Mathematics solution NCERT
Class 9 - Chapter 3: The World of Numbers
Q1: Prove that the following rational numbers are equal.
(i) 2/3 and 4/6
4/6 = (4 ÷ 2)/(6 ÷ 2)
= 2/3
Therefore,
2/3 = 4/6
(ii) 5/4 and 10/8
10/8 = (10 ÷ 2)/(8 ÷ 2)
= 5/4
Therefore,
5/4 = 10/8
(iii) -3/5 and -6/10
-6/10 = (-6 ÷ 2)/(10 ÷ 2)
= -3/5
Therefore,
-3/5 = -6/10
(iv) 9/3 and 3
9/3 = 3
Therefore,
9/3 = 3
Q2: Find the Sum
(i) 2/5 + 3/10
LCM of 5 and 10 = 10
2/5 = 4/10
4/10 + 3/10 = 7/10
Answer: 7/10
(ii) 7/12 + 5/8
LCM of 12 and 8 = 24
7/12 = 14/24
5/8 = 15/24
14/24 + 15/24 = 29/24
Answer: 29/24
(iii) -4/7 + 3/14
LCM of 7 and 14 = 14
-4/7 = -8/14
-8/14 + 3/14 = -5/14
Answer: -5/14
Q3: Find the Difference
(i) 5/6 - 1/4
LCM of 6 and 4 = 12
5/6 = 10/12
1/4 = 3/12
10/12 - 3/12 = 7/12
Answer: 7/12
(ii) 11/8 - 3/4
3/4 = 6/8
11/8 - 6/8 = 5/8
Answer: 5/8
(iii) -7/9 - (-2/3)
= -7/9 + 2/3
= -7/9 + 6/9
= -1/9
Answer: -1/9
Q4: Find the Product
(i) 2/3 × 3/10
(2 × 3)/(3 × 10)
= 6/30
= 1/5
Answer: 1/5
(ii) 7/11 × 5/8
(7 × 5)/(11 × 8)
= 35/88
Answer: 35/88
(iii) -4/7 × 5/14
(-4 × 5)/(7 × 14)
= -20/98
= -10/49
Answer: -10/49
Q5: Find the Quotient
(i) 2/3 ÷ 3/10
2/3 × 10/3
= 20/9
Answer: 20/9
(ii) 7/11 ÷ 5/8
7/11 × 8/5
= 56/55
Answer: 56/55
(iii) -4/7 ÷ 5/14
-4/7 × 14/5
= -56/35
= -8/5
Answer: -8/5
Q6: Show that (1/2 + 3/4) × 8/3 = (1/2 × 8/3) + (3/4 × 8/3)
LHS:
(1/2 + 3/4) × 8/3
= (2/4 + 3/4) × 8/3
= 5/4 × 8/3
= 10/3
RHS:
(1/2 × 8/3) + (3/4 × 8/3)
= 4/3 + 2
= 4/3 + 6/3
= 10/3
LHS = RHS
Hence Proved.
Q7: Simplify using the distributive property
7/9 (6/7 - 3/4)
Using distributive property:
= (7/9 × 6/7) - (7/9 × 3/4)
= 6/9 - 21/36
= 2/3 - 7/12
= 8/12 - 7/12
= 1/12
Answer: 1/12
Q8: Find the rational number x such that:
(5/6)(x + 3/5) = (5/6)x + 1/2
Solution:
Expand the left side:
(5/6)x + (5/6 × 3/5)
(5/6)x + 1/2
The equation becomes:
(5/6)x + 1/2 = (5/6)x + 1/2
Both sides are identical.
This means the equation is true for every value of x.
Answer: Any rational number can be the value of x.
Hence, x can be any rational number.