Speed Mathematics Simplified (56 page)

But the top 2 and bottom 8 are also divisible by the same number. Dividing both by 2, we have:

Now our answer comes out as 1/12. This is the same as our answer the first time we tried it, after reduction.

This process is traditionally called “cancellation.” It might more sensibly be called “reduction,” because that is what you really do. You do not cancel anything; you reduce numbers where you can.

Try the short-cut method of reduction on these:

Use your pad to play with these before checking the reduced forms below.

4/7 × 5/12 can be reduced by dividing the top 4 and bottom 12 by 4. The reduced form is 1/7 × 5/9, giving the answer 5/21.

3/5 × 5/9 has two reductions. The two 5's can both be divided by 5, which gives us 3/1 × 1/9. The top 3 and bottom 9 can both be divided by 3, giving 1/1 × 1/3. The answer must be 1/3.

2/3 × 6/7 offers only the bottom 3 and top 6, both to be divided by 3. Now the problem is 2/1 × 2/7, which gives 4/7.

5/6 × 3/20 offers two reductions. 5 goes into the top 5 and bottom 20, reducing the problem to 1/6 × ¾. 3 goes evenly into the bottom 6 and top 3, further reducing the problem to ½ × ¼. Answer, 1/8.

There is an important reason why I refuse to call this process “cancellation.” The technique is usually taught as an “X” process, from the top of one fraction to the bottom of another. It is definitely not necessarily so;
any
top and bottom (never a top and top or bottom and bottom, of course) will do, in the same fraction or in any of the fractions to be multiplied.

You can reduce 6/8 × 2/12 the same way you would 8/12 × 2/8, using any top and bottom that can be divided evenly by the same number. The only difference is that usually fractions are presented in arithmetic books already reduced for such problems. In our real-life figure work, they are not always so reduced for us. Look for reducing possibilities everywhere.

Dividing Fractions

Just as it may seem peculiar to multiply two quantities (if they are fractions) and get an answer smaller than either of them, so may it appear outrageous to divide one quantity into another and (if they are fractions) get an answer larger than either.

Keep firmly in mind that division is merely the reverse of multiplication, and review in your mind the reasons for the strange results you get in multiplication. In effect, the fraction divided is the answer to an imaginary multiplication, and the purpose of the division is to find the missing partner in the multiplication.

Let us start into the division of fractions with a simple example:

If we multiply by multiplying the respective tops and bottoms, then we might expect to divide by dividing them. In a problem this simple, we can indeed: 1 into 3 gives 3, and 2 into 4 gives 2: 3/2 is the answer.

Do not worry about that 3/2 yet. We will get into so-called improper fractions later.

The technique of simple division will theoretically work with any problem, but since every number does not “go into” every other number evenly we sometimes would end up with awkward decimal remainders and create some really difficult- to-handle answers.

This is why the standard trick of “inversion” has been developed. The trick has this rule:

To divide by a fraction, turn it upside down and multiply by it.

If this seems at all odd, reinforce your grasp of the reason why, as well as the rule, by considering that all division is merely an inversion of multiplication. When you multiply by 4, you count the other number ¼ times.

Another way of saying “divide by 27” is to say “multiply by 1/27.”

So another way to say “divide by ¾” is to invert the fraction and say “multiply by 4/3.”

The single greatest source of confusion to many people is remembering which fraction to invert. If you fully understand the why, you cannot ever again become confused. To make extra sure, run through the comparison once more.

In order to divide 28 by 14, would you set it up as

So in order to divide ½ by ¼, by would you set it up as

It is the fraction
by
which you divide that you invert—always.

Pull out your pad and do these examples with inversion:

Inverting the divider of the first problem gives us ¾ × 2/1. The answer is 6/4, which reduces to 3/2. If you used short-cut reduction, you would have converted the problem to 3/2 × 1/1 before multiplying.

The second problem becomes 7/8 × 5/2, which gives an answer of 35/16. This fraction cannot be reduced.

The third, when you invert the divider, becomes 5/6 × 3/2. This can be reduced to 5/2 × 1/2, with the answer 5/4.

Short-Cut Division

If you are sure of your technique, there is no need to rewrite such division problems with the divider inverted. You can do the inversion in your head by following this rule:

To divide, multiply the top of the fraction divided by the bottom of the divider, and put it on top. Multiply the bottom of the fraction divided by the top of the divider, and put it on the bottom.

In other words, you simply multiply each top by the other bottom. Keep your answer straight by using the fraction divided as your guide for the answer: the product of this top and the other bottom becomes the top of the answer. The entire process automatically inverts the divider without rewriting.

Here is an example:

Top of fraction divided (2) times the other bottom (4) is 8. Since you used the top of the fraction divided, this 8 goes on top of the answer. Bottom of fraction divided (3) times the other top (3) is 9. This goes on the bottom of the answer. The answer is 8/9.

Try one yourself: