination \(\frac { 1 }{ 3 }\) of the total students used unfair means and out of which \(\frac { 1 }{ 4 }\) caught red handed while cheating. If 5 students caught red handed then find the total number of students appeared in exam.

Why should we not use unfair means in an examination?

What values are being promoted?

Solution:

Let total number of students who appeared in the exam = x

Number of students who cheat = \(\frac { x }{ 3 }\)

Number of students who were caught red handed = \(\frac { x }{ 3 }\) × \(\frac { x }{ 4 }\) = \(\frac { x }{ 12 }\)

\(\frac { x }{ 12 }\) = 5

x = 12 × 5 = 60

Total number of students = 60

Cheating in the examination is a bad babbit

and its ruins the life of a student. So, it should be avoided.

**Higher Order Thinking Skills (HOTS)**

Question 1.

Area ol a square is 4 sq. in more than \(\frac { 2 }{ 3 }\) of the area of a rectangle. If the area of square is 64 sq. m, then find the dimensions of rectangle, given that breadth is \(\frac { 2 }{ 5 }\) of length.

Solution:

Area of a square = 64 sq. m

Area of rectangle = 64 – 4

= 60 × \(\frac { 3 }{ 2 }\) = 90 sq. m

Breadth = \(\frac { 2 }{ 5 }\) of length

Let length = x m

then breadth = \(\frac { 2 }{ 5 }\) x

Area = x × \(\frac { 2 }{ 5 }\) x = 90

⇒ \(\frac { 2 }{ 5 }\) x^{2} = 90

⇒ x^{2} = 90 × \(\frac { 5 }{ 2 }\) = 225 = (15)^{2}

⇒ x = 15

Length = 15 m

and breadth = \(\frac { 2 }{ 5 }\) × 15 = 6m

Hence, length of rectangle = 15 m

and breadth = 6 m

Question 2.

Rahul can do \(\frac { 2 }{ 7 }\) of a certain work in 6 days while Suresh can do \(\frac { 3 }{ 5 }\) of the same work in 9 days. They started work together but after 7 days Rahul left the work. Find in how many days Suresh can complete the remaining work?

Solution: