# MATH SHORTS

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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 = ½  and .25 = ¼

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 =
¼ (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 =
½ (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

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:

Beat the Calculator

To returnclick on the Patton Jr. High mascot

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