Speed Mathematics Simplified (16 page)

After you have done this, check your answer by the usual method of multiplying. If it checks out, good. If not, go back through the steps and see where you went wrong.

If it still does not come out right, compare your working with this description of the proper steps:

Step one: 4 x 4 is in the 10's. Put down 1.

Step two: 4 x 4 ends in 6. 7 x 4 is in the 20's. Add 6 and 2. Put down 8.

Step three: 7 x 4 ends in 8. Put down 8. The final answer is 188.

Now for the two special cases. Both are important, because examples involving them will crop up repeatedly in your work with numbers.

How to Handle Zeros

Sometimes, in going through the no-carry multiplying system, you will match a pair of digits whose product is less than ten. It might be 3 x 2. This product is 6. There is no left-hand, or tens, digit at all. In effect this product is in the zeros.

For this system, however, you must use a left-hand digit. Otherwise the answer will not come out right. So no-carry multiplication always depends on using a left-hand digit even if that digit is zero.

When you come across 3 x 2, you will consider it in the zeros, just as 3 x 4 is in the 10's, and 3 x 7 is in the 20's.

The reason for keeping this in mind is that your left-hand and right-hand product digits are essential to keeping your imaginary “carries” in proper order. Later, when we come to working with two or more digits in the multiplier, you will find them important for keeping your columns in line too. This is really no more difficult than remembering to put down the zero in 5 x 6 when working from right to left, and performs basically the same function.

Suppose, for instance, you faced this example:

Step one: 5 x 7 is in the 30's:

Step two (1): 5 x 7 ends in 5. 1 x 7 is in the
zeros.
5 plus zero is 5:

Step two (2): 1 x 7 ends in 7. 4 x 7 is in the 20's. 7 plus 2 is 9:

Step three: 4 x 7 ends in 8:

One other important point about products whose left-hand, or tens, digits are in the zeros should be kept in mind. Get in the habit of putting down a zero as the first digit of the answer if this is what the problem produces. It is not essential for one-line answers such as those in the above examples, but it is absolutely essential to getting two-line answers lined up properly.

This is what I mean:

Step one: 1 x 4 is in the zeros. Put down 0:

Step two: 1 x 4 ends in 4. 6 x 4 is in the 20's. Add 4 and 2. Put down 6:

Step three: 6 x 4 ends in 4. Put down 4:

Your answer is merely 64. The zero in front of it does not change its value. But when you come to multiplying by numbers of two or more digits, you will see the necessity of this technique. It is for precisely the same reason, as we said a page back, that in the traditional method you put down the zero of thirty or forty at the right of the answer in a two-line multiplication.

But since this is a new way of doing things, be sure to get into the habit of doing it this way whenever the problem works out like this. Try it on these two samples. Use your pad:

Be sure to do these. Simply reading through practice examples, intending to do them later, will not teach you how to do speed mathematics. Theory and practice go hand in hand.

Check your results and the steps you went through in the two samples above against this explanation:

First sample. Step one: 4 x 2 is in the zeros. Put down 0. Step two: 4 x 2 ends in 8. 9 x 2 is in the 10's. 8 plus 1 is 9. Step three: 9 x 2 ends in 8. Answer: 0 9 8.

Second sample. Step one: 3 x 2 is in the zeros. Put down 0. Step two (1): 3 x 2 ends in 6. 8 x 2 is in the 10's. 6 plus 1 is 7. Step two (2): 8 x 2 ends in 6. 6 x 2 is in the 10's. 6 plus 1 is 7. Step three: 6 x 2 ends in 2. Answer: 0 7 7 2.

If you neglected to put down the zeros in front of these answers, do (for the sake of your swift mastery of two-digit multipliers) go back and do them properly now. Simple repetition, pencil in hand, means a great deal in getting accustomed to new techniques such as these.

The new way of looking at half-products may come a little hard at first. You were taught to think “6 times 8 is 48.” Now, in two separate steps, you are learning to look at 6 times 8 and (for one step) think only “40's,” then (for another step) think only “8.” Don't worry about that part yet. It is really quite a simplification of the multiplication tables, and there is some practice ahead to help give you the knack.

Before going on to the final step in no-carry multiplying, get a firmer grip on the steps so far by doing these two examples. Turn to a clean page of your pad and try your teeth on these: