Fast Ways of Multiplications

Fast Ways of Multiplications

​Associative Property (Regroup):

​The way in which factors are grouped does not change the product.

a × (b × c) = a × b × c

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Associative Property (Regroup):

Example 1. Calculate:

(a). 4 × (5 × 9).

​Associative Property (Regroup):

​Example 1. Calculate:

(a). 4 × (5 × 9).

​Solution: 180

(a). 4 × (5 × 9) = (4 × 5) × 9 = 20 x 9 = 180.

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Associative Property (Regroup):

Example 1. Calculate:

(b). 2 × 7 × 5.

​Associative Property (Regroup):

​Example 1. Calculate:

(b). 2 × 7 × 5.

​Solution: 70

(b). 2 × 7 × 5 = (2 × 5) × 7 =10 × 7 = 70.

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Associative Property (Regroup):

Example 1. Calculate:

(c). 2 × 6 × 5.

​Associative Property (Regroup):

​Example 1. Calculate:

(c). 2 × 6 × 5.

​Solution: 60

(c). 2 × 6 × 5 = 2 × (6 × 5) = 2 × 30 = 60.

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Associative Property (Regroup):

Example 2. Find the product: 25 x (12 x 8).

​Associative Property (Regroup):

​Example 2. Find the product: 25 x (12 x 8).

​Solution: 2400

25 x (12 x 8)

= (25 x 8) x 12

= 200 x 12

= 2400.

​Distributive Property:

​a( b + c) = (a × b) + (a × c) = a × b + a × c.

(a – b) × c = (a × c) – (b × c) = a × c – b × c.

a × (b – c) = (a × b) – (a × c) = a × b – a × c

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Distributive Property:

Example 3. Calculate:

(a). 3 x (20 + 6).

​Distributive Property:

​Example 3. Calculate:

(a). 3 x (20 + 6).

​Solution: 78

(a). 3 x (20 + 6) = 3 x 20 + 3 x 6 = 60 + 18 = 78.

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Distributive Property:

Example 3. Calculate:

(b). (25 – 2) × 4

​Distributive Property:

​Example 3. Calculate:

(b). (25 – 2) × 4.

​Solution: 92

(b). (25 – 2) × 4 = 25 × 4 – 2 × 4 = 100 – 8 = 92.

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Distributive Property:

Example 3. Calculate:

(c). 4 × (50 – 5).

​Distributive Property:

​Example 3. Calculate:

(c). 4 × (50 – 5).

​Solution: 180

(c). 4 × (50 – 5) = 4 × 50 – 4 × 5 = 200 – 20 = 180.

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Distributive Property:

Example 4. Evaluate: 234 x 997 – 233 x 997.

​Distributive Property:

​Example 4. Evaluate: 234 x 997 – 233 x 997.

​Solution: 997

234 x 997 – 233 x 997

= (234 – 233) x 997

= 997.

​Some calculation skills

(1). (a + b + c) ÷ d = a ÷ d + b ÷ d + c ÷ d

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Some calculation skills

(1). (a + b + c) ÷ d = a ÷ d + b ÷ d + c ÷ d

Example 5. Evaluate the value of (12 + 48 + 63) ÷ 3.

​Some calculation skills

(1). (a + b + c) ÷ d = a ÷ d + b ÷ d + c ÷ d

​Example 5. Evaluate the value of (12 + 48 + 63) ÷ 3.

​Solution: 41

(12 + 48 + 63) ÷ 3

= 12 ÷ 3 + 48 ÷ 3 + 63 ÷ 3

= 4 + 16 + 21

= 20 + 21

= 41.

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Some calculation skills

(2). a ÷ (b × c) = a ÷ b ÷ c.

Example 6. 150 ÷ (5 × 5).

​Some calculation skills

(2). a ÷ (b × c) = a ÷ b ÷ c.

​Example 6. 150 ÷ (5 × 5).

​Solution: 6

150 ÷ (5 × 5)

= 150 ÷ 5 ÷ 5

=30 ÷ 5

= 6.

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Some calculation skills

(3). (a × b) ÷ c = (a ÷ c) × b

Example 7. Calculate: (48 × 5) ÷ 3

​Some calculation skills

(3). (a × b) ÷ c = (a ÷ c) × b

​Example 7. Calculate: (48 × 5) ÷ 3).

​Solution: 80

(48 × 5) ÷ 3

= (48 ÷ 3) × 5

=16 × 5

= 80.

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Some calculation skills

(4). a ÷ (b ÷ c) = a ÷ b × c

Example 8. Calculate: 48 ÷ (6 ÷ 5).

​Some calculation skills

(4). a ÷ (b ÷ c) = a ÷ b × c

​Example 7. Calculate: 48 ÷ (6 ÷ 5).

​Solution: 40

48 ÷ (6 ÷ 5)

= 48 ÷ 6 × 5

= 8 × 5

= 40.

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Some calculation skills

(5). (a – b) ÷ c = a ÷ c – b ÷ c

Example 9. (48 – 16) ÷ 8 = 48 ÷ 8 – 16 ÷ 8 = 6 – 2 = 4.

​Some calculation skills

(5). (a – b) ÷ c = a ÷ c – b ÷ c

​Example 9. (48 – 16) ÷ 8 = 48 ÷ 8 – 16 ÷ 8 = 6 – 2 = 4.

​Solution: 4

(48 – 16) ÷ 8

= 48 ÷ 8 – 16 ÷ 8

= 6 – 2

= 4.

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Some calculation skills

(5). (a – b) ÷ c = a ÷ c – b ÷ c

Example 10: Find the value of 8 x 7 ÷ 8 x 7.

​Some calculation skills

(5). (a – b) ÷ c = a ÷ c – b ÷ c

​Example 10. Find the value of 8 x 7 ÷ 8 x 7.

​Solution: 49

8 x 7 ÷ 8 x 7

= 8 ÷ 8 x 7 x 7

= 1 x 7 x 7

= 49.

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Some calculation skills

(5). (a – b) ÷ c = a ÷ c – b ÷ c

Example 11: (70 ÷ 4 + 90 ÷ 4) ÷ 4.

​Some calculation skills

(5). (a – b) ÷ c = a ÷ c – b ÷ c

​Example 11. (70 ÷ 4 + 90 ÷ 4) ÷ 4.

​Solution: 10

(70 ÷ 4 + 90 ÷ 4) ÷ 4

= [(70 + 90) ÷ 4] ÷ 4

= 160 ÷ 4 ÷ 4

= 40 ÷ 4 = 10.

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(6) Observe the pair of factors (2, 5), (4, 25), and (8, 125).

2 x 5 = 10 4 x 25 = 100 8 x 125 = 1,000.

Example 12: Calculate:

(1) 127x 4 x 25

​(6) Observe the pair of factors (2, 5), (4, 25), and (8, 125).

2 x 5 = 10 4 x 25 = 100 8 x 125 = 1,000.

​Example 12. Calculate:

(1) 127 x 4 x 25

​Solution: 12,700

(1) 127 x 4 x 25

= 127 x (4 x 25)

= 127 x 100

= 12,700.

Fill in the Blanks

(6) Observe the pair of factors (2, 5), (4, 25), and (8, 125).

2 x 5 = 10 4 x 25 = 100 8 x 125 = 1,000.

Example 12: Calculate:

(2) 25 x 7 x 125 x 4 x 8.

​(6) Observe the pair of factors (2, 5), (4, 25), and (8, 125).

2 x 5 = 10 4 x 25 = 100 8 x 125 = 1,000.

​Example 12. Calculate:

(2) 25 x 7 x 125 x 4 x 8.

​Solution: 700,000.

(2) 25 x 7 x 125 x 4 x 8.

= (25 x 4) x (125 x 8) x 7

= 100 x 1,000 x 7

= 100,000 x 7

= 700,000.

Fill in the Blanks

(6) Observe the pair of factors (2, 5), (4, 25), and (8, 125).

2 x 5 = 10 4 x 25 = 100 8 x 125 = 1,000.

Example 13: Calculate: 25 x 19 x 64 x 125.

​(6) Observe the pair of factors (2, 5), (4, 25), and (8, 125).

2 x 5 = 10 4 x 25 = 100 8 x 125 = 1,000.

​Example 13. Calculate: 25 x 19 x 64 x 125.

​Solution: 3,800,000.

25 x 19 x 64 x 125

= 25 x 19 x (4 x 8 x 2) x 125

= (25 x 4) x (125 x 8) x (19 x 2)

= (100) x (1,000) x (38)

= 3,800,000.

Fill in the Blanks

(6) Observe the pair of factors (2, 5), (4, 25), and (8, 125).

2 x 5 = 10 4 x 25 = 100 8 x 125 = 1,000.

Example 14: Calculate:

(1). 125 x 34 + 125 x 66.

​(6) Observe the pair of factors (2, 5), (4, 25), and (8, 125).

2 x 5 = 10 4 x 25 = 100 8 x 125 = 1,000.

​Example 14. Calculate:

(1). 125 x 34 + 125 x 66.

​Solution: 12,500

125 x 34 + 125 x 66

= 125 x (34 + 66)

= 125 x 100

= 12,500.

Fill in the Blanks

(6) Observe the pair of factors (2, 5), (4, 25), and (8, 125).

2 x 5 = 10 4 x 25 = 100 8 x 125 = 1,000.

Example 14: Calculate:

(2). 43 x 11 + 43 x 36 + 43 x 52 + 43.

​(6) Observe the pair of factors (2, 5), (4, 25), and (8, 125).

2 x 5 = 10 4 x 25 = 100 8 x 125 = 1,000.

​Example 14. Calculate:

(2). 43 x 11 + 43 x 36 + 43 x 52 + 43.

​Solution: 4,300

43 x 11 + 43 x 36 + 43 x 52 + 43.

= 43 x (11 + 36 + 52 + 1)

= 43 x (100)

= 4,300.

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(7) Multiplying two 2-digit numbers that have the same tens digit and their units digits sum to 10

Example 15: Calculate: 27 × 23.

​(7) Multiplying two 2-digit numbers that have the same tens digit and their units digits sum to 10

​Example 15. Calculate: 27 × 23

​Solution: 621

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(7) Multiplying two 2-digit numbers that have the same tens digit and their units digits sum to 10

Example 16: Calculate: 45 x 45

​(7) Multiplying two 2-digit numbers that have the same tens digit and their units digits sum to 10

​Example 16. Calculate: 45 x 45.

​Solution: 2,025

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(7) Multiplying two 2-digit numbers that have the same tens digit and their units digits sum to 10

Example 17: Calculate: 68 x 62.

​(7) Multiplying two 2-digit numbers that have the same tens digit and their units digits sum to 10

​Example 17. Calculate: 68 x 62.

​Solution: 4,216

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(7) Multiplying two 2-digit numbers that have the same tens digit and their units digits sum to 10

Example 18: Calculate: 85 x 85.

​(7) Multiplying two 2-digit numbers that have the same tens digit and their units digits sum to 10

​Example 18. Calculate: 85 x 85.

​Solution: 7,225.

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(7) Multiplying two 2-digit numbers that have the same tens digit and their units digits sum to 10

Example 19: Calculate: 29 x 21.

​(7) Multiplying two 2-digit numbers that have the same tens digit and their units digits sum to 10

​Example 19. Calculate: 29 x 21.

​Solution: 609

​(8). Multiplying a two or three-digit number by 11

​(1) Write down the units digit of the number. This will be the units digit of the product.

(2) Taking the digits on the left of the units digit in pairs, write down their sums. They will be the digits to the left of the units digit of the product.

(3) Write down the left most digit with any additional carry from step (2). This will be the left most digit of the product.

Fill in the Blanks

(8). Multiplying a two or three-digit number by 11

Example 20: Calculate: (1) 13 × 11

​(8). Multiplying a two or three-digit number by 11

​Example 20. Calculate: (1) 13 × 11

​Solution: 143

13 × 11 = 143

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(8). Multiplying a two or three-digit number by 11

Example 20: Calculate: (2) 63 × 11

​(8). Multiplying a two or three-digit number by 11

​Example 20. Calculate: (2) 63 × 11

​Solution: 693

63 × 11 = 693

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(8). Multiplying a two or three-digit number by 11

Example 20: Calculate: (3) 36 × 11

​(8). Multiplying a two or three-digit number by 11

​Example 20. Calculate: (3) 36 × 11

​Solution: 396

36 × 11 = 396

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(8). Multiplying a two or three-digit number by 11

Example 21: Calculate: (1) 48 × 11

​(8). Multiplying a two or three-digit number by 11

​Example 21. Calculate: (1) 48 × 11

​Solution: 528

48 × 11 = 528

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(8). Multiplying a two or three-digit number by 11

Example 21: Calculate: (2) 638 × 11

​(8). Multiplying a two or three-digit number by 11

​Example 21. Calculate: (2) 638 × 11

​Solution: 7018

638 × 11 = 7018

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(8). Multiplying a two or three-digit number by 11

Example 21: Calculate: (3) 787 × 11

​(8). Multiplying a two or three-digit number by 11

​Example 21. Calculate: (3) 787 × 11

​Solution: 8657

787 × 11 = 8657

(9). Multiply a two-digit number by 51 and a three-digit number by 501.

​Divide the number by 2. Write down the result. Follow the result with the original number.

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(9). Multiply a two-digit number by 51 and a three-digit number by 501.

Example 22: (1). 51 × 64.

(9). Multiply a two-digit number by 51 and a three-digit number by 501.

​Example 22. Calculate: (1). 51 × 64.

​Solution: 3,264

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(9). Multiply a two-digit number by 51 and a three-digit number by 501.

Example 22: (2). 501 × 326.

(9). Multiply a two-digit number by 51 and a three-digit number by 501.

​Example 22. Calculate: (2). 501 × 326.

​Solution: 163,326.

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PROBLEMS

Problem 1. Calculate:

(1) 17 × 125 × 8

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Problem 1. Calculate:

(2) 9 × 625 × 16

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Problem 1. Calculate:

(3) 44 × 25

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Problem 1. Calculate:

(4) 72 × 125

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Problem 1. Calculate:

(5) 12 × 25 × 5 × 4 × 8 × 125

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Problem 1. Calculate:

(6) 88 × 64 × 125 × 250

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Problem 2. Calculate:

(1) 2 × 7 × 5

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Problem 2. Calculate:

(2) 41 × 4 × 25

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Problem 2. Calculate:

(3) 24 × 25

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Problem 2. Calculate:

(4) 56 × 125 × 3

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Problem 2. Calculate:

(5) 35 × 25 × 2 × 8

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Problem 2. Calculate:

(6) 48 × 125

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Problem 2. Calculate:

(7) 25 × 8 × 5

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Problem 2. Calculate:

(8) 11 × 9 + 11

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Problem 2. Calculate:

(9) 125 × 54 + 125 × 46

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Problem 2. Calculate:

(10) 21 × 63 + 36 × 21 + 21

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Problem 3. Oral calculation:

(1) 53 × 57

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Problem 3. Oral calculation:

(2) 78 × 72

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Problem 3. Oral calculation:

(3) 24 × 26

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Problem 3. Oral calculation:

(4) 35 × 35

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Problem 4. Oral calculation:

(1) 23 × 27

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Problem 4. Oral calculation:

(2) 74 × 76

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Problem 4. Oral calculation:

(3) 46 × 44

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Problem 4. Oral calculation:

(4) 55 × 55

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Problem 4. Oral calculation:

(5) 61 × 69

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Problem 4. Oral calculation:

(6) 47 × 11

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Problem 4. Oral calculation:

(7) 353 × 11

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Problem 4. Oral calculation:

(8) 498 × 11

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Problem 4. Oral calculation:

(9) 33 × 33 + 33 × 33 + 33 × 33 + 33

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PROBLEMS

Problem 5. Fill the blank:

37 × 6 = ( )

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Problem 5. Fill the blank:

37 × 9 = ( )

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Problem 5. Fill the blank:

37 × 12 = ( )

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Problem 5. Fill the blank:

37 × 15 = ( )

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Problem 5. Fill the blank:

37 × ( ) = 666

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Problem 5. Fill the blank:

37 × ( ) = 888

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Problem 5. Fill the blank:

8547 × 26 = ( )

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Problem 5. Fill the blank:

8547 × 39 = ( )

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Problem 5. Fill the blank:

8547 × 78 = ( )

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Problem 5. Fill the blank:

8547 × ( ) = 999,999

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Problem 5. Fill the blank:

8547 × ( ) = 333,333

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Problem 5. Fill the blank:

8547 × ( ) = 444,444

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Problem 6. Fill the blank:

12345679 × 3 × 9 = ( )

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Problem 6. Fill the blank:

12345679 × 9 × ( ) = 444,444,444

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Problem 6. Fill the blank:

12345679 × ( ) = 999,999,999

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Problem 6. Fill the blank:

1,111× 1,111 = ( )

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Problem 6. Fill the blank:

11,111× 11,111 = ( )

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Problem 6. Fill the blank:

111,111× 111,111 = ( )

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Problem 6. Fill the blank:

1,111,111× 1,111,111 = ( )