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## Some Easy Methods to Multiply Two Numbers

Let’s visit some mathematical expressions typically formed thanks to the distributive laws of addition and multiplication. For the purposes of this tutorial, let’s restrict our domain to two-digit numbers.

(a + b) squared = a squared + b squared + 2 * a * b.

(ab) squared = a squared + b squared – 2 * a * b.

(a+b)*(c+d)=ac+ad+bc+bd.

(a + b) * (c – d) = ac – ad + bc – bd.

(ab) *(c+ d) = ac + ad -bc -bd.

(ab)(cd) = ac – ad – bc + bd.

Rewrite a number K as the sum or difference of two numbers in the form

(a +b) or (ab), where a is equal to the nearest multiple of 10 less than or just greater than the number K.

For example 63 would be rewritten as (60 + 3) and 68 would be rewritten as

(70 – 2). 73 would be rewritten as 70 + 3 and 79 would be rewritten as 80 – 1.

Let’s first calculate 47 * 47

Let’s use the formula (ab)*(ab)=a*a-2*a*b+b*b where a=50 and b=3.

So 47 * 47 = 2500 – 300 + 9 = 2209.

Let’s use these formulas (a+b) * (a + b) = a*a + 2 * a * b + b*b and calculate 73 * 73

73 * 73 = (70 + 3) ( 70 + 3 ) = 4900 + 420 + 9 = 5329.

26 * 26 = ( 30 – 4)(30 – 4) = 900 – 240 + 16 = 676.

Use (a +b)(c +d) to calculate 93 * 57 as

(90 + 3) * ( 50 + 7 ) = 4500 + 630 + 150 + 21, Simplify this further

writing this as 4500 + 600 + 30+100 + 50 + 20 + 1, 4500 + 700 + 101 which can be easily calculated as 5301

Let’s similarly calculate 47 * 69 using the expression for (ab) (cd). Rewrite the product in the form (50-3)(70-1). Expand individual terms like

3500 – 50 – 210 + 3, rewriting this as 3500 – 200 – 60 + 3; rewrite this as

3500 – 200 – 60 + 3; So 47 * 63 = 3243.

Using the expression (ab)(c+d) calculate 43 * 88, rewrite the original expression as (40 +3)*(90 – 2). Calculate the individual sums as 3600 – 80 + 270 – 6 = 3870 – 86 = 3870 – 70 – 16 = 3800 – 16 = 3784.

Let’s use the expression (ab)(cd) = ac – ad -bc + bd to quickly calculate 57 * 79. Rewrite this as (60 – 3) * (80 -1) = 4800 – 60 – 240 + 3 = 4800 – 300 + 3 = 4503.

By extending the analysis to 3-digit numbers, these simple sums of products can be easily formulated using the distributive laws of addition and multiplication, respectively.

Quickly calculate 153 * 94, rewriting 153 as 100 + 50 +3 and 94 as 90 + 4. Write the original product as (100+ 50 + 3)(90 + 4); the sum of the individual terms is 9000 + 400 + 4500 + 200 + 270 + 12. Sorting is by thousands, hundredths and ones. 13000 + 1000 + 300 + 82; quickly calculate the sum as 14382.

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