Speed Mathematics Simplified (39 page)

This method is called “breakdown.”

Could you use the method we first mentioned, proportionate change, to find the tip? Yes, though multiplication works in a different fashion from division. In division you double both numbers to keep the relationship the same. In multiplication, to keep the relationship the same you must cut one of them in half if you double the other. ½ × 2 is 1—so the problem will have the same answer.

To use proportionate change on this tip, you would double the 15 to 30, and cut the $4.00 in half to get $2.00. 30% of $2.00 is (3 × 2) 60¢ again.

Which is the better short cut? Neither. Each one fits certain combinations of numbers better than the other does, and it is helpful to know both so you can select the easier of the two for any one case.

This is the central fact about short cuts. There are literally hundreds of short cuts, from the quick method of squaring a number ending in 5 (when did you last have to square a number ending in 5?) to multiplying by 11 in one operation (a little more useful, but still pretty specialized). Of the many available, only four types of short cut are really applicable to enough problems to be worth the trouble of learning, unless for the reward of knowing a number oddity to impress people.

These four short cuts have certain similarities and certain differences. Learn them well, learn how to recognize which is most valuable in any one case, and you can add to your already advanced handling of basic numbers the extra advantage of frequent “overleaps” in lightning calculation.

Dig out your pad again, or better yet get a fresh one, and prepare to enter another fascinating aspect of numbers.

13

BREAKDOWN

I
T IS hardly likely that you would ever multiply a number by ten by putting down the number with ten under it, and multiplying out digit by digit like this:

Instead, you know that in order to multiply any number by ten you simply add a zero. If the number has a decimal point in it, you move the decimal point to the right instead of adding the zero.

984 × 10 is 9840.

653.92 × 10 is 6539.2.

Elementary as this is, the principle is basic to many of the short cuts in number work. In many cases, we can save time by multiplying or dividing a number by ten, a hundred, or even a thousand before even beginning work.

In division, of course, you remove a zero (or move the decimal point one place to the left) in order to divide by ten.

2390 divided by 10 is 239.

718.64 divided by 10 is 71.864.

In order to avoid any possible confusion, make sure you understand that any whole number is
presumed
to have a decimal after it. We shall get more deeply into the subject in the chapter on decimals, but for the moment let's point out that 75 can be considered to be 75.00. Then, if we divide by 10, we move that presumed decimal one place to the left. 75 divided by 10 is 7.5.

Each digit, you remember, increases tenfold in value as it moves one place to the left. So to multiply by a hundred, we add
two
zeros (ten times ten), or move the decimal point two places to the right.

984 × 100 is 98400.

653.92 × 100 is 65392.

When dividing by a hundred, we also move the decimal point two places—to the left.

984 divided by 100 is 9.84.

653.92 divided by 100 is 6.5392. We would most likely round it off to 6.54.

Undoubtedly none of this is new to you. It is merely a refresher. But the refresher is important, because the more easily and automatically you can
think
this multiplication or division by ten or a hundred, the more quickly and confidently you will handle the short cuts that involve such division or multiplication as a basic part.

Our second step into the breakdown short cut is through another obvious technique that may well be second nature to you already.

In dealing with many numbers, you probably know already how to multiply by 9 in the “round off and adjust” method. Rather than multiplying by 9, you multiply by 10—and subtract 1.

Compare the two methods:

The working in these two examples is not dramatically different, but they are cited to illustrate a point and to lead into more sophisticated examples. Once again, for the sake of your number sense, try to “feel” the identity of the two expressions above of precisely the same situation.

Just as you probably already knew this special dodge in handling 9, it is likely that you have used in the past the same sort of approach in handling numbers very near 100.

If you have to multiply 238 by 99, surely you would not bother to set up the whole problem and multiply it out line by line. You just subtract one 238 from a hundred 238's, as this comparison demonstrates:

If you were required to multiply by 101, on the other hand, you would simply add one 238 to a hundred 238's. This, after a very moderate amount of practice, you easily do in your head. After a few tries, you should be able to “see” the answer as 24038.

Not very often is your work as extra-simple as multiplying by 99 or 101. But the principle works in a surprising variety of cases, and is the “round off and adjust” special subdivision of our first general short cut: breakdown.

The over-all rule for breakdown is this: break one of your numbers down into
two
easier-to-handle numbers.

Thus we broke 9 down into 10 and 1.

We broke 99 down into 100 and 1.

We can also—here is where the method becomes far more generally useful—break 45 into 50 less 5. 5, you note, is exactly
of 50. Or we break 44 down into 40 plus 4—the 4 being exactly
of 40.

Stop for a moment and try the first example:

It is especially helpful when you can break down a number into two parts of which one is an even fraction of the other, such as 50 and 5. You cannot always do this, of course, which is why we also use other short cuts.

The exact breakdown may well depend on the relationship between the numbers to be multiplied. In some cases one breakdown will make sense, in other cases quite a different breakdown.

Note how it varies in these two cases:

Which breakdown of 18 might you use in the first example? The number 18 can be broken into 12 plus 6 (½ of 12), into 9 plus 9 (two equal parts), into 20 minus 2 (
of 20).