Speed Mathematics Simplified (17 page)

We will not go through the explanation of these in detail. Do them thoughtfully and carefully, working at this point for full understanding and accuracy rather than speed. Speed will follow because you are now working in what is essentially a simpler and more logical manner.

Once you have done the two examples, check your results by repeating the two problems according to your old method. If the answers check out, fine. If not, study the steps in detail to find out where you went wrong.

Now for the final step.

Recording Tens

So far, all our examples have been carefully selected to avoid one special situation that is really more complicated in the traditional method than it is in no-carry multiplying. But the situation does need a technique to handle it, and we have a simple and automatic one.

This example will demonstrate the special situation. Go through it step by step and find the new problem:

Step one: 8 x 6 is in the 40's:

Step two: 8 x 6 ends in 8. 9 x 6 is in the 50's. 8 plus 5 is—

STOP! You cannot put down a single digit standing for the sum of 8 and 5. This goes over ten. In our new way to add, we do not even try to add them. Instead, we subtract the complement of 8 (2) from 5 and put down 3:

But this is not quite right. When we use a complement, we must also record a ten. How do we record a ten here?

One of the secrets of this simplified mathematics is that we let the tens take care of themselves. We record them in adding, or cancel them in subtracting. We never, never try to remember them. That would be inefficient.

In multiplying, then, we will simply use the same written symbol we use in adding. We will underline. In this case, an underline will raise the value of the underlined digit by one—just as, in subtracting, a slash reduces the value of the slashed digit by one. Since the underline is in effect
carried back
from the 3 (the underline represents the 1 in 13, which is one place to the left), we will underline the digit one place to the left.

So our answer now looks like this:

Step three: 9 x 6 ends in 4:

Since the underline raises the value of the underlined digit by one, we read our answer like this:

Is this correct? Check the problem and see. Just as important, or even more important, see if the logic of it is clear.

If this seems the least bit complicated, review in your mind the schoolbook approach to this problem. Here is the thinking you were instructed to do: “6 x 9 is 54. Put down the 4, carry the 5. 6 x 8 is 48. We carried a 5. 8 plus 5 is 13. Put down the 3. Carry one from the 13. Add the 1 of the 13 to the 4 of 48. 1 plus 4 is 5. Put down 5.” Which, once you are equally familiar with both approaches, is really simpler?

Let's go through this new process once more in detail:

Step one: 4 x 3 is in the 10's:

Step two (1): 4 x 3 ends in 2. 6 x 3 is in the 10's. 2 plus 1 is 3 :

Step two (2) : 6 x 3 ends in 8. 8 x 3 is in the 20's. 8 and 2 are complements. Zero, record:

Step three: 8 x 3 ends in 4:

Until you are thoroughly accustomed to reading slashed and underlined digits accurately without hesitation, it is good practice to rewrite such answers: