So rather than continue the hijack, I thought I’d start a thread about mental math tricks.
My go-to trick is to reduce a multiplication of multiple digit numbers to repeated multiplications of smaller numbers - I’d rather avoid mental addition when possible. So when someone asked me to multiple 12 times 128 (which did happen!) I double 128 twice (256, then 512) and then triple the result to get 1536. I’m sufficiently practiced that I find doubling or tripling a number in my head to be pretty easy - but if the number is long, I’ll “chunk” the longer number into simpler subgroups - for example, I’d think of 32536 as 7500 plus 108, or 7608 (notice - only one carry required). Double checking is also helpful. For 617, I can triple 17 to get 51, and then double to get 102, or recognize 17 as about a 1/6th of a hundred, and just say “about a 100” as the answer (if all I need is an approximation).
Well, sure, many of us use such tricks, but the problem is that people who don’t (or refuse to try to) understand math can’t grasp the logic behind the tricks.
I’ve often said that elementary math education should include logical thinking, and that fractions, decimals, and per centages should be presented as a single concept. If kids were taught to truly understand the logic behind 1/2 = .5 = 50%, then maybe their minds wouldn’t go blank and panic at the sight of “%” as adults.
The key is to break it down to problems you know off the top of your head. So I see a number like 17 and I think “20 is easier to work with… lets do 20 x 6”. Then I know my answer will be 3 x 6 more than the real answer. So I end up with 120 - 18 = 102.
Another simple breakdown would be (10 x 6) + (7 x 6)
Yep, that’s exactly how I do 17x6 in my head. I actually went for the first method instinctively, but the second (which is just regular longhand multiplication) is perhaps easier. My brain prefers the first for some reason.
Multiplying by 5 is the same as dividing by 2 and moving the decimal over a spot. Not always needed, but if you have something like 1284x5, I can quickly tell you it’s 6420 by dividing by 2 instead of working it out the usual way.
I use all sorts of tricks, including the a^2-b^2 mentioned above. If I want to multiply by 64, I will just double 6 times. If I want to multiply by 63, I will multiply by 64, then subtract. I know all the squares of numbers up to 32, which is often useful. If I want to find the prime decomposition of a number, I use various tricks. E.g. to find if a number has either 29 or 31 as a divisor, I try to divide by 2931 = 899, which is near enough to 900 to make it relatively easy and then, assuming the remainder is not divisible by either 29 or 31, it follows that the original number wasn’t. Once lying in bed in a rented vacation house whose address was 1763 I decided to find its prime decomposition. I went all the way up to 4143 before discovering that it was that. The next morning I mentioned that the address was the product of a pair of twin primes and my son answered immidiately that it must 41*43. He’d estimated the square root. I guess it worked since I soon fell asleep. Beats counting sheep.
It’s a lot easier to multiply 128 by ten and by two and add.
Similrly, X times 78, is X times 80 minus 2X. To get X times 80, double it three times and add a zero. Then double X and subtract that. 78x137 = 137 > 274 > 548 > 1096 > 10960, then subtract 274. To avoid carrying, subtract 300 (10660) and add 26 = 10686. Reduce one of the numbers to 2s and 5s, ,which are much easier to multiply by.
I do Celsius to Fahrenheit by doubling, subtracting 1/10, and adding 32. So 31C to 62 minus 6.2, rounded to 6, for 56, then add 32 for 88. The other way, subtract 32, halve the result, add 1/10. 88F to 56 to 28 plus 2.8 rounded = 31.
I find doubling to be really easy (though that could be that I just have practiced it many times (and of course, the fellow who asked me the question blundered by making 128 one of the numbers to multiply, since that is a power of two, while I can double twice with no thought at all).
I had a former boss who would get annoyed with me because I could do cost estimates in my head faster than she could do it on a calculator. She couldn’t understand orders of magnitude, which is not a difficult mental exercise. It’s helpful when you never bothered to learn your multiplication tables past 12. She would say something like “So if we have 25 units and we put $13,000 into each one for renovations. . .” In my head I’m thinking “20 times 13 is 260 and add about another quarter of that, so about $320,000.” No matter how many times I did this, she would insist on continuing with her pocket calculator to reach nearly the same answer.
Richard Feynman was good at arithmetic but not as good as Hans Bethe who taught Feynman how to square numbers near 50:
(50 ± x)[sup]2[/sup] = 2500 ± 100x + x[sup]2[/sup]
Double the units digit and subtract the tens digit (or the tens and hundreds) from the results. If the result is divisible by seven, the number is divisible by seven. e.g 84 – 4*2=8 - 8 = 0
And thirteen.
Multiply the units digit by four and then add it to the tens digit. If it’s divisible by 13, you’re set. e.g., 52. 2 * 4 = 8 + 5 = 13
Decimal conversion of ninths or elevenths are easy. Ninths is repeating decimal of multiples of 11 (ex: 4/9 = .44444), and elevenths is repearing myltiples of nine ( 4/11 = .36363636363.)
Also sevenths is a repeating decimal of multiples of seven times 2-4-8, but starting in different places. 1/7= .142857,142857,142857… and 3/7 = .42857,142857,142857…
I know how to tell if a number is divisible by 3. Add up the digits, keep doing it until you get to a manageable number. If that’s divisible by 3 so was the original number.
123,456,789 = 45, divisible by 3.
12,345,679 = 37; not divisible by 3.
When adding up a long list of numbers, I go down the columns grouping them into clusters that add up to 9, 10 or 11, and mentally add those in my head. It reduces a list of 50 numbers to only about 20, which can be done twice as fast. Sometimes I have to jump down the list to borrow a 1 or a 2 and skip it when I come to it. So my mind goes down the list saying 10, 20, 31, 41, 50, 59, 70, 80, 89, 98 . . . When I get to a hundred, I start over and use fingers to add up the hundreds.
Also, keep in mind how much precise accuracy will be required. If I’m going to divide my sum by another number, for a percentage,which will be rounded off, an exact tabulation is not required and I can save a lot of time if I’m close enough.
Seeing the title of this thread reminds me of my principal. I taught seventh-grade math and she came to me one day and told me I should just be teaching the kids math short cuts so it would be easier for them and skip doing explanations and practice. She didn’t understand when I told her that short cuts usually came after you understand what you’re doing.