NCERT Exemplar Problems Class 7 Maths – Rational Numbers

Question 1:
A rational number is defined as a number that can be expressed in the form p/q, where p and q are integers and
(a) q = 0 (b) q = 1
(c) q ≠ 1 (d) q ≠ 0
Solution :
(d) By definition, a number that can be expressed in the form of p/q, where p and q are integers and q≠0, is called a rational number.

Question 2:
Which of the following rational numbers is positive?

Solution :
(c) We know that, when numerator and denominator of a rational number, both are negative,
it is a positive rational number.
Hence, among the given rational numbers $\left( \frac { -3 }{ -4 }\right)$ is positive.

Question 3:
Which of the following rational numbers is negative?

Solution :

Question 4:
In the standard form of a rational number, the common factor of numerator and denominator is always
(a) 0 (b) 1 c) -2 (d)2
Solution :
(b) By definition, in the standard form of a rational number, the common factor of numerator and denominator is always1
Note: Common factor means, a number which divides both the given two numbers.

Question 5:
Which of the following rational numbers is equal to its reciprocal?
(a) 1 (b) 2 c) 1/2 (d)0
Solution :

Question 6:
The reciprocal of 1/2 is
(a) 3 (b) 2 c) -1 (d)0
Solution :
(b) Reciprocal of $\frac { 1 }{ 2 } =\frac { 1 }{ \frac { 1 }{ 2 } }$ =2

Question 7:
The standard form of $\frac { -48 }{ 60 }$ is

Solution :

Question 8:
Which of the following is equivalent to 4/5 ?

Solution :

Note: If the numerator and denominator of a rational number is multiplied/divided by a non-zero integer, then the result we get, is equivalent rational number.

Question 9:
How many rational numbers are there between two rational numbers?
(a) 1 (b) 0
(c) unlimited (d) 100
Solution :
(c) There are unlimited numbers between two rational numbers.

Question 10:
In the standard form of a rational number, the denominator is always a
(a) 0 (b) negative integer
(c) positive integer (d) 1
Solution :
(c) By definition, a rational number is said to be in the standard form, if its denominator is a positive integer.

Question 11:
To reduce a rational number to its standard form, we divide its numerator and denominator by their
(a) LCM (b) HCF
(c) product (d) multiple
Solution :
(b) To reduce a rational number to its standard form, we divide its numerator and denominator by their HCF.

Question 12:
Which is greater number in the following?
(a) – $\frac { 1 }{ 2 }$ (b) 0 (c) $\frac { 1 }{ 2 }$ (d)-2
Solution :

Fill in the Blanks

In questions 13 to 46, fill in the blanks to make the statements true.

Question 13:
$\frac { -3 }{ 8 }$ is a rational number
Solution :
The given rational number $\frac { -3 }{ 8 }$ is a negative number, because its numerator is negative integer.
Hence, $\frac { -3 }{ 8 }$ is a negative rational number.

Question 14:
is a____rational number.
Solution :
The given rational number 1 is positive number, because its numerator and denominator are positive integer.
Hence, 1 is a positive rational number.

Question 15:
The standard form of $\frac { -8 }{ 36 }$ is______ .
Solution :

Question 16:
The standard form of $\frac { 18 }{ -24}$ is______ .
Solution :

Question 17:
On a number line, $\frac { -1 }{ 2 }$ is to the______of Zero(0).
Solution :

Note All the negative numbers lie on the left side of zero on the number line

Question 18:
On a number line, $\frac { 3}{ 4}$ is to the______of Zero(0).
Solution :
On a number line, $\frac { 3 }{ 4 }$ is to the right of Zero(0).

Note All the positive numbers lie on the right side of zero on the number line.

Question 19:
$\frac { -1 }{ 2 }$ is _____ than $\frac { 1 }{ 5 }$ .
Solution :

Question 20:
$\frac { -3 }{ 5 }$ is _____ than 0.
Solution :

Question 21:
$\frac { -16 }{ 24 }$ and $\frac { 20 }{ -16 }$ represent_______ rational numbers.
Solution :

Question 22:
$\frac { -27 }{ 45 }$ and $\frac { -3 }{ 5 }$ represent_______ rational numbers.
Solution :

Question 23:
Additive inverse of $\frac { 2 }{ 3 }$ is_____.
Solution :
Since, additive inverse is the negative of a number.
Hence, additive inverse of $\frac { 2 }{ 3 }$ is $\frac { -2 }{ 3 }$ .
Note Additive inverse is a number, which when added to a given number, we get result as zero.

Question 24:
$\frac { -3 }{ 5 }$ + $\frac { 2 }{ 5 }$ = _____.
Solution :

Question 25:
$\frac { -5 }{ 6 }$ + $\frac { -1 }{ 6 }$ = ______.
Solution :

Question 26:
$\frac { 3 }{ 4 }\times \left( \frac { -2 }{ 3 }\right)$ = _____.
Solution :

Question 27:
$\frac { -5 }{ 3 }\times \left( \frac { -3 }{ 5 }\right)$ = _____.
Solution :

Question 28:
Given, $\frac { -6 }{ 7 } =\bar { 42 }$
Solution :

Question 29:
$\frac { 1 }{ 2 }$ = $\frac { 6 }{ - }$
Solution :

Question 30:
$\frac { -2 }{ 9 }$ – $\frac { 7 }{ 9 }$ = _____
Solution :

In questions 31 to 35, fill in the boxes with the correct symbol ‘<‘,'<‘ or ‘=’.

Question 31:
$\frac { 7 }{ -8 } \Box \frac { 8 }{ 9 }$
Solution :

Question 32:
$\frac { 3 }{ 7 } \Box \frac { -5 }{ 6 }$
Solution :

Question 33:
$\frac { 5 }{ 6 } \Box \frac { 4 }{ 8 }$
Solution :

Question 34:
$\frac { -9 }{ 7 } <\frac { 4 }{ -7 }$
Solution :

Question 35:
$\frac { 8 }{ 8 } \Box \frac { 2 }{ 2 }$
Solution :

Question 36:
The reciprocal of_______ does not exist.
Solution :
The reciprocal of zero does not exist, as reciprocal of 0 is 1/0, which is not defined.

Question 37:
The reciprocal of 1 is_______
Solution :
The reciprocal of 1=1/1
Hence, the reciprocal of 1 is 1.

Question 38:
$\frac { -3 }{ 7 } \div \left( \frac { -7 }{ 3 }\right)$ =________
Solution :

Question 39:
$0\div \left( \frac { -5 }{ 6 }\right)$ =_________
Solution :

Question 40:
$0\times \left( \frac { -5 }{ 6 }\right)$ =_________
Solution :
Hence, $0\times \left( \frac { -5 }{ 6 }\right)$ =0
Because, zero multiplies by any number result is zero.

Question 41:
_____ x $\left( \frac { -2 }{ 5 }\right)$ =1
Solution :

Question 42:
The standard form of rational number – 1 is_______.
Solution :
∴ HCF of given rational number -1 is 1.
For standard form = -1 +1 = -1
Hence, the standard form of rational number -1 is -1.

Question 43:
If m is a common divisor of a and b, then $\frac { a }{ b } =\frac { a+m }{ - }$
Solution :

Question 44:
If p and q are positive integers, then $\frac { p }{ q }$ is a______ rational number and $\frac { p }{ -q }$ is a_____ rational number.
Solution :
if p and q are positive integers, then p/q is a positive rational number, because both numerator and denominator are positive and $\frac { p }{ -q }$ is a negative rational number, because denominator is in negative

Question 45:
Two rational numbers are said to be equivalent or equal, if they have the same_______form.
Solution :
Two rational numbers are said to be equivalent or equal, if they have the same simplest form.

Question 46:
If p/q is a rational number, then q cannot be_____________
Solution :
By definition, if B is a rational number, then q cannot be zero.

True/False

In questions 47 to 65, state whether the following statements are True or False.

Question 47:
Every natural number is a rational number, but every rational number need not be a natural number.
Solution :
True
e.g. 1/2 is a rational number, but not a natural number.

Question 48:
Zero is a rational number.
Solution :
True
e.g. Zero can be written as 0 = 0/1. We know that, a number of the form $\frac { p }{ q }$ , where p, q are integers and q ≠ 0 is a rational number. So, zero is a rational number.

Question 49:
Every integer is a rational number but every rational number need not be an integer.
Solution :
True
Integers…. – 3,-2,-1, 0,1,2, 3,…
Rational numbers:
$1,\frac { -1 }{ 2 } ,0,\frac { 1 }{ 2 } 1,\frac { 3 }{ 2 } ,$ ……
Hence, every integer is rational number, but every rational number is not an integer.

Question 50:
Every negative integer is not a negative rational number.
Solution :
False
Because all the integers are rational numbers, whether it is negative/positive but vice-versa is not true.

Question 51:
If $\frac { p }{ q }$ is a rational number and m is a non-zero integer, then
$\frac { p }{ q } =\frac { p\times m }{ q\times m }$
Solution :
True
e.g. Let m = 1,2, 3,…

Note: When both numerator and denominator of a rational number are multiplied/divide by a same non-zero number, then we get the same rational number

Question 52:
If $\frac { p }{ q }$ is a rational number and m is a non-zero common divisor of p and q, then
$\frac { p }{ q } =\frac { p\div m }{ q\div m }$
Solution :

Question 53:
In a rational number, denominator always has to be a non-zero integer.
Solution :
Basic definition of the rational number is that, it is in the form of $\frac { p }{ q }$ , where q ≠ 0. It is because any number divided by zero is not defined.

Question 54:
If $\frac { p }{ q }$ is a rational number and m is a non-zero integer, then $\frac { p\times m }{ q\times m }$ is a rational number not equivalent to $\frac { p }{ q }$ .
Solution :

Question 55:
Sum of two rational numbers is always a rational number.
Solution :
True
Sum of two rational numbers is always a rational number, it is true.
$\frac { 1 }{ 2 } +\frac { 2 }{ 3 } =\frac { 3+4 }{ 6 } =\frac { 7 }{ 6 }$

Question 56:
All decimal numbers are also rational numbers.
Solution
True
All decimal numbers are also rational numbers, it is true.
$0.6=\frac { 6 }{ 10 } =\frac { 3 }{ 5 }$

Question 57:
The quotient of two rationals is always a rational number.
Solution :
False
The quotient of two rationals is not always a rational number.
e.g. 1/0.

Question 58:
Every fraction is a rational number.
Solution :
True
Every fraction is a rational number but vice-versa is not true.

Question 59:
Two rationals with different numerators can never be equal.
Solution :
False

Question 60:
8 can be written as a rational number with any integer as denominator.
Solution :
8 can be written as a rational number with any integer as denominator, it is false because 8 can be written as a rational number with 1 as denominator i.e.8/1.

Question 61:
$\frac { 4 }{ 6 }$ is equivalent to $\frac { 2 }{ 3 }$
Solution :
True

Question 62:
The rational number $\frac { -3 }{ 4 }$ lies to the right of zero on the number line.
Solution :
False

Question 63:
The rational number $\frac { -12 }{ 15 }$ and $\frac { -7 }{ 17 }$ are on the opposite sides of zero on the number line.
Solution :

Question 64:
Every rational number is a whole number.
Solution :
False
e.g. $\frac { -7 }{ 8 }$ is a rational number, but it is not a whole number, because whole numbers are 0,1,2….

Question 65:
Zero is the smallest rational number.
Solution :
False
Rational numbers can be negative and negative rational numbers are smaller than zero.

Question 66:
Match the following:

Solution :

Question 67:
Write each of the following rational numbers with positive denominators.
$\frac { 5 }{ -8 } ,+\frac { 15 }{ 28 } \frac { -17 }{ 13 }$
Solution :

Question 68:
Express $\frac { 3 }{ 4 }$ as a rational number with denominator:
(a)36 (b) — 80
Solution :

Question 69:
Reduce each of the following rational numbers in its lowest form
(i) $\frac {- 60 }{ 72 }$
(ii) $\frac {91 }{ -364 }$
Solution :

Question 70:
Express each of the following rational numbers in its standard form

Solution :

Question 71:
Are the rational numbers $\frac {-8 }{ 28 }$ and $\frac {32 }{ -12 }$ equivalent? Give reason.
Solution :

Question 72:
Arrange the rational numbers $\frac { -7 }{ 10 } ,\frac { 5 }{ -8 } ,\frac { 2 }{ -3 } ,\frac { -1 }{ 4 } ,\frac { -3 }{ 5 }$ in ascending order.
Solution :

Question 73:
Represent the following rational numbers on a number line.
$\frac { 3 }{ 8 } ,\frac { -7 }{ 3 } ,\frac { 22 }{ -6 }$
Solution :

Question 74:
If $\frac { -5 }{ 7 }$ = $\frac {\times }{ 28 }$ find the value of x.
Solution :

Question 75:
Give three rational numbers equivalent to
(i) $\frac { -3 }{ 4 }$
(ii) $\frac { 7 }{ 11 }$
Solution :

Question 76:
Write the next three rational numbers to complete the pattern:

Solution :

Question 77:
List four rational numbers between $\frac { 5 }{ 7 }$ and $\frac { 7 }{ 8 }$ .
Solution :

Question 78:
Find the sum of

Solution :

Question 79:
Solve:

Solution :

Question 80:
Find the product of

Solution :

Question 81:
Simplify:

Solution :

Question 82:
Simplify:

Solution :

Question 83:
Which is greater in the following?

Solution :

Question 84:
Write a rational number in which the numerator is less than ‘-7 x 11′ and the denominator is greater than ’12+ 4’.
Solution :

Question 85:
If x = $\frac { 1 }{ 10 }$ and y = $\frac { -3 }{ 8 }$ , then evaluate x + y, x-y, xxy and x ÷ y.
Solution :

Question 86:
Find the reciprocal of the following:

Solution :

Question 87:
Complete the following table by finding the sums.

Solution :

Question 88:
Write each of the following numbers in the form p/q, where p and q are integers.
(a) six-eighths (b) three and half
(c) opposite of 1 (d) one-fourth
(e) zero (f) opposite of three-fifths
Solution :

Question 89:
$\frac { p }{ q }$ = $\frac { \Box }{ \Box }$
Solution :

Question 90:
Given that, $\frac { p }{ q }$ and $\frac { r }{ s }$ are two rational numbers with different denominators and both of them are in standard form. To compare these rational numbers, we say that

Solution :

Question 91:
In each of the following cases, write the rational number whose numerator and denominator are respectively as under:
(a) 5-39 and 54-6 (b) (- 4) x 6 and 8 ÷ 2
(c) 35 ÷ (- 7) and 35 -18 (d) 25 +15 and 81÷40
Solution :

Question 92:
Write the following as rational numbers in their standard forms.

Solution :

Question 93:
Find a rational number exactly halfway between

Solution :

Question 94:

Solution :

Question 95:
What should be added to $\frac { -1 }{ 2 }$ to obtain the nearest natural number?
Solution :

Question 96:
What should be subtracted from $\frac { -2 }{ 3 }$ to obtain the nearest integer?
Solution :

Question 97:
What should be multiplied with $\frac { -5 }{ 8 }$ to obtain the nearest integer?
Solution :

Question 98:
What should be divided by $\frac { -1 }{ 2 }$ to obtain the greatest negative integer?
Solution :

Question 99:
From a rope 68 m long, pieces of equal size are cut. If length of one piece is $4\frac { 1 }{ 4 }$ m, find the number of such pieces.
Solution :

Question 100:
If 12 shirts of equal size can be prepared from 27 m cloth, what is length of cloth required for each shirt?
Solution :

Question 101:
Insert 3 equivalent rational numbers between

Solution :

Question 102:
Put the (✓), wherever applicable

Solution :

Question 103:
‘o’ and ‘b’ are two different numbers taken from the numbers 1-50. What is the largest value that $\frac { a-b }{ a+b }$ can have? What is the largest $\frac { a+b }{ a-b }$ can have?
Solution :

Question 104:
150 students are studying English, Maths or both. 62% of the students are studying English and 68% are studying Maths. How many students are studying both?
Solution :

Question 105:
A body floats $\frac { 2 }{ 9 }$ of its volume above the surface. What is the ratio of the body submerged volume to its exposed volume? Rewrite it as a rational number.
Solution :

In questions 106 to 109, find the odd one out of the following and give reason.

Question 106:

Solution :

Question 107:

Solution :

Question 108:

Solution :

Question 109:

Solution :
From the above given rational numbers, we can see that $\frac { -7 }{ 3 }$ is in its lowest form while others have common factor in numerator and denominator.

Question 110:
What’s the Error? Chhaya simplified a rational number is this manner $\frac { -25}{ -30 }$ = $\frac { -5}{ -6 }$ What error did the student make?
Solution :