Download A. Multiplying Two 2-digit Numbers: 47 x 38

0011 0010 1010 1101 0001 0100 1011
RAPID Math
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• Divisibility Rules
– 2 All even numbers (ending in 0,2,4,6 or 8)
– 3 The sum of the number’s digits is divisible by 3
0011 0010 1010 1101 0001 0100 1011
– 4 The last two digits of the number form a 2-digit
number divisible by 4
– 5 The number ends in a 5 or 0
– 6 Divisible by both 2 and 3
– 7 Take the last digit, double it, and subtract it from
Usually
the digits that remain. Repeat until you get to a
easier to
divide by
7
number that you know is/is not divisible by 7.
– 8 The last three digits of the number form a 3-digit
number divisible by 8
– 9 The sum of the number’s digits is divisible by 9
– 10 The number ends in a 0
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11Alternately add and subtract the digits from left to right.
(You can think of the first digit as being 'added' to zero.) If
the result (including 0) is divisible by 11, the number is also.
0011 0010 1010 1101 0001 0100 1011
Example: to see whether 365167484 is divisible by 11, start
by subtracting:
[0+]3-6+5-1+6-7+4-8+4 = 0; therefore 365167484 is
divisible by 11.
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12 If the number is divisible by both 3 and 4, it is also
divisible by 12.
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13 Delete the last digit from the number, then subtract 9
times the deleted digit from the remaining number. If what
is left is divisible by 13,
then so is the original number.
• For what single digit value of n is the
number n5,3nn,672 divisible by 11?
0011• 0010
1010rewrite
1101 0001 the
0100digits
1011
Let’s
of the number,
alternating subtraction and addition signs
between the digits as follows: n – 5 + 3 – n + n
– 6 + 7 – 2 = n – 3. If n – 3 is divisible by 11,
then the entire original number will be
divisible by 11. This means we need to find a
digit n, such that n – 3 is equal to 11, 22, 33,
etc, and don’t forget 0!! If n = 3, then n – 3 = 0
which is divisible by 11.
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Multiplication by 5, 50, 25 etc.
0011 0010 1010 1101 0001 0100 1011
• Halving numbers is also very easy, so rather
than multiply by 5 we can put a 0 onto the
number and halve it, because 5 is half of 10.
So for, 44 × 5 we find half of 440 which is
220 so 44 × 5 = 220
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Compute
0011 0010 1010 1101 0001 0100 1011
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68 × 5 =
87 × 5 =
452 × 5 =
27 × 50 =
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Squaring Numbers that End in
5
0011 0010 1010 1101 0001 0100 1011
• Squaring is multiplication in which a
number is multiplied by itself: so 75 × 75 is
called "75 squared" and is written 75².
• The formula By One More Than the One
Before provides a beautifully simple way of
squaring numbers that end in 5.
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• Example of it, In the case of 75², we simply multiply the 7
(the number before the 5) by the next number up, 8. This
gives us 56 as the first part of the answer, and the last part
is simply 25 (5²).
Compute??
0011 0010 1010 1101 0001 0100 1011
• 75²=
• 35²=
• 65²=
• 85²=
Note: Example 4
Also since 4½= 4.5, the same method applies
to squaring numbers ending in ½.
So 4½² = 20¼, where 20 = 4×5 and ¼=½².
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Compute?
0011 0010 1010 1101 0001 0100 1011
• 305² = 93025 where 930 = 30×31
• So compute:• 205²=
• 605²=
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Multiplying Two Numbers Using
the Difference of Two Squares
46 x 54
0011 0010 1010 1101 0001 0100 1011
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Square the average of the two numbers
Average = 50
502 = 2500
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Multiplying Two Numbers Using
the Difference of Two Squares
46 x 54
0011 0010 1010 1101 0001 0100 1011
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Square half the difference of the two
numbers
54 – 46 = 8
Half of 8 is 4
42 = 16
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Have you ever heard of
a²- b²
0011 0010 1010 1101 0001 0100 1011
WHEN THE SUM of two numbers multiplies
their difference -(a + b)(a − b)
-- then the product is the difference of their
squares:
(a + b)(a − b) = a2 − b2
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This is a very handy formula for fast
computation!!!
Multiplying Two Numbers Using the
Difference of Two Squares
0011 0010 1010 1101 0001 0100 46
1011 x 54
Subtract the two numbers to get your answer
502 – 42 =
2500 – 16 =
2484
46 x 54 = 2484
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Practice
0011 0010 1010 1101 0001 0100 1011
36 x 44
1584
28 x 32
896
14 x 36
504
67 x 83
5561
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To Multiply Two Numbers Ending in 5
and Differing by 10
0011 0010 1010 1101 0001 0100 1011
75 x 85
• Write down 75
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• In front of the 75 write the product of the tens
digit of the smaller number and the sum of the
tens digit of the larger number and 1
7 x 9 = 63
75 x 85 =
6375
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Practice
35 x 45
1575
85 x 95
8075
65 x 75
4875
0011 0010 1010 1101 0001 0100 1011
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Multiplying Two Numbers Squared
2 x 32
8
0011 0010 1010 1101 0001 0100 1011
• Multiply the numbers then square
8 x 3 = 24
82 x 32 = 242
= 576
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Multiply A Number By 9
37 x 9
0011 0010 1010 1101 0001 0100 1011
Multiply the number by 10 37 x 10 = 370
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Subtract the original number from the number
above
370 – 37
= 363
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Multiplying/Dividing by Factors
0011 0010 1010 1101 0001 0100 1011
Sometimes you can rapidly work a problem
by multiplying/dividing by factors of the
second number
144 x 15 => 144 x 3 = 432
144 x 15 = 2160
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432 x 5 =2160
Multiplying/Dividing by Factors
0011 0010 1010 1101 0001 0100 1011
Practice problems:
16023
237 x 49
7532
296 x 28
41104
734 x 56
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Checking Your Work By
Casting Out 9’s
0011 0010 1010 1101 0001 0100 1011
To check your work by “casting out
nines” you:
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First add the digits together
Then keep adding the digits together
till you get a one digit answer
Checking Your Work By Casting Out
9’s
0011 0010 1010 1101 0001 0100 1011
Example:
13579
+ 24680
38259
1+3+5+7+9=25; 2+5=7
2+4+6+8+0=20; 2+0=2
3+8+2+5+9=27; 2+7=9
Thus the answer checks!
BUT WAIT!
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9
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Checking Your Work By Casting Out
9’s
0011 0010 1010 1101 0001 0100 1011
It Gets Easier!
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Now we get to actually “casting out
nines”
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When adding, leave out all nines and
numbers that add to nine
Checking Your Work By Casting Out
9’s
0011 0010 1010 1101 0001 0100 1011
Example:
13579 leave out 9 and 3+7 1+5=6
+ 24680 2+4+6+8+0=20; 2+0=2
38259 leave out 9 3+8+2+4=17; 7+1
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Unfortunately, this only shows mistakes 8 out of 9
times, but it is still a quick check.
Casting out Nines
0011 0010 1010 1101 0001 0100 1011
65324
+ 89173
154497
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