Speed Mathematics Simplified (40 page)

For the first example, the most convenient breakdown of 18 might well be 12 plus 6—because most of us have dealt enough in grosses to know almost by instinct that 12 × 12 is 144. So 18 × 12 is 144 plus ½ of 144 (72)—which we can see as 216.

For the second example, however, most of us could not quickly “see” the answer to 62 × 12. If we break down 18 into 9 plus 9, we can quickly multiply 62 × 9 and then add the answer to itself. Furthermore, we can multiply by 9 using 10 minus 1. This is two-step breakdown. Complex as it may seem at first glance, a very quick and simple way of solving this example would be to handle it as “620 minus 62—doubled.”

If we chose the third breakdown, our number work would be surprisingly similar to that involved in the second. 20 minus
2 is identical to 10 minus 1, doubled; only the order of operation is changed.

The advantages of this sort of breakdown show up more dramatically, of course, in longer numbers. Try one of the last two breakdowns of 18 on the following problem. Use your pad and pencil:

Let us choose the (10 minus 1) doubled breakdown for 18 in this case. Here is how the work should look:

As with many of the demonstrations, the short-cut nature of the method is not as striking at first sight as you will find it in actual practice. Often you will use almost as many figures, and as many operations. But you are using basically simpler combinations: multiplying by 10 instead of by 9; subtracting instead of doing another digit-by-digit multiplication; doubling instead of adding two lines.

Just for comparison, here is how the 20 minus 2 breakdown for 18 works in the same problem:

In this case, it is presumed that you can jot down twice any figure at sight, and add a 0 at the end to get the effect of multiplying by 20.

There is virtually no limit to the breakdowns you can find. You can break down a number into two parts that add up to the original number (such as 12 plus 6 in 18, or 10 plus 5 in 15) or two parts of which you subtract one from the other to get the number (such as 100 minus 1 for 99, 20 minus 2 for 18, 60 minus 6 for 54).

How would you break down 81? Depending on the number you needed to multiply, you could make it 80 plus the original number, or 90 minus
of the
product
(since 90 minus 9 is 81, and 9 is
of 90).

Your proportions need not always be
. They can be ½, 1/3, ¼, or any other convenient fraction. The key is to find a
convenient
fraction, or there is no sense in using the breakdown method.

See if you can recognize convenient breakdowns for these numbers:

Of 39, we can make 40 minus 1. Of 26, we would make 25 plus 1. In another short cut, incidentally, you will find a far easier way to use a number such as 25 than by multiplying by 2 and then 5 and adding. 77 is obviously 70 plus
of the product, while 63 is 70 minus
of the product. We can tackle 125 in several ways; for this use, we can consider it 100 plus ¼ of the product. 720 is 800 minus
the product.

In the choice of short-cut methods, and in the best use of each, you have great flexibility. There is no substitute for number sense here, for it is in finding the relationships that your key to method selection lies. There are so many variations, so many slightly different approaches, that it is up to you to select the fastest and easiest in each case.

Try these problems on your pad, finding an appropriate breakdown for each: