4. 1^st method of multiplication

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The 1^st method of multiplication is based on the equalities :

(b + c) x a = (b x a) + (c x a)
(b + c) / a = (b / a) + (c / a)
a x (b + c) = (a x b) + (a x c)

and

(a + b) x (c + d) = (a x c) + (a x d) + (b x c) + (b x d)

As well as the remarkable identities

(a + b)² = a² + 2 x a x b + b²
(a - b)² = a² - 2 x a x b + b²
(a + b) x (a – b) = a² - b²

Using the 1^st equalities

Calculate : 36 x 7

For this, 36 must be broken down into 30 and 6.

36 x 7	= (30 + 6) x 7
	= 30 x 7 + 6 x 7
	= 210 + 42
	= 252

Calculate : 45 x 19

45 x 19	= 45 x (20 - 1)
	= 45 x 20 - 20
	= 900 - 45
	= 855

Calculate : 58 / 2

58 / 2	= (60 - 2) / 2
	= (60 / 2) - (2 / 2)
	= 30 - 1
	= 29

Calculate : 134 / 2

134 / 2	= (120 + 14) / 2
	= (120 / 2) + (14 / 2)
	= 60 + 7
	= 67

Calculate : 522 / 3

I break down 522 into 300 and 222 and then 222 into 210 and 12.

522 / 3	= (300 + 210 + 12) / 3
	= (300 / 3) + (210 / 3) + (12 / 3)
	= 100 + 70 + 4
	= 174

522 / 3 = (300 + 210 + 12) / 3 = (300 / 3) + (210 / 3) + (12 / 3) = 100 + 70 + 4 = 174

Using the equality (a + b) x (c + d) = (a x c) + (a x d) + (b x c) + (b x d)

For example : 54 x 38 :

54 x 38	= (50 + 4) x (30 + 8)
	= (50 x 30) + (50 x 8) + (4 x 30) + (4 x 8)
	= 1500 + 400 + 120 + 32
	= 1532 + 520
	= 2052

In mental calculation, we proceed as follows:

1 - I multiply the tens 5x3=15

I memorise 15 (1 in thousands and 5 in hundreds)

2 - I multiply the units 4x8=32

I add 32 which gives 1532

3 - I multiply diagonally 5x8=40 and 4x3=12

I add 40+12=52 which I put in hundreds and tens: 5 in hundreds and 2 in tens

4 - I calculate the sum 1532+520=2052 and this is the result.

Another example : 68 x 47 :

68 x 47	= (70 - 2) x (50 - 3)
	= (70 x 50) - (70 x 3) - (2 x 50) + (2 x 3)
	= 3500 - 210 - 100 + 6
	= 3506 - 310
	= 3506 - 306 - 4
	= 3200 - 4
	= 3196

Use of remarkable identities

(a + b)² = a² + 2 x a x b + b²
(a - b)² = a² - 2 x a x b + b²
(a + b) x (a – b) = a² - b²

Example: 34 x 34

34 x 34	= (30 + 4)2
	=302 + 2 x 30 x 4 + 42
	= 900 + 16 + 240
	= 1156

Example 45 x 45

Let's use for this the identity:

a² – b² = (a + b) x (a – b)

452 - 52	= (45 + 5) x (45 - 5)
45 x 45 - 25	= 50 x 40
45 x 45	= 2000 + 25
	= 2025

Another example of the use of this formula

a² – b² = (a + b) x (a – b)

Calculate : 56 x 64

Note that the distance of the two values from 60 is 4.

56 x 64	= (60 - 4) x (60 + 4)
	= 602 - 42
	= 3600 - 16
	= 3584

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