MATH SHORTS



Ever want to do math in your head, faster than a speeding calculator?

Well then, that's the challenge and here's how ...

Say you want to multiply 375 x 32; well, that's a little tough, but if you 
know decimal equivalents, such as .375 = 3/8, you're well on your way.

Take 32 and multiply it times 3/8.  Now the denominator 8 will cancel the 
whole number 32 and make it 4.  Now take 4 x the numerator 3 and that gives 
you 12.  Well, since the original number was changed to a decimal and 3 places 
were lost, add those three places back on to the answer, 12, and you get 12,000 - 
the answer for 32 x 375.  Isn't that a simple strategy?  Let's try another, 
but first you need to know ...


DECIMAL EQUIVALENTS AND HOW THEY ARE RELATED TO THEIR FRACTION COUNTERPARTS
You know for example .50 = and .25 = Notice the relationship between the fractions themselves and their equivalent decimals: .25 is half of .50 and 1/4 is half of 1/2! Now, half of 1/4 (.25) is ____ ? Right, 1/8. So, what is half of .25? Too hard? Then ask yourself what is half of 25? Right, 12 1/2! Or that's 12.5. Since we're looking at a decimal with originally 2 places (.25), that means half of .25 = .125 Zounds, do you know what that means? You can determine the decimal equivalents of fractions by merely taking half of the fractions you already have memorized! OK, so here's what we know so far: 1.00 = 1 .75 = 3/4 .50 = 1/2 .25 = ? Now let's try some more: half of .25 was ___ ? so it's equivalent fraction is ___ ? OK, .25 = 1/4 and .125 = 1/8 So what is 3/8 as a decimal? Take the first fact: what does 1/4 + 1/8 equal? You say 3/8? Ah ha!! So add their decimal equivalents, .25 + .125 = .375 Bingo!! 3/8 = .375 So what would be the decimal equivalent of 5/8? Well, let's see 1/2 + 1/8 = 5/8; and 1/2 = .50 and 1/8 = .125 So, .50 + .125 = ? Now you're catching on! 7/8 = ? Harder yet, 1/16 = ? Well, let's see 1/8 = .125 and half of 1/8 is 1/16; so 1/2 of .125 is ___ Too hard?? Think... what is half of 125? (125 = 100 + 25, so 1/2 of 100 = 50 and half of 25 = 12.5) So half of 125 is, is... yes, you got it! 62.5!! Now the original decimal had 3 places, so lets put those back in plus one more: .0625 and there you have it! 1/16 = 9/16 = 1/8 (2/16) = .125 10/16 (5/8) = .625 3/16 = 11/16 = (2/8) = .25 12/16 (3/4) = .75 5/16 = 13/16 = 6/16 (3/8) = .375 14/16 (7/8) = .875 7/16 = 15/16 = (4/8) = .5 Now, let's get down and multiply! 125 x 16 = (1/8 x 16 = 2), now add 3 places = 2,000 3125 x 32 1875 x 48 = (5/16 x 32 = ?), now add 4 places = (3/16 x 48 = 9), now add 4 places = 90,000 375 x 64 = 875 x 32 = 4375 x 16 = 25 x 20 = 75 x 24 = 625 x 48 = 75 x 44 = 50 x 98 = 8125 x 80 = Now, try these. 25 x 48 = 625 x 32 = 125 x 48 = 875 x 72 = 375 x 24 = 9375 x 32 =
More Mental Strategies
Check this out: Any 2-digit number ending in 5 is rather easy to square, that is, multiply the number times itself. The product will always end in 25. Now take the first digit (in the 10's place) and multiply it times the next higher digit, and place that number before the 25. (e.g. 25 x 25 = ?; 2 x 3 = 6, so the answer is 625. 35 x 35 = ?, 3 x 4 = 12, so the answer: 1225) Squaring 5's 15 x 15 = 65 x 65 = 25 x 25 = 75 x 75 = 35 x 35 = 85 x 85 = 45 x 45 = 95 x 95 = 55 x 55 =


The Difference of 2 Perfect Squares, or Multiplying Differences Try this one: 32 x 28 Too hard? Nanu, not with a strategy!! Check this out: What if we rewrote it like this: (30 + 2)(30 - 2) Now, take 30 x 30 = 900, and 2 x 2 = 4, and find the difference: 896 and there you have it! Now try 47 x 53 rewritten = (50 - 3)(50 + 3), multiply 50 x 50 = 2500, and then 3 x 3 = 9, and find the difference: 2500 - 9 = 2491! Holy products, it works!! Caution: this only works with numbers equally distant from a 10's number. In our first example both numbers were 2 away from each side of 30 and in our second, 3 away from each side of 50. 26 x 34 = 46 x 54 = 21 x 19 = 68 x 72 = 42 x 38 = 76 x 84 = 57 x 63 = 93 x 87=
To Double or Not A wild way to multiply by doubling one number and taking half of the other! Begin with your number on the left and take half of that number and each succeeding number until you get down to 1, casting off all remainders. (It's shorter if you put the smallest number on the left, no matter what order the original multiplication problem was written.) Place these in a vertical column under the first number. With the number on the right, double it and each succeeding number until you get down to the 1 from the first column. Cross out all the even pairs of numbers in both columns. Now add together the remaining numbers in the right column. That sum equals the product of the original two numbers multiplied together! Example 1: 17 * 41 Take half of 8 82 Double the number the number 4 162 in this column in this column 2 328 each time you go down each time you go down 1 656 Now eliminate the underlined pairs of numbers since they are both even. Next add the numbers remaining in the right column: (41 + 656 = 697, the answer to 17 * 41) Example 2: 25 * 53 Take half of 12 106 Double the number the number 6 212 in this column in this column 3 424 each time you go down each time you go down 1 848 Now eliminate the underlined pairs of numbers since they are both even. Next add the numbers remaining in the right column: (53 + 424 + 848 = 1325, the answer to 25 * 53) Example 3: 43 * 426 Take half of 21 852 Double the number the number 10 1704 in this column in this column 5 3408 each time you go down each time you go down 2 6816 1 13632 Now eliminate the underlined pairs of numbers since they are both even. Next add the numbers remaining in the right column: (426 + 852 + 3408 + 6816 + 13632 = 18318, the answer to 43 * 426)
Communal Numbers Now here's one for addition: Ever been asked to add numbers on the spot and try carrying sums in your head? Well, stop it!! Try this instead with the following example, 85 + 35 = (rearrange it to your advantage: you know 85 + 5 is 90, now add the 30 and you get the intended sum of 120!! Wasn't that a bit easier?? 127 + 38 = (Think: 127 + 3 = 130, now add 35, you get ____ ) 134 + 136 = (Think: 34 + 6 = 40, 140 + 130 = ____ ) 97 + 68 = 49 + 81 = 58 + 62 = 54 + 96 = 89 + 46 = More advanced: 49 + 83 = (49 + 1 = 50, 50 + 80 = 130, then add 2, or 132)

For help in factoring polynomials:

Try factoring by type!


more SHORTS:


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