Speed Mathematics Simplified (51 page)

There is another way of handling the proportions, incidentally, and this is to change the answer rather than the number divided. In division, you can change the divider—divide into the original number divided—, then change the answer in the same way you changed the divider.

Our example above now becomes 70
× 2. “See” the answer; it is the same one we got before. According to the numbers and the change involved, this is sometimes easier.

Doubling is only one of the proportionate changes you can use. You can triple, quadruple, or multiply by any number you choose. Or you can cut in half, in thirds, in quarters, as you will. Remember that in division and multiplication you cannot add or subtract, however; the change must be in the nature of a multiplication or division. And remember to compensate in the other number or in the answer.

In its simplest terms, here is an illustration of a problem clearly calling for one specific proportionate change:

The simplest change here is to triple the divider. This gets rid of the fraction and reduces the whole divider to one working digit to boot. You divide by 100—and multiply the number divided or the answer by 3 to compensate.

The general rule becomes obvious from this example and the former ones. Use the proportionate change that will get rid of any fraction or turn the last digit into a 0—when possible, of course.

When dealing with the number 45 as a divider or multiplier, we double it—to form the easier-to-handle 90.

When dealing with 33 1/3, we triple—because it gets rid of the fraction, but also because in this case it turns the number into 100.

How would this rule apply to the divider 3½?

In order to get rid of the fraction, you double it to 7. If you see it in decimal terms—3.5—the .5 is also the signal to double, and you still change it to 7.

How about the number 1.25 (or 12.5, 125, etc.)? If you double it, you have 2.5. This is simpler to handle than 1.25, of course, but it is still in two digits. Double it again, on quadruple the original number, and it becomes the easy-to-handle number 5. You then compensate, if you are dividing, by multiplying the number divided or the answer by 5 to keep the change proportionate.

Changing Downward

Proportionate change does not always mean multiplying. It can also mean changing in the opposite direction. Consider this problem:

How can you most easily change this divider into a single digit number? You could do it by multiplying by 5, which converts 18 to 90. In this case, however, that is the hard way. Instead, cut it in half. Half of 18 is 9. Keep the change proportionate by cutting the number divided or the answer in half too. 9
gives 40. Or 9
× ½ gives 40.

Here is another example:

No multiplication of the divider will change it to a single digit number. But cutting it to 1/3 will; 1/3 of 21 is 7. So divide 7 into 1/3 of 168 (56) or into 168 and compensate by dividing the answer by 3.

It may have struck you that dividing a number to convert it is really only a new facet of the factor short cut. It is indeed. When we cut a divider to ½ or 1/3 or some other fraction, we are really factoring it—but you will note that
the rest of our handling is a little different, and your frame of mind as you look at the problem is quite different. You are thinking of change—not factors.

Some of the following numbers can be simplified by changing upward, some by changing downward. Play with them a bit until you feel you have the simplest form of each.

Try them yourself before reading on.

The quickest way to convert each of these in the proportionate-change short cut is:

Multiplying

It has already been pointed out that proportionate change applies as easily to multiplying as it does to dividing. In multiplying, however, you reverse the compensation. If you double the multiplier to simplify it, then you cut the other number or the answer in half. If you use 1/3 of the multiplier, then you triple the other number or the answer.

Here is an example:

Try this one on your pad or in your head:

Cover the answer with your pad until you have finished.

To simplify 45, you naturally double it to 90. Note here that if you divide 695 by 2, you will have a remainder. This will be wiped out when you multiply by 90; if it is not, then you went astray somewhere. In such a case, however, it is usually easier to divide the answer by 2 rather than the number multiplied.