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The shortest path

Shortcuts are a great way to do a lot of math in your head. You know the guy in the office that always responds when you ask “What’s 25 times 56?” He’s using shortcuts, and so can you. The best thing about these so-called tricks? They always have a real math basis, or they wouldn’t work! Here are a few multiplication shortcuts. Learn them, and you’ll see how often the situations for which they are useful show up.

What’s the square of [some number > 20]? Does this come up for you very often? Well, maybe not, but from the square you can get great estimates on other products, and you’ll be surprised how often you come across the problems when you can easily solve them. Now, no one really expects you to memorize all the squares to 100 but, with a few shortcuts, you can compute most of them fairly easily.

  • The square of any number that ends in 5 (e.g. 45, 95, 245), represented as (X5)2, is ((X * (X+1))*100) + 25. This looks complex, but isn’t. If you’re squaring 85, X is 8, so the answer is (8*9)*100 + 25, or 7200+25 = 7225. 1152 is (11*12)=132, *100+25 = 13225. See?
  • If you know the square of X, the square of X+1 is just X2 + X + X + 1; i.e. if you know 302 is 900, then 312 is 900 + 30 + 31 = 961. The reverse also works; i.e. if you know 202 is 400, then 192 = 400 – 20 – 19 = 361. Hmm, those both end in 61… is that another pattern? (I haven’t found one yet).
  • The square of the numbers near 100 can be computed as a difference from 1002. Since 96 is 4 less than 100, 962 = 1002 – (200*4) + 42, or 10000 – 800 + 16 = 9216. 972 = 10000 – 600 + 9 = 9409, etc. For number over 100 (like 104), it’s the same, but the 200 factor is positive: 1042 = 10000 + (4*200) + 16 = 10816.

Ok, but what if I don’t want a square? Fine, not all of life’s math problems involve squares. As it turns out, there are some shortcuts for finding the product of different numbers, too.

  • Multiplying any number X by 50 or 25 is easy because they go so neatly into 100. Simply compute ((X/2)*100) to multiply by 50, and (((X/2)/2)*100) for multiply by 25. e.g. 468*50 = 468/2 * 100 = 23400; 468*25 = 468/2/2 * 100 = 234/2 * 100 = 11700. It’s easy to see how this would also work for any multiple of 500, 250, etc.
  • Multiply any two numbers that end in 5. If two number ending in 5 (125, 65) are represented as X5 and Y5 where X and Y are the digits preceding the final 5 (12, 6), then the product is ((X*Y) + ((X+Y)/2))*100 + 25. In English, multiply X and Y, then add the average, multiply the sum by 100 and add 25. e.g. 125 * 65 = ((12*6) + (12+6)/2) * 100 + 25 = (72+9)*100 + 25 = 8125. Note, for this one it’s possible for the average of X and Y to be a non-integer (with .5 at the end)… if that happens, the product will end in 75 instead of 25, for obvious reasons. Extra credit if you can prove the first squares shortcut above using this one.
  • Multiply two numbers that are an even, “short” distance from each other. If you have two numbers X and Y, and X – Y is even and “small”, then you can easily compute the product using the difference of squares. e.g. to multiply 48 and 52, with an average of 50 (A) and a distance from the average of 2 (D), the product will be the difference of the squares of A and D; 48 * 52 = (50-2)*(50+2) = (A-D)*(A+D) = A2 – D2 = 502 – 22 = 2500 – 4 = 2496.

Study these a bit, and you’ll start to see uses for them show up in your job, your hobbies, or your kids’ homework. Most of these are just shortcuts I use from day to day or figured out for fun. I’ll sometimes see a pattern after 2 or 3 tries at a similar set of problems, then I’ll spend time proving my shortcut (or disproving, as the case may be) so I can use it on the rest and future problems. Try to prove any of the above shortcuts, and comment with your proof. I’d love to see it.

The Count

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