Q-1: Which of the following will not represent zero?
(a) 1 + 0
(b) $0\times0$
(c) $\cfrac{0}{2}$
(d) $\cfrac{(10 – 10)}{2}$
Solution:
(a) 1 + 0 = 1
Hence, it does not represent zero.
(b) $0\times0=0$
Hence, it represents zero.
(c) $\cfrac{0}{2} = 0$
Hence, it represents zero.
(d) $\cfrac{(10 – 10)}{2} = \cfrac{0}{2} = 0$
Hence, it represents zero.
Q-2: If the product of two whole numbers is zero, can we say that one or both of them will be zero? Justify through examples.
Solution:
If the product of two whole numbers is zero, definitely one of them is zero.
Example:
$0\times4=0$ or
$2\times0=0$ or
$0\times0=0$
$\therefore$ If the product of two whole numbers is zero then we can say that one or both of them will be zero.
Q-3: If the product of two whole numbers is 1, can we say that one or both of them will be 1? Justify through examples.
Solution:
If the product of two whole numbers is 1, both numbers should be equal to 1
Example: 1 × 1 = 1
But 1 × 5 = 5 and is not equal to 1
Hence, it’s clear that the product of two whole numbers will be 1, but only in situations where both numbers are 1.
Q-4: Find using distributive property:
(a) 728 × 101
(b) 5437 × 1001
(c) 824 × 25
(d) 4275 × 125
(e) 504 × 35
Solution:
(a) 728 × 101
= 728 × (100 + 1)
= 728 × 100 + 728 × 1
= 72800 + 728
= 73528
(b) 5437 × 1001
= 5437 × (1000 + 1)
= 5437 × 1000 + 5437 × 1
= 5437000 + 5437
= 5442437
(c) 824 × 25
= (800 + 24) × 25
= (800 + 25 – 1) × 25
= 800 × 25 + 25 × 25 – 1 × 25
= 20000 + 625 – 25
= 20000 + 600
= 20600
(d) 4275 × 125
= (4000 + 200 + 100 – 25) × 125
= (4000 × 125 + 200 × 125 + 100 × 125 – 25 × 125)
= 500000 + 25000 + 12500 – 3125
= 534375
(e) 504 × 35
= (500 + 4) × 35
= 500 × 35 + 4 × 35
= 17500 + 140
= 17640
Q-5: Study the pattern:
1 × 8 + 1 = 9
1234 × 8 + 4 = 9876
12 × 8 + 2 = 98
12345 × 8 + 5 = 98765
123 × 8 + 3 = 987
Write the next two steps. Can you say how the pattern works?
(Hint: 12345 = 11111 + 1111 + 111 + 11 + 1)
Solution:
Working pattern:
(1) x 8 + 1 = 9
(12) x 8 + 2 = (11 + 1) x 8 + 2 = 98
(123) x 8 + 3 = (111 + 11 + 1) x 8 + 3 = 987
(1234) x 8 + 4 = (1111 + 111 + 11 + 1) x 8 + 4 = 9876
(12345) x 8 + 5 = (11111 + 1111 + 111 + 11 + 1) x 8 + 5 = 98765
$\therefore$ the next two steps are
(123456) x 8 + 6 = (111111 + 11111 + 1111 + 111 + 11 + 1) x 8 + 6 = 987654
(1234567) x 8 + 7 = (1111111 + 111111 + 11111 + 1111 + 111 + 11 + 1) x 8 + 6 = 987654
