Speed Mathematics Simplified (61 page)

A surprising number of people have trouble determining the proper “bottom” to decimal-form fractions, because they never learned this simple trick. For practice in using it, read the following decimals as fractions by saying aloud both the top and bottom of each fraction, just as we might say .3 as
:

If you keep firmly in mind the visualization of the point as a 1 followed by as many 0's as there are digits, you should be able to read the above decimals at sight as 67/100 (hundredths); 20/100 (this is the same as 2/10, but it has a slightly different meaning in decimals. That comes later.); 432/1,000 (thousandths); 5/10; 25/100; 6478/10,000 (ten-thousandths); 4/1,000.

Mixed Numbers

Expressing mixed numbers (a whole number plus a fraction) is much easier in decimals than it is in fractional form. Whatever part of the number is in front of (to the left of) the point is a whole number. The part to the right of the point expresses the fraction.

So 22.4 is read as “22 and 4/10ths.”

It is often read, too, as “22 point 4.” There is nothing wrong with this. It is a short hand way of reading and saying the number, but it does not drive home the actual quantity involved as firmly as does reading the decimal as a full fraction.

1.43 is read as “1 and 43/100ths.”

Read aloud the number 45.67.

If you read it properly as “45 and 67/100 (hundredths),” go ahead to read the following numbers. Say the full number followed by the fractional part in terms of tenths, hundredths, thousandths, or whatever:

If you hesitated over any of these, particularly the one-thousandth or eight-hundredths, it would be a good idea to review the last few pages before going on.

Adding Decimals

There is no trick at all to adding numbers with decimals in them if you keep the basic rule in mind: line up the points. If you were adding 10,342 to 61, you would line up the right-hand ends of these numbers. The point in a number with a decimal fraction is just as clearly and firmly the end of the whole number as is the end when the digits come to a stop there.

Using this rule, set up the following numbers for addition:

Cover the arrangement below with your pad until you have done your part.

Using the points as the ends of the whole numbers, you line up the above numbers for addition like this:

That is really all there is to it. Elementary, but very important. Once you have lined up the numbers properly, you simply go ahead and add. Ignore the points, except to put a point in your answer in line with the column. Tens carried back across the point as you add behave just as if there were no point there, which is one of the great advantages of using decimals. They enable you to handle the fractional parts right
along with your whole numbers, instead of creating them separately.

Having dismissed addition this easily, we can say the same thing about subtraction: keep your points in line, and “borrow” (or slash) across the point as if it were not even there. One example will make this clear:

In both addition and subtraction, you can pretend the points are invisible—as long as you line them up, and make sure to put one in your answer in line with the others.

Multiplying Decimals

When you come to multiplying decimals, you do not bother to line up the points because you have another way of placing the point properly in your answer. Refer back to the chapter on no-carry multiplication, if need be, to refresh your understanding of the following rule:

Add the digits in the two numbers multiplied. Starting with the very left top digit (including the 0 if it is a 0), count this many digits for the answer.

In multiplying decimals, add only the following special qualifier to the general rule:

—to the left of the point.

That “to the left of the point” applies both to the numbers multiplied, and to the answer as well.

It works like this:

How did we place that point in the answer? Each of the two numbers multiplied has two digits to the left of the point. So our answer should have four digits before the point. We start at the very left of the top line with the 0 that does not show up in the final answer but that does have to be counted.

In every other aspect of the problem, we simply ignore the points altogether. You can prove it out by nines-remainders or elevens-remainders, ignoring the points for this purpose too
except
that you start at the point in figuring odd and even digits for an eleven-remainder. If you use continuous subtraction, just keep right on subtracting as you go past the point.

In the example above, you could also have used the classic “point off as many places from the right as there are places to the right of the point in the two numbers multiplied.” To rely on this method, however, would rob you of the rapid-estimating nature of no-carry multiplying. Work from left to right instead of right to left, and you can do just as much of any problem as you need to in order to get the accuracy required in that particular situation.

Dividing Decimals

For dividing decimals, we cannot improve on the usual rule: move the point in the divider (if any) all the way to the right. Put a point in your answer as many places to the right of the point in the number divided (if any) as you moved the point in the divider.

If this means adding 0's to the number divided in order to move your point far enough, go ahead and add them.

Here are two examples:

Try it yourself. Where will the point in the answer appear for each of the following problems?