**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 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 =2

**Question 7: **The standard form of 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) – (b) 0 (c) (d)-2

**Solution :**

**Fill in the Blanks**

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

**Question 13: ** is a rational number

**Solution :**

The given rational number is a negative number, because its numerator is negative integer.

Hence, 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 is______ .

**Solution :**

**Question 16: **The standard form of is______ .

**Solution :**

**Question 17: **On a number line, is to the______of Zero(0).

**Solution :**

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

*Note***Question 18: **On a number line, is to the______of Zero(0).

**Solution :**

On a number line, 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: ** is _____ than .

**Solution :**

**Question 20: ** is _____ than 0.

**Solution :**

**Question 21: ** and represent_______ rational numbers.

**Solution :**

**Question 22: ** and represent_______ rational numbers.

**Solution :**

**Question 23: **Additive inverse of is_____.

**Solution :**

Since, additive inverse is the negative of a number.

Hence, additive inverse of is .

**Note**Additive inverse is a number, which when added to a given number, we get result as zero.

**Question 24: ** + = _____.

**Solution :**

**Question 25: ** + = ______.

**Solution :**

**Question 26: ** = _____.

**Solution :**

**Question 27: ** = _____.

**Solution :**

**Question 28: **Given,

**Solution :**

**Question 29: ** =

**Solution :**

**Question 30: ** – = _____

**Solution :**

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

**Question 31: **

**Solution :**

**Question 32: **

**Solution :**

**Question 33: **

**Solution :**

**Question 34: **

**Solution :**

**Question 35: **

**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: ** =________

**Solution :**

**Question 39: ** =_________

**Solution :**

**Question 40: ** =_________

**Solution :**

Hence, =0

Because, zero multiplies by any number result is zero.

**Question 41: **_____ x =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

**Solution :**

**Question 44: **If p and q are positive integers, then is a______ rational number and 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 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 , 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:

……

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 is a rational number and m is a non-zero integer, then

**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 is a rational number and m is a non-zero common divisor of p and q, then

**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 , where q ≠ 0. It is because any number divided by zero is not defined.

**Question 54: **If is a rational number and m is a non-zero integer, then is a rational number not equivalent to .

**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.

**Question 56: **All decimal numbers are also rational numbers.

**Solution**

**True**

All decimal numbers are also rational numbers, it is true.

**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: ** is equivalent to

**Solution :**

**True**

**Question 62: **The rational number lies to the right of zero on the number line.

**Solution :**

**False**

**Question 63: **The rational number and 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. 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.

**Solution :**

**Question 68: **Express 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)

(ii)

**Solution :**

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

**Solution :**

**Question 71: **Are the rational numbers and equivalent? Give reason.

**Solution :**

**Question 72: **Arrange the rational numbers in ascending order.

**Solution :**

**Question 73: **Represent the following rational numbers on a number line.

**Solution :**

**Question 74: **If = find the value of x.

**Solution :**

**Question 75: **Give three rational numbers equivalent to

(i)

(ii)

**Solution :**

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

**Solution :**

**Question 77: **List four rational numbers between and .

**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 = and y = , 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: ** =

**Solution :**

**Question 90: **Given that, and 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 to obtain the nearest natural number?

**Solution :**

**Question 96: **What should be subtracted from to obtain the nearest integer?

**Solution :**

**Question 97: **What should be multiplied with to obtain the nearest integer?

**Solution :**

**Question 98: **What should be divided by 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 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 can have? What is the largest 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 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 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 = What error did the student make?

**Solution :**