Speed Mathematics Simplified (25 page)

The first answer digit, by inspection, is 7. Now we multiply and subtract digit by digit:

One: 8 x 7 is in the 50's, and 5 from 5 is 0.

Two: 8 x 7 ends in 6, and 6 from 9 is 3.

The remainder so far is 336. In order to produce the next answer digit, we mentally bring down the 3 and divide 8 into 33. We put down the answer digit 4. Now we verify and produce the remainder:

One: 8 x 4 is in the 30's, and 3 from 3 is 0.

Two: 8 x 4 ends in 2, and 2 from 3 is 1.

The remainder at this point is 16. In the illustration above, we have already mentally brought down the 6 and put down the next answer digit, 2. Is there any remainder?

One: 8 x 2 is in the 10's, and 1 from 1 is 0.

Two: 8 x 2 ends in 6, and 6 from 6 is 0.

The problem comes out even. A little later on, we shall get into fractional and decimal remainders.

Now try one entirely on your own. It will be an easy one, to get the technique firmly bedded in your habits before going on to more complicated problems. Do this one on your pad:

After you have finished, check your working against this step-by-step explanation:

First digit: 7. One: 6 x 7 is in the 40's, and 4 from 4 is 0. Two: 6 x 7 ends in 2, and 2 from 5 is 3. Remainder (by mentally bringing down the 6), 36.

Second digit: 6. One: 6 x 6 is in the 30's, and 3 from 3 is 0. Two: 6 x 6 ends in 6, and 6 from 6 is 0.

Automatic “Borrowing*

The demonstrations so far have been chosen for simplicity. They are simple both because you are dividing by single digits and because there is no canceling (“borrowing”) involved in the subtraction.

Consider this case:

This problem will involve canceling. Yet because you have learned to use canceling in the answer instead of “borrowing” in the larger number, you will find it no trick at all to adapt what you already know to the smooth and efficient working of this kind of division.

The first answer digit, by inspection, is 2. We get the remainder so far in our usual way:

One: 9 x 2 is in the 10's, and 1 from 2 is 1.

Two: 9 x 2 ends in 8, and 8 from 2 is—

STOP! Larger from smaller. Do not subtract. Add the complement of 8 (2) to 2 and slash left:

This answer should look perfectly normal after your work with left-to-right subtraction. It is simply 4. The (slashed) 1 has been reduced by the slash by one in value, to 0.

Mentally bringing down the next 2, you “see” the answer of 9 into 42 as 4 and put this down as the second answer digit. Now for the remainder:

One: 9 x 4 is in the 30's, and 3 from 4 is 1.

Two: 9 x 4 ends in 6, and 6 from 2 is—larger from

smaller. Do not subtract. The complement of 6 (4) plus 2 is 6, and slash:

See if you can finish this problem yourself. Mentally bring down the proper digit and see the answer. After you have worked it out, check against this explanation:

The next answer digit is 7—9 into 63.

One: 9 x 7 is in the 60's, and 6 from 6 is 0.

Two: 9 x 7 ends in 3, and 3 from 3 is 0.

Try one example that involves “borrowing” before going on. Use your pad and do this problem just as we did the one above:

Do this with your pad and pencil before checking against the working figures below. A full understanding of the steps in shorthand division is essential before we get into longer problems. Once you have gone through the routine several times, you will find that you can handle any division with confidence.

Check your work and your answer with this finished problem. Here is how it should look:

That is all there is to it. No single part of this process is complicated. It is all based on techniques you have already mastered in earlier parts of this book. But the combination of them is new. If you have any trouble assembling the parts into a smooth-working whole, then go back and re-check the weak parts.

Make very sure you have everything so far down pat, because we are now going to add the third and final element in shorthand division that makes it just a bit more complex. You have already learned to handle this step in multiplication, but the mental processes will have to stretch one more notch when you apply it to division.

Stop now and make sure you are ready for the next step by doing these two problems on your pad:

When you have done these, compare your results with these models: