Q1. Prove that the following rational numbers are equal: (i) 2/3 and 4/6 (ii) 5/4 and 10/8 (iii) −3/5 and −6/10 (iv) 9/3 and 3
Answer:
Solution:
Two rational numbers a/b and c/d are equal if ad = bc.
(i) 2/3 and 4/6: 2 × 6 = 12, 3 × 4 = 12, so they are equal.
(ii) 5/4 and 10/8: 5 × 8 = 40, 4 × 10 = 40, so they are equal.
(iii) −3/5 and −6/10: (−3) × 10 = −30, 5 × (−6) = −30, so they are equal.
(iv) 9/3 and 3: 9/3 can be written as 9/3 and 3 as 3/1. 9 × 1 = 9, 3 × 3 = 9, so they are equal.
Answer: All pairs are equal as verified by cross-multiplication.
Q2. Find the sum: (i) 2/5 + 3/10 (ii) 7/12 + 5/8 (iii) −4/7 + 3/14
Answer:
Solution:
(i) 2/5 + 3/10
LCM of 5 and 10 = 10
= 4/10 + 3/10 = 7/10
(ii) 7/12 + 5/8
LCM of 12 and 8 = 24
= 14/24 + 15/24 = 29/24
(iii) −4/7 + 3/14
LCM of 7 and 14 = 14
= −8/14 + 3/14 = −5/14
Answer: (i) 7/10, (ii) 29/24, (iii) −5/14
Q3. Find the difference: (i) 5/6 − 1/4 (ii) 11/8 − 3/4 (iii) −7/9 − (−2/3)
Answer:
Solution:
(i) 5/6 − 1/4
LCM of 6 and 4 = 12
= 10/12 − 3/12 = 7/12
(ii) 11/8 − 3/4
LCM of 8 and 4 = 8
= 11/8 − 6/8 = 5/8
(iii) −7/9 − (−2/3)
= −7/9 + 2/3
LCM of 9 and 3 = 9
= −7/9 + 6/9 = −1/9
Answer: (i) 7/12, (ii) 5/8, (iii) −1/9
Q4. Find the product: (i) 2/3 × 3/10 (ii) 7/11 × 5/8 (iii) −4/7 × 5/14
Answer:
Solution:
(i) 2/3 × 3/10 = (2 × 3)/(3 × 10) = 6/30 = 1/5
(ii) 7/11 × 5/8 = (7 × 5)/(11 × 8) = 35/88
(iii) −4/7 × 5/14 = (−4 × 5)/(7 × 14) = −20/98 = −10/49
Answer: (i) 1/5, (ii) 35/88, (iii) −10/49
Q5. Find the quotient: (i) 2/3 ÷ 3/10 (ii) 7/11 ÷ 5/8 (iii) −4/7 ÷ 5/14
Answer:
Solution:
(i) 2/3 ÷ 3/10 = 2/3 × 10/3 = (2 × 10)/(3 × 3) = 20/9
(ii) 7/11 ÷ 5/8 = 7/11 × 8/5 = (7 × 8)/(11 × 5) = 56/55
(iii) −4/7 ÷ 5/14 = −4/7 × 14/5 = (−4 × 14)/(7 × 5) = −56/35 = −8/5
Answer: (i) 20/9, (ii) 56/55, (iii) −8/5
Q6. Show that: (1/2 + 3/4) × 8/3 = 1/2 × 8/3 + 3/4 × 8/3.
Answer:
Solution:
This is the distributive law of multiplication over addition.
LHS = (1/2 + 3/4) × 8/3
1/2 + 3/4 = 2/4 + 3/4 = 5/4
LHS = 5/4 × 8/3 = (5 × 8)/(4 × 3) = 40/12 = 10/3
RHS = 1/2 × 8/3 + 3/4 × 8/3
= 8/6 + 24/12 = 4/3 + 2 = 4/3 + 6/3 = 10/3
LHS = RHS, hence verified.
Answer: The equality holds, verifying the distributive law.
Q7. Simplify the following using the distributive property: 7/9 (6/7 − 3/4).
Answer:
Solution:
Using distributive property: a(b − c) = ab − ac
= (7/9 × 6/7) − (7/9 × 3/4)
= (42/63) − (21/36)
Simplify: 42/63 = 2/3, 21/36 = 7/12
= 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.
Answer:
Solution:
Using distributive property on LHS:
LHS = 5/6 × x + 5/6 × 3/5
= 5x/6 + (5 × 3)/(6 × 5)
= 5x/6 + 15/30
= 5x/6 + 1/2
Given: LHS = 5x/6 + 1/2 = RHS = 5x/6 + 1/2
Both sides are identical, so the equation holds for all rational numbers x.
Answer: The equation is an identity; it is true for any rational number x.