Multiplication by Five ----------------- ======================================== The Rule: take half the neighbour; add 5 if the "number" is odd. This is another simple lesson, note that there is now one added twist "add 5 if the 'number' is odd." This is another frequently used method in this multiplication system, and it always and only refers to the "number" (see definitions) in its raw state... and has nothing to do with the result of doubling or halving the "number." Note that if the Rule were written in the proper order of execution, it would read: if the "number" is odd add 5; either way add half the neighbour. Obviously that version is not as easy to understand, so keep in mind that the semicolon in the rule separates steps that may actually be executed in a different order, but are written in that order for clairity. Now, take an easy example of 6381 times 5. 0 6 3 8 1 x 5 --------- 5 - look at the number and see that it is odd, say [ 5 ] then look for the neighbour (assume zero since there is no neighbour) and add "half" of that (zero again) to the five... Your mental stops should have been: [ 5 ] answer: 5. 0 6 3 8 1 x 5 --------- 0 5 - look at the number (8) and see that it is not odd, then look for the neighbour (1) and add half of that ("half" of 1 is zero)... Your mental stops should have been: [ 0 ] answer: 0 0 6 3 8 1 x 5 --------- 9 0 5 - look at the number (3) see that is it odd, say [ 5 ] then add "half" of the neighbour to the 5, and say [ 9 ]. Your mental stops should have been: [ 5, 9 ] answer: 9 continue the pattern... I will continue to give abbreviated commentary. 0 6 3 8 1 x 5 --------- 1 9 0 5 - even, add half of 3... say [ 1 ]. Your mental stops should have been: [ 1 ] answer: 1 0 6 3 8 1 x 5 --------- 3 1 9 0 5 - even, add half of 6... say [ 3 ]. Your mental stops should have been: [ 1 ] answer: 1 See... what did I tell you? Now try the next one on your own. 0 4 5 8 9 7 2 x 5 ------------- the product is 2,294,860. You've already learned my usage of "number", "neighbour", and "carry". Now I'll introduce you to to the concept of "half" a number. "Half" with quotation marks because it is a simplified half... throw out fractions. Thus "half" of 5 would be 2, "half" of 3 is 1, and "half" of 1 is zero. Of course half of 4 is still 2, and so on with all even numbers. Try to do this instantly... don't look at 6 and say "half of 6 is 3", instead look at 6 and say 3. Try doing that now on the following digits. 8,6,4,5,3,0,9,2,1,5,3,7,4,9,0,8,1,7,2,6 Simple huh? Let's move on... Multiplication by Six ------------------- ========================================= The Rule: To each "number" add "half" of the neighbour; plus 5 if the "number" is odd. Note that you do not add 5 if the "neighbour" is odd... you only have to do that if the "number" is odd. Take an easy example, 2563 times 6. 0 2 5 6 3 x 6 --------- 8 - add 5 to the 3 because the "3" is odd; there is no neighbour, so 8 is the answer. 0 2 5 6 3 x 6 --------- 7 8 - 6 plus "half" of 3 is 7; the "number", 6, is even so no further steps are taken... 0 2 5 6 3 x 6 --------- `3 7 8 - 5 (odd) plus 5 (the number) plus "half" of 6 is 13. 0 2 5 6 3 x 6 --------- 5`3 7 8 - the "carry" plus 2 plus "half" of 5 is 5. 0 2 5 6 3 x 6 --------- 1 5`3 7 8 - 0 plus "half" of 2 is 1. Of course all this explanation is only to make this simple method as crystal clear as possible. With a reasonable amount of practice in your daily life this method will become second nature. Try these two yourself: 0 1 2 5 3 x 6 --------- 0 2 1 3 4 8 9 6 x 6 --------------- The answer to the first problem is 7,518. And the answer to the second is 12,809,376. Thus far all of the numbers I have used as multiplicands have been long numbers, but you can use the same methods for single digit multiplicands. Try this one, many people had problems memorizing it for some reason (but there will be no need for your children to spend many unhappy hours in memorization if you teach them what I'm teaching you) : 0 9 x 6 --- `4 - 5 (odd) plus 9 is 14; no "neighbour" 0 9 x 6 --- 5`4 - the "carry" plus 0 plus "half" of 9 is 5 Some comments on proper mental methods ------------------- ========================================================== It is important that you start off with the proper mental methods of calculation. Just as in learning to speed read you must learn to see whole words and lines of text rather than o n e l e t t e r at a time, so too should you learn to look at a number and see its "half" rather than have to think about it. The methods I'm teaching you become second nature to accountants and mathematicians whose livelyhood involves numbers, but have not yet become so for you... that will be the most difficult part of what you are going to learn. So you must practice, practice, practice. Another important practice is only saying only the result of adding a number or taking half of its neighbour, like this: 0 1 2 8 x 6 ------- 6 8 The 6 is the 2 plus half the 8. but do not say "half of 8 is 4, and 2 and 4 is 6." Instead, look at the 2 and the 8, see that hald of 8 is 4, and say to yourself "2,6." At first this will be difficult, so it may be better to say to yourself "2,4,6." Another point to practice is the step of adding the 5 when the number (not the neighbour) is odd. Take this case: 0 2 3 6 x 6 ------- `1 6 The 1 is the 1 of 11, as the (`) shows, and the 11 is 3 plus 5 (because 3 is odd) plus 3 (half of 6). The correct procedure, at first, is to say "5,8,3,11." After some practice it should be cut down to simply "8,11." The 5 that come sin because 3 is odd should be added first, otherwise you might forget it. In the same way, when there is a (`) for a carried 1 (or more rarely (") for a carried 2), this should be added before the neighbour (for times 11) is added or half the neighbour (for times 6). Once again, if it is left for after this step it may be completely forgotten. One final example to bring the rest of the mental steps together: 0 8 3 4 x 6 ------- 5`0`0 4 Look at the 8 and say "9,", adding the dot (`); then say "10," adding "half" the 3. At first it is better to look at the 8 and say "9," adding the dot, then say "1" for "half" of 3, then "10,", and write the zero and "carry" dot. When there is a dot and also a 5 to be added (because of oddness), say "6" instead of "5" and then add the number itself. This cuts out a step and is easy to get used to. Try some problems of your own. Multiplication by Seven ----------------- ========================================= The Rule: Double the number and add half the neighbour; add 5 if the number is odd. Just like multiplication by six only different right? I will continue to present the solutions in the same manner you should be solving them yourself. Practice solving them in this manner yourself before looking at the solution as this helps to develop your concentration... the secret to learning. Take the easy example of 452 times 7. 0 4 5 2 x 7 ------- 4 - (even), 4 (double 2), (no neighbour), 4. 0 4 5 2 x 7 ------- `6 4 - 5 (odd), 15 (double 5), 1 (half of 2), 16. 0 4 5 2 x 7 ------- `1`6 4 - (even), 9 (double 4 plus the carry), 2 (half of 5), 11. 0 4 5 2 x 7 ------- 3`1`6 4 - (even), 1 (double zero plus the carry), 2 (half of 4), 3. Like I said, easy. Now you try the next one on your own: 0 8 3 1 5 x 7 --------- 0 4 1 2 0 5 9 3 x 7 --------------- You didn't forget that half of 1 is zero did you? The solutions are 58,205 and 28,844,151 aren't they.. Of course... Now I'll help you practice some of the mental steps you have learned. Some practive in proper mental methods ------------------- ========================================================== You've already practiced taking half of a number, but though it should be simple for you, you haven't practiced looking at a number (8 for example) and saying its double... do try that now: 2,5,3,8,5,3,6,9,5,3,6,8,9,6,4,2, 1,0,3,5,8,6,5,4,2,6,8,9,7,4 Great! Now practice looking a number, saying its double and adding the neighbour... this is how you multiply by 12. 2 5 3 8 5 3 6 9 5 1 4 8 9 6 1 2 4 0 9 5 8 6 5 4 2 6 8 9 7 4 3 1 6 3 5 8 3 5 1 6 6 5 9 4 5 2 3 6 Almost done... now, using the same pairs of numbers practice looking at the number, saying its double and adding half the neighbour. This is used in "times 7" problems. Finally look at the following numbers, and practice saying "5," then say 5 plus double the number. 5, 3, 9, 7, 1, 9, 5, 7, 3, 1, 3, 5, 1, 9, 5, 3 There are all kinds of ways you can practice what you have learned without having to actually sit at a table with pencil and paper... try multiplying the price of gas by 6, 7, 11, or 12 with the grease of your finger on the dust on your back window while filling your tank... try doing so with your birthdate, ssn, home or work phone numbers, or drivers liscence number in your head... while waiting at a stoplight do so with liscense plate numbers... see if you can think of some more... and practice, practice, practice. enjoy...Last updated 12 Dec 95