Natural Numbers and Whole Numbers

Q1: Fill in the blanks

i. \(42 \times 0 =\) ________

Step 1: Multiplying any number by 0 gives 0. \[ 42 \times 0 = 0 \] Answer: 0

ii. \(592 \times 1 =\) _______

Step 1: Multiplying any number by 1 gives the number itself. \[ 592 \times 1 = 592 \] Answer: 592

iii. \(328 \times 573 =\) ______ \(\times 328\)

Step 1: By the commutative property of multiplication, \[ 328 \times 573 = 573 \times 328 \] Answer: 573

iv. \(229 \times\) _______ \(= 578 \times 229\)

Step 1: Again, by commutativity, the missing number is 578: \[ 229 \times 578 = 578 \times 229 \] Answer: 578

v. \(32 \times 15 = 32 \times 6 + 32 \times 7 + 32 \times\) _______

Step 1: Sum of 6 and 7 is 13, so to make 15, \[ 15 = 6 + 7 + 2 \] Check that \(6 + 7 + 2 = 15\). Thus, the missing number is 2. Rewrite the multiplication as: \[ 32 \times 15 = 32 \times 6 + 32 \times 7 + 32 \times 2 \] Answer: 2

vi. \(23 \times 56 = 20 \times 56 +\) ________ \(\times 56\)

Step 1: \[ 23 = 20 + 3 \] Therefore, \[ 23 \times 56 = 20 \times 56 + 3 \times 56 \] Answer: 3

vii. \(83 \times 54 + 83 \times 16 = 83 \times (\) ________) \(= 83 \times\) _______ \(= \) _______

Step 1: Use distributive property: \[ 83 \times 54 + 83 \times 16 = 83 \times (54 + 16) \] Calculate sum inside bracket: \[ 54 + 16 = 70 \] Then multiply: \[ 83 \times 70 = 5810 \] Answer: 54 + 16; 70; 5810

viii. \(98 \times 273 – 75 \times 273 =\) (______) \(\times 273 =\) _______ \(\times 273\)

Step 1: By distributive property: \[ 98 \times 273 – 75 \times 273 = (98 – 75) \times 273 \] Calculate inside bracket: \[ 98 – 75 = 23 \] Therefore: \[ = 23 \times 273 \] Answer: 98 – 75; 23

Q2: Evaluate

i. \(984 \times 102\)

Step 1: Express 102 as \(100 + 2\): \[ 984 \times 102 = 984 \times (100 + 2) \]Step 2: Use distributive property: \[ = 984 \times 100 + 984 \times 2 \]Step 3: Calculate each term: \[ 984 \times 100 = 98,400 \] \[ 984 \times 2 = 1,968 \]Step 4: Add the results: \[ 98,400 + 1,968 = 100,368 \]Answer: 100,368

ii. \(385 \times 1004\)

Step 1: Express 1004 as \(1000 + 4\): \[ 385 \times 1004 = 385 \times (1000 + 4) \]Step 2: Use distributive property: \[ = 385 \times 1000 + 385 \times 4 \]Step 3: Calculate each term: \[ 385 \times 1000 = 385,000 \] \[ 385 \times 4 = 1,540 \]Step 4: Add the results: \[ 385,000 + 1,540 = 386,540 \]Answer: 386,540

iii. \(446 \times 10002\)

Step 1: Express 10002 as \(10,000 + 2\): \[ 446 \times 10002 = 446 \times (10,000 + 2) \]Step 2: Use distributive property: \[ = 446 \times 10,000 + 446 \times 2 \]Step 3: Calculate each term: \[ 446 \times 10,000 = 4,460,000 \\ 446 \times 2 = 892 \]Step 4: Add the results: \[ 4,460,000 + 892 = 4,460,892 \]Answer: 4,460,892

Q3: Evaluate using properties:

i. \(548 \times 98\)

Step 1: Express 98 as \(100 – 2\): \[ 548 \times 98 = 548 \times (100 – 2) \]Step 2: Use distributive property: \[ = 548 \times 100 – 548 \times 2 \]Step 3: Calculate each term: \[ 548 \times 100 = 54,800 \] \[ 548 \times 2 = 1,096 \]Step 4: Subtract: \[ 54,800 – 1,096 = 53,704 \]Answer: 53,704

ii. \(924 \times 988\)

Step 1: Express 988 as \(1000 – 12\): \[ 924 \times 988 = 924 \times (1000 – 12) \]Step 2: Use distributive property: \[ = 924 \times 1000 – 924 \times 12 \]Step 3: Calculate each term: \[ 924 \times 1000 = 924,000 \] \[ 924 \times 12 = 11,088 \]Step 4: Subtract: \[ 924,000 – 11,088 = 912,912 \]Answer: 912,912

Q4: Evaluate using properties:

i. \(679 \times 8 + 679 \times 2\)

Step 1: Use distributive property: \[ 679 \times 8 + 679 \times 2 = 679 \times (8 + 2) \]Step 2: Calculate inside the bracket: \[ 8 + 2 = 10 \]Step 3: Multiply: \[ 679 \times 10 = 6,790 \]Answer: 6,790

ii. \(284 \times 12 – 284 \times 2\)

Step 1: Use distributive property: \[ 284 \times 12 – 284 \times 2 = 284 \times (12 – 2) \]Step 2: Calculate inside the bracket: \[ 12 – 2 = 10 \]Step 3: Multiply: \[ 284 \times 10 = 2,840 \]Answer: 2,840

iii. \(55873 \times 94 + 55873 \times 6\)

Step 1: Use distributive property: \[ 55873 \times 94 + 55873 \times 6 = 55873 \times (94 + 6) \]Step 2: Calculate inside the bracket: \[ 94 + 6 = 100 \]Step 3: Multiply: \[ 55873 \times 100 = 5,587,300 \]Answer: 5,587,300

iv. \(7984 \times 15 – 7984 \times 5\)

Step 1: Use distributive property: \[ 7984 \times 15 – 7984 \times 5 = 7984 \times (15 – 5) \]Step 2: Calculate inside the bracket: \[ 15 – 5 = 10 \]Step 3: Multiply: \[ 7984 \times 10 = 79,840 \]Answer: 79,840

v. \(8324 \times 1945 – 8324 \times 945\)

Step 1: Use distributive property: \[ 8324 \times 1945 – 8324 \times 945 = 8324 \times (1945 – 945) \]Step 2: Calculate inside the bracket: \[ 1945 – 945 = 1000 \]Step 3: Multiply: \[ 8324 \times 1000 = 8,324,000 \]Answer: 8,324,000

vi. \(3333 \times 987 + 13 \times 3333\)

Step 1: Rewrite to use distributive property: \[ 3333 \times 987 + 13 \times 3333 = 3333 \times 987 + 3333 \times 13 = 3333 \times (987 + 13) \]Step 2: Calculate inside the bracket: \[ 987 + 13 = 1000 \]Step 3: Multiply: \[ 3333 \times 1000 = 3,333,000 \]Answer: 3,333,000

Q5: Find product of the:

i. greatest number of three digits and smallest number of five digits.

Step 1: Greatest number of three digits = 999
Smallest number of five digits = 10,000
Step 2: Product = \(999 \times 10,000\)
Step 3: Multiply: \[ 999 \times 10,000 = 9,990,000 \]Answer: 9,990,000

ii. greatest number of four digits and the greatest number of three digits.

Step 1: Greatest number of four digits = 9,999
Greatest number of three digits = 999Step 2: Product = \(9,999 \times 999\)Step 3: Calculate \(9,999 \times 999\) by rewriting 999 as \(1000 – 1\): \[ 9,999 \times (1000 – 1) = 9,999 \times 1000 – 9,999 \times 1 = 9,999,000 – 9,999 = 9,989,001 \]Answer: 9,989,001

Q6: Fill in the blanks

i. \((437+3)\times(400-3)=397\times\)_____ =______

Step 1: Calculate the values inside the parentheses: \[ (437 + 3) = 440 \\ (400 – 3) = 397 \]Step 2: Rewrite the expression: \[ (437+3) \times (400-3) = 440 \times 397 \]Step 3: Given in question as: \[ (437+3) \times (400-3) = 397 \times \_\_\_\_ = \_\_\_\_ \]Since multiplication is commutative, \[ 440 \times 397 = 397 \times 440 \]So, the blank is \(440\).
Step 4: Calculate the product \(397 \times 440\): \[ 397 \times 440 = 397 \times (400 + 40) = 397 \times 400 + 397 \times 40 \\ = 158,800 + 15,880 = 174,680 \]Answer: 440 and 174,680

ii. \(66 + 44 + 22 = 11 \times\) (_______) \(= 11 \times\) ______ = ______

Step 1: Factor 11 from each term: \[ 66 = 11 \times 6, \quad 44 = 11 \times 4, \quad 22 = 11 \times 2 \]Step 2: Rewrite sum: \[ 66 + 44 + 22 = 11 \times 6 + 11 \times 4 + 11 \times 2 = 11 \times (6 + 4 + 2) \]Step 3: Calculate inside bracket: \[ 6 + 4 + 2 = 12 \]Step 4: Final multiplication: \[ 11 \times 12 = 132 \]Answer: 6 + 4 + 2, 12, 132