Dear friends, I must apologize to you for this long break. Now let us start from where we left off. We were discussing about using both hands for calculations without making use of any calculator.

Recap

What we did was, we gave values to the left and right fingers. When two numbers are to be multiplied, we just joined the two fingers that represent the numbers, and for between 6x6 to 10x10, we gave all the fingers touching and below 10 and for the free fingers units. Then we multiplied the values of the free fingers of hands and added that product to the number of tens ( i.e. the number of fingers joined and those below them) as an example when we have to multiply 8 by 9 we joined LM and RI as first step. There are two fingers free on the left hand ( LI and LT) and only 1 finger (RT) on the right. So the product of free fingers is 2 and we added it to the number of tens 7 (three from left hand and four from right hand, thus a total of 7). Thus the product is 72.

You might have already mastered it and even otherwise think it is not a problem. Now comes the vantage over calculator. Let us suppose we want products between 11x11s to 15x15s. Remember we skipped in between products 6x11s to 6x15s. I will come to it after this to make a point clear to you. So bear with me for few minutes.

Now, let us turn to our problem of 11x 11s to 15x15s.

As we have only ten fingers, and as we go up we associate larger values to each finger, correspondingly we have to change the values of fingers joined and below, as also the values of the free fingers. How we change is all that you need to pay your attention for. If you got it, the answer is very very simple.

Just look. When we are multiplying figures from 6x6 to 10x 10, we started with values above 5 for each hand. Then the sum of these two is 10. so we assigned to the fingers joined and below them a value of 10.

Now for values between 11x11 and 15 x15, we start giving to each finger on the right and left a value above 10. So what value should we give by analogy to the fingers joined and below them now? Right. 10+10= 20. And what about the values of free fingers? You must add 5 to the number of free fingers on the right as well as left.

Let us do it by example.

Example 1. Suppose you want 12x12

What you join?
LR and RR.
So there are 4 fingers joined and below, their value is 4x20 = 80.
Then there are 3 free fingers each on two hands. So their value is 5+3 in each case and their product is 64. now adding 64 to 80 we get 144 which is exactly what it is.

Example 2. Take for example 14x 12
We join LI and RR.
There are 6 fingers joined or below.
Their value is 120. and 1 finger on the left, its value is 6 and 3 fingers on the right, their value is 8 and the product is 6x8=48 and add 120 + 48= 168 and 14x12 is that.
And now you can master up to 15x15.

Let us get back to the problem I left out deliberately. That is from 6x11s to 6x15s. In the cases I described above, the range of multiplication in both the hands is same. That means, we are assigning the same set of values, say from 6 to 10 or 11 to 15 for the fingers of both hands. There by they are evenly matched.

But when you multiply 6 by 11, you have to take care while assigning values to the joined fingers and below them, and the free fingers. (Following the order that the order of multiplication that the left number is represented by the left hand and the right number by the right hand I am giving the following explanation.)

Whenever a large number is multiplied by a small number, we can say that the large number is pulled down or the small number is pulled up. Applying the same thing, we assign the values as follows:

1. All fingers joined and below them will be assigned values 10+5 = 15.
   (Please note one hand starts with 5+ and another with 10+ so the fingers joined and below are assigned 15)

2. Free fingers opposite to the lesser number will be units and to the larger number we add

3. There is no change in the other procedure.

Let us do by example to be very clear.

Example 3: Take for example 7x 13.

As usual we join LR and RM.
We have 5 fingers joined or below.
Each has a value of 15 and so their total is 15x5= 75.
Now since left hand is lesser number, and there are 3 free fingers, their value is 3+5= 8 where as the right hand has 2 free fingers and their value is units only.
Thereby their product is 8x2=16.
Then add 75 +16 to get 91 which is 7x13.

Example 4: 6x14

We join LL and RI.
There are 5 fingers joined and below.
Their value is 5x15= 75.
For the left hand there are 4 free fingers but their value is 4+5=9 where as there is only 1 free finger on the right and its value is 1.
Their product is 8.
The final value is 75 + 9=84.
And now you have mastered from 6x6 to 15 x15.

Master this before you venture into the higher plane. Good luck     

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