Speed Mathematics Simplified (41 page)

We have already covered the most convenient breakdown for 15. It was inserted here to remind you of the repetitive character of many useful breakdowns. In multiplying 895 by 10 and adding half the product, you would think simply “8950, plus 4475, is 13425.”

In dealing with 473 × 38, you run into another fraction in your breakdown. 38 is 2 less than 40. 2, in turn, is
of 40. So, to multiply by 38, you can multiply by 40 and subtract
of the product.
is just ½ of
.
You do it like this:

Note two instructive points about this example:

First, you can (and should) jot down the answer to 40 times 473 from left to right without copying the original number. It is simply 4 × 473, digit by digit in the no-carry method, plus one zero.

Second, you can (and should) jot down from left to right the division of 18920 by 20 without any strain. You simply divide by 2 and start one place to the right when you put down the answer, which also divides automatically by 10. The combination results in a division by 20.

The third breakdown, 682 × 27, breaks down the 27 into 30 minus
the product. Here again, you multiply 682 by 3 as you jot down the result and add one zero to make the multiplication by 30 instead of by 3. Under it you write the same digits one place to the right, without the zero, which automatically divides by 10, and then subtract:

Short cuts are a variety of methods, not a single system. There are many problems to which you can find short cuts, others to which you cannot, without doing more work than simplified arithmetic would involve. It comes down to recognizing the short cut that makes sense in a flash, because if you brood for more than an instant or two on whether or not to use a short cut at all, in that time your new systems of basic arithmetic could have finished most of the problem.

Try recognizing breakdown possibilities in these numbers:

Some of these begin to pioneer new breakdown possibilities that we have mentioned but not yet fully demonstrated.
Yet your own good number sense should show you interesting ways in each case.

50, for instance, is exactly half of 100. It is entirely up to you whether you find it easier and quicker to multiply by 5 and add a zero, or to add two zeros and divide by 2. Simple as it may seem, this is a perfectly valid short cut.

45 has been mentioned before, as 50 times the number, less
of the product. If you have not noticed it before, 45 is also 30 times the number, plus ½ the product. Which is better? Neither. It depends on the relationships of the numbers with which you are working, and on your own preferences.

24 is a new one. 24 is 20 times the number (double it and add a zero), plus
of the product. Jotting down
is simple: double the product, but start one place to the right. This divides by 10 and multiplies by 2 at the same time. If this seems at all obscure, follow the working in this example: