Chapter 16 - Proportionate Change

The fourth generally useful type of short cut has no traditional name. Because the phrase most accurately describes what we do, we will call it proportionate change.

The technique is simply that: proportionate change. You change one number of a problem into a simpler form in any way you wish (double it, triple it, cut it to one-third, or whatever) but change the other number in proportion so the essential relationship remains the same.

For instance:

The conversion here should be quite obvious, one glance at the problem shows us that 45 can be converted into a one-digit (plus 0) number by doubling. So we double it, without hesitation, to the more easily handled 90.

The proportionate part of the rule is simpler for division than it is for multiplication. In division, you do to the number divided precisely the same thing you did to the divider. In multiplication, you do to the other number exactly the opposite of whatever you did to the first number.

In the example above, you double 45 to 90. To keep the proportion, you now double the 180 to 360. The answer, simply by inspection, is obviously 4—9 into 36.

Try one yourself:

Start by examining the divider for any simple change that will convert it to a one-digit number—plus a 0 if necessary. Doubling 35 changes it to 70. Do the same thing to the number divided, which changes it to 420. Most of us would find it rather difficult to "see" the answer to 35 /210, but the answer to 70 / 420 should be a matter of reading ea and sy as "easy."

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 / 210 x 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, ori 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 / 3 6 0 gives 40. Or 9 / 7 2 0 x 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.

Here are both workings:

Usual way                    Two proportionate change ways 

When you change your multiplier by cutting it in half or into another fraction, then you compensate by multiplying the other number or the answer by the same amount. Again, it is just the reverse of your compensation in division.

In order to master the point thoroughly, it would not hurt to work out all three forms of this problem. It will help you "feel" the identity of the end results, no matter how the numbers were twisted and turned in working out those results.

Proportionate change is especially valuable in dealing with fractions of all kinds. Even when a proportionate change cannot reduce one of the numbers you must handle to a single digit, it can often simplify it to a remarkable degree.

Would you rather divide by 4 ⅓—or by 13, and multiply the answer by 3?

Is it easier to multiply by 6 ⅝—or by 53, and divide the answer by 8? For an even more dramatic example, you would prefer to handle 6 ¼ as 25—and compensate by multiplying by 4 (in division) or dividing by 4 (in multiplication).

Try out the idea on these numbers:

If you multiply each of these numbers by the quantity that will convert it into a whole number, you get the following results (The multiplier is in parentheses.):

In each of the above cases, of course, you compensate in the other number or in the answer with the same multiplier. Do whichever comes more easily.

Run through one whole problem now. Cover the answer with your pad as you work this out with proportionate change:

Surely it is much simpler with the short cut than without it. We double 7 ½ to make 15, which we see at once goes into 300 exactly 20 times—times 2 is 40. Or 15 goes into 600 precisely 40 times.

Now do a multiplication with this technique. Move your pad over the answer and solve this problem with proportionate change:

Use whichever proportionate change suits you best, but do it before checking with the answer.

The logical conversion for 1 ¾ is to multiply it by 4 and divide the other number or the answer by the same factor. Your answer either way is 497. The two workings are these:

Proportionate change is very largely a special application of factoring, and contains some elements of aliquots as well—as you have no doubt observed. It is such a special application, particularly in compensating in the other number and in often making a number larger, that it is classically considered a separate short-cut method.

As an exercise in number sense, consider the essential identity of doubling 35 to make 70 and factoring it into 7 and 5. In multiplication, if you double 35 to 70, you divide the other number or the answer by 2 in order to compensate. Now, dividing by 2 is an aliquot approach to multiplying by 5, is it not? And we picked up an extra 0 when we doubled 35 to 70 —which corresponds to the seemingly missing 0 if we consider a division by 2 to be an aliquot for 5.

The various short cuts overlap and are overlapped by the others in many respects. The basic number relationships remain constant; we are merely using different conversions to make those relationships more visible and easier to handle.

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