Super Fast and Easy: Two-Digits Multiplication

Multiplying Two-Digit Numbers: The Other Way

While some of our kids or students easily give the answer in the above multiplication, others still  find it difficult to multiply such in a small amount of time. This post may help them to do such. With proper guidance of the teacher or the parent, they can do it.

Let us first recall one topic in Algebra which is the Product of Two Binomials

(a +b)² = a² + b² + 2ab

(a-b) ²  = a² + b² - 2ab

(a + b)  (c + d) = ac + ad + bc + bd.

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

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

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

Let us rewrite a number k as a sum or difference of two numbers in the form

(a +b) or (a-b)

where a is equal to the nearest multiple of 10 less than or just greater than the number K.

For examples:

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 us first calculate 47 * 47

Let us use the formula

(a-b) * (a-b) = a * a - 2 * a * b + b *b where a = 50 and b = 3.

So 47 * 47 = 2500 - 300 + 9 = 2209.

Let us calculate 73 * 73

Let us use these formula (a+b) * (a + b) = a*a + 2 * a * b + b*b

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 by writing this as,

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

5301

Let us similarly calculate 47 * 69 using the expression for (a-b) (c-d).

Let us rewrite the product as

=(50-3) (70-1).                        Expanding the individual terms as

=3500 - 50 - 210 + 3,              rewriting this as

=3500 - 200 - 60 + 3;              rewriting this as

=3500 - 200 - 60 + 3;              So

=3243.

Using the expression (a-b)(c+d) let us calculate 43 * 88,

Rewrite the original expression as

(40 +3)*(90 - 2). Compute individual sums as

= 3600 - 80 + 270 - 6

= 3870 - 86

= 3870 - 70 - 16

= 3800 - 16

= 3784.

Let us use the expression (a-b) (c-d) = ac - ad -bc + bd to calculate 57 * 79 quickly.

Rewriting this as

(60 - 3) * (80 -1)

= 4800 - 60 - 240 + 3

= 4800 - 300 + 3

= 4503.

This can be extended to 3 digit numbers, where the sums of products can be easily formulated using distributive laws of addition and multiplication respectively.

Let us try to calculate 153 * 94 quickly, by rewriting 153 as 100 + 50 +3 and 94 as 90 + 4.

Let us write the original product as

(100+ 50 + 3) (90 + 4);                                   the sum of individual terms is

9000 + 400 + 4500 + 200 + 270 + 12.            Sort is by thousands, hundredths and Units place.

13000 + 1000 + 300 + 82;                              quickly calculate the sum as

14382.