Now let us move a bit further and find out the products of different numbers.
Suppose you need to multiply 95 and 97 or 51 and 59 or 35 and 39 etc. We have a wonderful system in place which helps us find the product of any pair of multiplicands.
Let us start with 95 and 97.
The rule says that whenever you come across pairs of multiplicands like above, write the multiplicands one below the other and follow the procedure detailed below:
Step 1: Take the nearest multiple of 10 as base
Step 2: Subtract the base from the multiplicands individually and put the value with sign to their right.
Step 3: Cross add the multiplicands and the differences check they are equal. When the base is 10 to 90, this value represents 10s or 90s as the case may be.
Step 4: Multiply the differences and put them in the units place. (It goes without saying that whenever the product exceeds ten there will be carry over of 1)
Yes. The process looks so long and cumbersome. But in reality it is damn easy and has a self check for the value obtained.
Let us see. The given example is to fine the product of 95 and 97. By observation we can take 100 as base.
Put 95 and 97 one over the other
The difference of the numbers with base looks like this:
You can see by cross adding the multiplicands and the differences, here 97-5=92 as also 95-3=92. Since we have taken 100 as base what we got this value is 100s. Then multiplying (-5) and (-3) we get 15. Thus the product of 95 and 97 is 9215.
There is another interesting cross check. You add 95 and 97 and subtract 100 from it. You get 92. You add 5 and 3 the differences and subtract from 100, you still get 92. Thus when we take any base, remember clearly that the value we get in these operations is that many times the base we have chosen.
So when you remember it you will make necessary adjustments to get 10s or 100s. Let us go to the second example: The process reads as follows:
Take 50 as base. Write the given numbers 51 and 59 as follows:
and as already explained the 50 on the left side is 50s as such the real value in 10s is 50X5= 250 or 25 100s by adjustment . So the final value is 2500+ 09= 2509
Let us see one more: 35X39
Take 40 as base by observation. The procedure looks like this:
Since the value 34 represents 40s the value in tens is 34X4 = 136 So the final value is 1365
Am I clear?
Let us recap what we did and what our operations mean. Whenever we are subtracting the base from the given number, we are trying to find out how far is our number from the base. Thus when two such numbers are multiplied, we are getting the shortage from the product of the bases. But in the number system, this product place is nothing but the x2 term. That is why when we take 50 as base we are getting 50s and 40 as base 40s. When we take 100 as base, in the first case for example, we got 100s directly. Because we are working with the decimal system and our bases are not 10 or 100 but different numbers for operational convenience, we are making necessary adjustments to get back to decimal system.You can verify the old examples by this method as well.
Here base being 90, the value is (90X8 =720 in tens or 72 in hundreds) 7216
Since 120 is the base here the final value is (13X12 = 156 in hundreds) 15609
See this simple example:
Here no adjustment is necessary since 10 is the base. So 8x8= 64. Did you get it right? You see here another example;
Here gain no adjustment necessary since 10 is the base. What about this example?
You can clearly see that the product is 4 short of 10 square since 10 is our base. Hence the value is 100 - 4= 96. Now, do you get it right?
Please feel free to ask me any doubts. Because if you master this simple technique, you will see that you can understand and handle different number systems with ease.