Speed Mathematics Simplified (57 page)

Keep your top and bottom straight by matching to the fraction divided rather than to the divider, and you can read the answer at sight. It is 5/8.

Beware of one possible misunderstanding here. When you divide in this fashion, you
cannot reduce
in the normal fashion by dividing tops and bottoms simultaneously. This is because you would invert the divider if you rewrote it before multiplying, so in essence the top of the divider becomes its bottom and vice versa. You can, if you take care to keep track of the proper tops and bottoms, reduce by dividing both tops or both bottoms by any number that will go into them evenly, because you invert the divider in multiplying anyway.

Adding Fractions

Adding and subtracting fractions is, surprisingly, more work than multiplying or dividing them. The reason is simple, and is based on the fact that it makes no difference in what order you multiply a series of numbers—but it makes a big difference in what order you multiply and add.

Consider this quantity:

Does it make any difference whether you treat this as 2 × 3, plus 4—or as 2 × the sum of 3 + 4? Try it and see. One handling gives you 10. The other gives you 14.

It is critically important to add and multiply in the proper portions. 2 × (3 + 4) is not the same as (2 × 3) + 4. Examine the two expressions carefully and you will discover the cause for the difference. In the first handling, the 4 gets multiplied by the 2 after it has been added to the 3. In the second, the 4 never gets multiplied by the 2 at all. So the end result is quite different.

Another way of approaching the special rules for adding and subtracting fractions is to remember that each fraction is, depending on its bottom number, in its own special number system—one not accounted for in our regular digits and expressible in our digit system only as a division problem. 1/3, 2/3, and 3/3 are all quantities based on one-third of 1. But ¼, 2/4, 3/4, and 4/4 are quantities based on one-fourth of 1. Thirds and fourths are not in the same number system at all, and trying to add or subtract combinations of the two is like adding gallons and litres.

The first job in adding or subtracting fractions, then, is to get them all into the same number system, Fortunately, it is not hard at all.

There is a very simple way of converting different fractions into the same system. We just multiply the bottoms and adjust the tops. We can even forget that forbidding schoolroom phrase “lowest common denominator,” because we do not need it. All we have to do is multiply.

Here is how it works:

First, in order to determine the number system in which we can express both quantities, we multiply the bottoms. 4 × 3 is 12. This 12 will be the bottom of the answer, because it is a number system that can express both fourths and thirds accurately.

Before we can add, however, we must convert each fraction to this new system. ¾ is ¾, but it is not 3/12. How many twelfths is it? The simplest way to convert is to multiply each top by the
other
bottom, because this is the number by which we multiplied the bottom and, as we know, multiplying top and bottom by the same number does not change the value of the fraction.

3 × 3 is 9, so ¾ is 9/12. We do not worry about that in working, however. All we care about at the moment is the 9. For the second fraction, we multiply 2 × 4 and get 8. Now we add the two products, and this becomes the top of the answer. 9 + 8 is 17. The answer is 17/12.

Once again, look at these four expressions and try to feel their identity:

This answer is a fraction larger than 1. We will get to the handling of such fractions soon. First, let us finish addition and subtraction.

Try the simplified rule on the following addition. The rule, in one sentence, reads: To add fractions, multiply the bottoms for the new bottom; multiply each top by the other bottom and add these products for the new top.

You do not have to go through the entire step-by-step visualization above each time you do it. Just multiply the bottoms for the bottom of the answer. Multiply each top by the other bottom and add the products for the top of the answer.

For the problem above, our bottom is 2 × 3 or 6. 1 × 3 is 3, plus 2 × 2 is 4, gives 7 as the top. The answer is 7/6.

Try a few more with this technique. It is really simpler and faster than worrying about common denominators:

Work out and reduce where possible the answers to these on your pad.

The answers, in order, are 17/20, 16/12, 10/12, and 26/20. The second answer—16/20—can be reduced to 4/3.

Adding More Than Two

The rule becomes just a little more complicated when you add three or more fractions. You have to reduce all of them to the same number system.

The rule is not very much more complicated, however. Let us take it in two steps:

      
1. Multiply all the bottoms together. This will be the bottom of the answer.

      
2. Multiply each top by
all
the bottoms
except
its own, and add all the products. This will be the top of the answer.

This rule is precisely the same as the rule for adding two fractions, generalized to handle any number of fractions. Here is an example:

The first step is to multiply all the bottoms together. 2 × 3 is 6, × 5 is 30. The bottom of the answer is 30.

The second step is to multiply each top by all the bottoms except its own. 1 × 3 is 3, × 5 is 15. 2 × 2 is 4, × 5 is 20. 3 × 2 is 6, × 3 is 18. Add 15 and 20 and 18 to get the top of the answer: 53. The answer is 53/30.

Examine carefully the steps in this addition, and you will see that in each case we are really multiplying each fraction's top and bottom by the same number: the products of the bottoms of all the other fractions. This translates all the fractions into the same number system and adjusts all the tops at the same time, without changing the quantity of each fraction.

Do one on your own with this method:

First, find the bottom of the answer. 4 × 3 is 12, × 5 is 60.

Now for the top. 3 × 3 is 9, × 5 is 45. 2 × 4 is 8, × 5 is 40. 2 × 4 is 8, × 3 is 24. Adding 45, 40, and 24, we get 109 as the top of the answer…109/60.

Special Cases

There is a further short cut in adding a series of fractions in which some of them are already in the same terms. The usual method is to hunt through all the bottoms for the “lowest common denominator,” which takes a bit of figuring and then adjustment of each top.

It is far easier simply to add all like fractions first; then add the resulting unlike fractions in the method just described.

In order to add like fractions (all thirds, say, or all fifths), you simply add the tops. Do nothing to the bottoms. The sum 1/5 and 2/5 is the sum of the tops—3—over the same bottom: 3/5.

Here is how to handle a typical situation: