Some easy methods to multiply two numbers

Let’s visit some mathematical expressions typically formed due 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.

Let’s rewrite a number K as a 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 slightly 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.

First let’s 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 this formula (a+b) * (a + b) = a*a + 2 * a * b + b*b and let’s 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). Let’s rewrite the product as (50-3) (70-1). Expanding individual terms like

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

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

Using the expression (ab)(c+d) let’s calculate 43 * 88, rewrite the original expression as (40 +3)*(90 – 2). Calculate individual sums such 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 calculate 57 * 79 quickly. Rewriting this as (60 – 3) * (80 -1) = 4800 – 60 – 240 + 3 = 4800 – 300 + 3 = 4503.

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

Let’s calculate 153 * 94 quickly, rewriting 153 as 100 + 50 +3 and 94 as 90 + 4. Let’s 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 Units. 13000+1000+300+82; quickly calculate the sum as 14382.