Fill in the blanks:
(i) 42 × 0 = ...............
(ii) 592 × 1 = ...............
(iii) 328 × 573 = ............... × 328
(iv) 229 × ............... = 578 × 229
(v) 32 × 15 = 32 × 6 + 32 × 7 + 32 × ...............
(vi) 23 × 56 = 20 × 56 + ............... × 56
(vii) 83 × 54 + 83 × 16 = 83 × (...............) = 83 × ............... = ...............
(viii) 98 × 273 - 75 × 273 = (...............) × 273 = ............... × 273
Answer
(i) Any number multiplied by 0 gives 0.
42 × 0 = 0.
(ii) Any number multiplied by 1 gives the number itself.
592 × 1 = 592.
(iii) By the Commutative Property of Multiplication: a × b = b × a.
328 × 573 = 573 × 328.
(iv) By the Commutative Property of Multiplication: a × b = b × a.
229 × 578 = 578 × 229.
(v) Since 6 + 7 + 2 = 15, we have:
32 × 15 = 32 × 6 + 32 × 7 + 32 × 2.
(vi) Since 23 = 20 + 3, we have:
23 × 56 = 20 × 56 + 3 × 56.
(vii) Using the Distributive Law of Multiplication over Addition:
83 × 54 + 83 × 16 = 83 × (54 + 16) = 83 × 70 = 5810.
(viii) Using the Distributive Law of Multiplication over Subtraction:
98 × 273 - 75 × 273 = (98 - 75) × 273 = 23 × 273.
Evaluate (using distributive property):
(i) 984 × 102
(ii) 385 × 1004
(iii) 446 × 10002
Answer
(i) 984 × 102
⇒ 984 × (100 + 2)
Using the Distributive Law of Multiplication over Addition:
⇒ 984 × 100 + 984 × 2
⇒ 98400 + 1968
⇒ 100368.
Hence, 984 × 102 = 100368.
(ii) 385 × 1004
⇒ 385 × (1000 + 4)
Using the Distributive Law of Multiplication over Addition:
⇒ 385 × 1000 + 385 × 4
⇒ 385000 + 1540
⇒ 386540.
Hence, 385 × 1004 = 386540.
(iii) 446 × 10002
⇒ 446 × (10000 + 2)
Using the Distributive Law of Multiplication over Addition:
⇒ 446 × 10000 + 446 × 2
⇒ 4460000 + 892
⇒ 4460892.
Hence, 446 × 10002 = 4460892.
Evaluate using properties:
(i) 548 × 98
(ii) 924 × 988
Answer
(i) 548 × 98
⇒ 548 × (100 - 2)
Using the Distributive Law of Multiplication over Subtraction:
⇒ 548 × 100 - 548 × 2
⇒ 54800 - 1096
⇒ 53704.
Hence, 548 × 98 = 53704.
(ii) 924 × 988
⇒ 924 × (1000 - 12)
Using the Distributive Law of Multiplication over Subtraction:
⇒ 924 × 1000 - 924 × 12
⇒ 924000 - 11088
⇒ 912912.
Hence, 924 × 988 = 912912.
Evaluate using properties:
(i) 679 × 8 + 679 × 2
(ii) 284 × 12 - 284 × 2
(iii) 55873 × 94 + 55873 × 6
(iv) 7984 × 15 - 7984 × 5
(v) 8324 × 1945 - 8324 × 945
(vi) 3333 × 987 + 13 × 3333
Answer
(i) 679 × 8 + 679 × 2
Using the Distributive Law of Multiplication over Addition: a × b + a × c = a × (b + c).
⇒ 679 × (8 + 2)
⇒ 679 × 10
⇒ 6790.
Hence, 679 × 8 + 679 × 2 = 6790.
(ii) 284 × 12 - 284 × 2
Using the Distributive Law of Multiplication over Subtraction: a × b - a × c = a × (b - c).
⇒ 284 × (12 - 2)
⇒ 284 × 10
⇒ 2840.
Hence, 284 × 12 - 284 × 2 = 2840.
(iii) 55873 × 94 + 55873 × 6
Using the Distributive Law of Multiplication over Addition:
⇒ 55873 × (94 + 6)
⇒ 55873 × 100
⇒ 5587300.
Hence, 55873 × 94 + 55873 × 6 = 5587300.
(iv) 7984 × 15 - 7984 × 5
Using the Distributive Law of Multiplication over Subtraction:
⇒ 7984 × (15 - 5)
⇒ 7984 × 10
⇒ 79840.
Hence, 7984 × 15 - 7984 × 5 = 79840.
(v) 8324 × 1945 - 8324 × 945
Using the Distributive Law of Multiplication over Subtraction:
⇒ 8324 × (1945 - 945)
⇒ 8324 × 1000
⇒ 8324000.
Hence, 8324 × 1945 - 8324 × 945 = 8324000.
(vi) 3333 × 987 + 13 × 3333
⇒ 3333 × 987 + 3333 × 13
Using the Distributive Law of Multiplication over Addition:
⇒ 3333 × (987 + 13)
⇒ 3333 × 1000
⇒ 3333000.
Hence, 3333 × 987 + 13 × 3333 = 3333000.
Find the product of the:
(i) greatest number of three digits and smallest number of five digits.
(ii) greatest number of four digits and the greatest number of three digits.
Answer
(i) The greatest number of three digits = 999.
The smallest number of five digits = 10000.
Product = 999 × 10000 = 9990000.
Hence, the required product is 9990000.
(ii) The greatest number of four digits = 9999.
The greatest number of three digits = 999.
Product = 9999 × 999
⇒ 9999 × (1000 - 1)
⇒ 9999 × 1000 - 9999 × 1
⇒ 9999000 - 9999
⇒ 9989001.
Hence, the required product is 9989001.
Fill in the blanks:
(i) (437 + 3) × (400 - 3) = 397 × ............... = ...............
(ii) 66 + 44 + 22 = 11 × (...............) = 11 × ............... = ...............
Answer
(i) Here, 437 + 3 = 440 and 400 - 3 = 397.
So, (437 + 3) × (400 - 3) = 440 × 397 = 397 × 440 [by commutative property]
⇒ 397 × 440 = 174680.
(ii) Since 66 = 11 × 6, 44 = 11 × 4 and 22 = 11 × 2, using the distributive property:
⇒ 66 + 44 + 22 = 11 × 6 + 11 × 4 + 11 × 2
⇒ 11 × (6 + 4 + 2) = 11 × 12 = 132.