Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.3 Solutions: The World of Numbers
Rational numbers help us work with fractions, negative fractions and integers in one system. Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.3 Solutions focus on this part of The World of Numbers, where students practise equivalent rational numbers and the four basic operations on fractions.
In Ganita Manjari Class 9 Chapter 3 Exercise 3.3, students solve questions on equality of fractions, sums, differences, products, quotients and the distributive property. These Class 9 Maths Chapter 3 Exercise 3.3 Solutions explain every step clearly so students can revise rational numbers Class 9 concepts in a simple, textbook-based way.
NCERT Solutions for Class 9 Maths Chapter 3 Exercise Set 3.3
1. Prove that the following rational numbers are equal:
(i) 2/3 and 4/6
Solution:
Two rational numbers a/b and c/d are equal if:
a × d = b × c
For 2/3 and 4/6:
2 × 6 = 12
3 × 4 = 12
Since both products are equal:
2/3 = 4/6
Answer: The rational numbers 2/3 and 4/6 are equal.
(ii) 5/4 and 10/8
Solution:
For 5/4 and 10/8:
5 × 8 = 40
4 × 10 = 40
Since both products are equal:
5/4 = 10/8
Answer: The rational numbers 5/4 and 10/8 are equal.
(iii) −3/5 and −6/10
Solution:
For −3/5 and −6/10:
(−3) × 10 = −30
5 × (−6) = −30
Since both products are equal:
−3/5 = −6/10
Answer: The rational numbers −3/5 and −6/10 are equal.
(iv) 9/3 and 3
Solution:
Write 3 as 3/1.
Now compare 9/3 and 3/1.
9 × 1 = 9
3 × 3 = 9
Since both products are equal:
9/3 = 3
Answer: The rational numbers 9/3 and 3 are equal.
2. Find the sum:
(i) 2/5 + 3/10
Solution:
Find a common denominator.
LCM of 5 and 10 is 10.
2/5 = 4/10
So,
2/5 + 3/10 = 4/10 + 3/10
= 7/10
Answer: 7/10
(ii) 7/12 + 5/8
Solution:
LCM of 12 and 8 is 24.
7/12 = 14/24
5/8 = 15/24
So,
7/12 + 5/8 = 14/24 + 15/24
= 29/24
Answer: 29/24
(iii) −4/7 + 3/14
Solution:
LCM of 7 and 14 is 14.
−4/7 = −8/14
So,
−4/7 + 3/14 = −8/14 + 3/14
= −5/14
Answer: −5/14
3. Find the difference:
(i) 5/6 − 1/4
Solution:
LCM of 6 and 4 is 12.
5/6 = 10/12
1/4 = 3/12
So,
5/6 − 1/4 = 10/12 − 3/12
= 7/12
Answer: 7/12
(ii) 11/8 − 3/4
Solution:
Convert 3/4 to denominator 8.
3/4 = 6/8
So,
11/8 − 3/4 = 11/8 − 6/8
= 5/8
Answer: 5/8
(iii) −7/9 − (−2/3)
Solution:
Subtracting a negative rational number is the same as adding the corresponding positive rational number.
−7/9 − (−2/3) = −7/9 + 2/3
Convert 2/3 to denominator 9.
2/3 = 6/9
So,
−7/9 + 6/9 = −1/9
Answer: −1/9
4. Find the product:
(i) 2/3 × 3/10
Solution:
Multiply numerators and denominators.
2/3 × 3/10 = (2 × 3) / (3 × 10)
Cancel the common factor 3.
= 2/10
= 1/5
Answer: 1/5
(ii) 7/11 × 5/8
Solution:
7/11 × 5/8 = (7 × 5) / (11 × 8)
= 35/88
Answer: 35/88
(iii) −4/7 × 5/14
Solution:
−4/7 × 5/14 = (−4 × 5) / (7 × 14)
= −20/98
Simplify by dividing numerator and denominator by 2.
= −10/49
Answer: −10/49
5. Find the quotient:
(i) 2/3 ÷ 3/10
Solution:
To divide by a fraction, multiply by its reciprocal.
2/3 ÷ 3/10 = 2/3 × 10/3
= 20/9
Answer: 20/9
(ii) 7/11 ÷ 5/8
Solution:
7/11 ÷ 5/8 = 7/11 × 8/5
= 56/55
Answer: 56/55
(iii) −4/7 ÷ 5/14
Solution:
−4/7 ÷ 5/14 = −4/7 × 14/5
Cancel 14 ÷ 7 = 2.
= −4 × 2 / 5
= −8/5
Answer: −8/5
6. Show that: (1/2 + 3/4) × 8/3 = 1/2 × 8/3 + 3/4 × 8/3.
Solution:
First solve the left-hand side.
LHS = (1/2 + 3/4) × 8/3
Convert 1/2 to denominator 4.
1/2 = 2/4
So,
LHS = (2/4 + 3/4) × 8/3
= 5/4 × 8/3
= 40/12
= 10/3
Now solve the right-hand side.
RHS = 1/2 × 8/3 + 3/4 × 8/3
= 8/6 + 24/12
= 4/3 + 2
= 4/3 + 6/3
= 10/3
Since:
LHS = RHS = 10/3
Answer: Hence proved.
7. Simplify the following using the distributive property: 7/9 (6/7 − 3/4).
Solution:
Use the distributive property:
a(b − c) = ab − ac
So,
7/9 (6/7 − 3/4) = 7/9 × 6/7 − 7/9 × 3/4
Now simplify each term.
7/9 × 6/7 = 6/9 = 2/3
7/9 × 3/4 = 21/36 = 7/12
So,
7/9 (6/7 − 3/4) = 2/3 − 7/12
Convert 2/3 to denominator 12.
2/3 = 8/12
Therefore:
8/12 − 7/12 = 1/12
Answer: 1/12
8. Find the rational number x such that: 5/6 (x + 3/5) = 5/6 x + 1/2.
Solution:
Given equation:
5/6 (x + 3/5) = 5/6 x + 1/2
Use the distributive property on the left-hand side.
5/6 (x + 3/5) = 5/6 × x + 5/6 × 3/5
= 5x/6 + 15/30
= 5x/6 + 1/2
So, the left-hand side becomes:
5x/6 + 1/2
This is exactly the same as the right-hand side:
5/6 x + 1/2
Therefore, the equation is true for every rational number x.
Answer: The equation is true for all rational numbers x.
Related Links – Class 9 Maths Chapter 3