Speed Mathematics Simplified (63 page)

How do we place the decimal? Remember the decimal rule. The answer has the same number of digits (to the left of the point) as do the two numbers multiplied (to the left of the point). 298 has three places, .36 has none, so the answer has three digits to the left of the point including the first digit of the first partial product, even if it is a 0. The answer is 107.28.

Do one more:

Cover the solution with your pad until you have finished this to your satisfaction.

8% is the equivalent of .08, and our solution looks like this:

Note that there seems to be a spare 0 in the answer. This is to aid the placing of the point in the answer, since the multiplier (.08) has in effect
minus one
places before the point. If we include the 0 in .08 in writing our answer, the correct
handling of the point is automatic. We place it two spaces to the right because there are two places to the left of the points in the numbers multiplied.

Finding What Per Cent A Number I
₈

Often you need to find what per cent one number is of another. You might have, for instance, the two numbers 15 and 75, and be required to express one of them as a percentage of the other.

The important thing is to make very sure which number is which. Do you want to know what per cent 15 is of 75, or what per cent 75 is of 15? It makes a big difference.

Recall at this point that a per cent is only a special way of writing a decimal, and that a decimal is a special form of fraction. So in either of the above cases, you are really being asked to show a fraction in percentage form.

If you want to know what per cent 15 is of 75, you need to convert into decimal (and therefore percentage) form the fraction 15/75. If you are required to state what per cent 75 is of 15, you again must convert into decimal and percentage form the fraction 75/15.

Another way of keeping your relationships absolutely straight, in case this conversion does not lock itself memorably in your mind, is that one of the numbers always follows the word
of.
You always ask “what per cent is this number
of
that?” The number
following the “of”
is always the
base
—the base of which you are figuring a percentage—and the
base
is always the
bottom
of the fraction.

You know perfectly well how to convert any fraction to decimal form. You divide the top by the bottom. To convert this decimal fraction to a per cent, move the decimal point two places to the right.

What per cent is 15 of 75?

The fraction to which we want a percentage answer is 15/75. Using the other key, the number following “of' is 75, and the base is the bottom—again, 15/75. Now convert:

Move the point two places to the right, and we have the answer 20%. 15 is 20% of 75.

Turn the relationship around. What per cent is 75 of 15? Here the fraction expressing the relationship is 75/15. Or, again, the number following “of” is 15 and therefore the base and the bottom. Divide:

In order to convert this in turn to a percentage, move the point two places to the right—adding 0's as necessary. So 75 is 500% of 15.

500% means that for each 100 parts of the other number, you have 500 parts of this one. Wiping out the 100's, you see that 500% is the same as five times as much.

Try one on your own now. Cover the explanation below with your pad and work out both sides of this relationship:

20 is what per cent of 50?

50 is what per cent of 20?

For the first comparison, the number following the “of,” and therefore our base, is 50. The fraction is 20/50. Dividing by the bottom, we get

We move the point two places to the right, and find that 20 is 40% of 50.

Reversing the question, we have a base of 20—the number following the “of.” The fraction is 50/20. The division is

Again we move the point two places to the right. 50 is 250% of 20.

In these examples, we have not bothered to reduce each fraction to its simplest form before dividing because showing the division with the original numbers in the question seems to
make the process clearer. In practice, of course, you would consider these numbers 2 and 5 rather than 20 and 50.

Finding An Unknown Base

One of the most baffling operations in percentage seems to be finding an unknown base. If you have a clear grasp of the relationships, however, it becomes quite easy.

An example of this situation might be the question, “90 is 45% of what?”

We know the number that is a percentage of another. We know the percentage. But we do not know the base.

Let us approach the method through logical conversion of the methods we already understand. Once you know why, you are not likely to forget how.

We have three numbers: 90, 45%, and “what.” The number (unknown) following “of” is “what,” so “what” is the base.

The fraction, therefore, is

We know the answer to the fraction, but we do not know the fraction itself. In order to convert a fraction to a decimal, and therefore a percentage, we divide the top by the bottom. So we will set up the problem, along with the answer we know:

Now, if someone asked you, without confusing matters by including words such as percentage and decimals, the question, “What divided into 90 gives the answer .45?” you would answer without a second thought, “Divide .45 into 90 and find out.”

Divider multiplied by answer must give number divided. Number divided, divided by the answer, must give the divider.

So we simply divide the number we have by the percentage, and we find the base: