Speed Mathematics Simplified (42 page)

This breakdown example emphasizes the value we put at the beginning of this chapter on being able to handle multiplication and division by 10, 100, etc., without hesitation or strain. This is the key to handling breakdowns such as the one above of doubling a number and then doubling the result—but multiplying by 10 in the first case and dividing by 10 in the second.

Our next number is 33. This is obviously 30, plus
of the product.

63 is based on the opposite principle. 63 can easily be broken down into 70, minus
of the product.

82 is a bit different. We will break 82 down into 80, plus
of the product. This is not difficult to handle. You simply divide the product by 4, but start writing your answer one extra place to the right. This divides by 4 and by 10 all at once—dividing by 40. See how it works:

In a division such as the one above, you have an automatic running check on your accuracy because the division must come out even. If it does not, you know you have made a mistake. This is because two whole numbers, when multiplied, must give a whole-number answer. So if your division has a remainder, you are warned to recheck it.

There is no clear-cut advantage in this particular problem to breaking down 82 in the fashion we did, rather than into 80 plus twice the original number (which is merely a simpler expression of what our regular multiplication does). In one case you divide by 4 and 10; in the other you double. If the product of 80 × 5555 were part of the problem, however, it would be very tempting to divide the first product of 44440 by 4 and 10 to get the second line. Once again, which breakdown is best depends on how the numbers relate to each other.

By and large, the major value of breakdown is in permitting you to use easier-to-handle operations and digits. Breaking down 78 into 80 minus twice the other number, for instance, lets you substitute a simple doubling for a multiplication by 8 at the second step.

Breakdown—like any short-cut technique—is valuable to you only as you learn to handle it easily and well. Do not dismiss it out of hand if your first reading of a particular problem leaves you more baffled than enlightened, but on the other hand do not force yourself to use a particular short cut that after a few tries does not spring into your mind naturally and obviously. The purpose is to save work, not make it.

Longer Numbers

The easiest numbers to break down are usually those with two digits. But this does not mean that much longer numbers
cannot also be broken down, frequently with dramatic results.

Take the multiplier 297, for instance. The nearest one-digit number that can form the base of your breakdown is 300. The difference between 297 and 300 happens to be a very convenient
of the product.

Note the same feature in the numbers 396—495—594. For each of them, you can substitute a multiplication by the next even hundred and subtract
of the product, instead of multiplying through by three digits and then adding all three lines.

In reverse, the same short cut is possible with 303—which you have probably used in the past without special instruction. There is no need to multiply twice by 3; merely copy the first product again, two places to the right, and add.

Now that you have learned to add or subtract
or
,
or
, and so on, the possible range becomes considerably larger. You might handle 306, for instance, by doubling the first product two places to the right, rather than multiplying by 6.

As the breakdowns become more complex, so does the saving of time in using them. When you break down a three-digit number into two one-digit parts, you save a full digit in your work while at the same time performing a basically simpler operation.

For instance, consider the multiplier 784. There is no simple relationship between 700 and 84, so you do not break it down that way. 784 is just 16 less than 800, however, and 16 is exactly
of 800. So, instead of multiplying digit by digit by 784, we can multiply by 800 and subtract
of the product. Now the short cuts become visibly dramatic: