I have the following math trick which is multiplying numbers

I have the following math trick which is multiplying numbers up to 20.

Take 15 x 13 for an example.

Always place the larger number of the two on top in your mind.
Then draw the shape of Africa mentally so it covers the 15 and the 3 from the 13 below. Those covered numbers are all you need.
First add 15 + 3 = 18
Add a zero behind it (multiply by 10) to get 180.
Multiply the covered lower 3 x the single digit above it the \"5\" (3x5= 15)
Add 180 + 15 = 195.

My questions are:

1) i need to mathematical prove that prove this method does work for multiplication numbers up to 20x20.

2) Find an improvement of the algorithm or describe a real-world problem or completely different mathematical problem it can be used to solve.

.Please show your work!

Solution

1)

This is a very interesting trick deviced by you for multipying two digits numbers upto 20 (i.e, numbers from 10 to 19).

Proof is as follows

Suppose the two numbers be ab and cd.

Therefore, ab = 10a + b

and cd = 10c + d

So the product of two numbers would be

ab * cd = (10a + b) * ( 10c + d)

             = 100a*c + 10a*d + 10c*b + b*d                                                                (1)

According to your trick solution is

ab * cd = (ab + d)*10 + bd

             = (10a + b + d)*10 +b*d                                                                                 (2)

Now for a=1 and c= 1

Equation (1) will reduce to 10(d+10 +b) +b*d

Equation (2) will also reduce to 10(10 + b + d) + b*d

So, your formula will work perfectly in this case.

2)

There is an imrpovement which you can do to cover all 2 digit numbers, which is like this

For your problem of multiplying 15X13, do like this

Step 1: Multiply 5X3 = 15

Step 2: Multiply unit digit of one number to tens digit of another number for both unit digits and add them then multiply the sum by 10, which in this case would be (5*1 + 3*1)*10 = 80

Step 3: Multiply tens digits and multiply them by 100, which for this case will be 1*1*100 = 100

Step 4: Add numbers obtained from previous 3 steps to get the answer which in your case would be 15+80+100 =195

You can use your method for general life situations like if you are buying 14 articles and cost of each article is 13 $.