Multiples of 9 Using Finger Trick

Some find it hard to memorize the multiplication table. For multiples of 9, learn this finger trick and you will find that multiplying 9 with any number will be quick and easy.

This trick is done by spreading your fingers as shown in the image, then fold the corresponding finger for the multiplier of 9. Left side of the folded finger will be the tens digit while right side of the folded finger will be the ones digit.

1 x 9 = ?
Folding 1st finger, no finger remains on left side, while 9 fingers remain on the right side. That makes 0 and 9, so 1 x 9 = 9. The answer is obviously 9, this is just for the sake of example.

2 x 9 = ?
Folding 2nd finger, 1 finger remains on left side, while 8 fingers on right side. So 2 x 9 = 18.

3 x 9 = ?
Folding 3rd finger, 2 fingers remain on left side, while 7 fingers on right side. So 3 x 9 = 27.

4 x 9 = ?
Folding 4th finger, 3 fingers remain on left side, while 6 fingers on right side. So 4 x 9 = 36.

5 x 9 = ?
Folding 5th finger, 4 fingers remain on left side, while 5 fingers on right side. So 5 x 9 = 45.

6 x 9 = ?
Folding 6th finger, 5 fingers remain on left side, while 4 fingers on right side. So 6 x 9 = 54.

7 x 9 = ?
Folding 7th finger, 6 fingers remain on left side, while 3 fingers on right side. So 7 x 9 = 63.

8 x 9 = ?
Folding 8th finger, 7 fingers remain on left side, while 2 fingers on right side. So 8 x 9 = 72.

9 x 9 = ?
Folding 9th finger, 8 fingers remain on left side, while 1 finger on right side. So 9 x 9 = 81.

10 x 9 = ?
Folding 10th finger, 9 fingers remain on left side, while no finger on right side. So 10 x 9 = 90.
It's obviously 90. I just show here that this trick is applicable from 1 to 10.

Quick and easy, right?

#Matamata

Multiplying Any Number with 5

Multiplying any number with 5 can be done easily by adding digit "0" after it, then divide it by 2.

Example:
You are buying 5 pieces of something worth 245 Php. How much is the total cost?

245 x 5 = ?

Add "0" to 245, then divide it by 2.
245 x 5 = 2450/2  = 1225 Php


More examples:

14 x 5 = ?
14 x 5 = 140/2 = 70

160 x 5 = ?
160 x 5 = 1600/2 = 800

887 x 5 = ?
887 x 5 = 8870/2 = 4435

1268 x 5 = ?
1268 x 5 = 12680/2 = 6340

#Matamata

Multiplying Special 2-digit Numbers

42 x 48 = ?

Notice that their tens digits are the same, 4. And their ones digits sums up to 10, that is, 2 plus 8 equals 10. For special numbers like these, there is a shortcut to get the product.

To get the answer, first multiply the tens digit by itself plus 1, that is 4 x (4+1) or simply put 4 x 5.
4 x 5 = 20
20 will be the first digits of the answer. To get the remaining two digits, multiply  the ones digits 2 and 8.
2 x 8 = 16
So we have 20 and 16, put them together, we have 2016 as the answer.
42 x 48 = 2016

More examples:


53 x 57 = ?
53 x 57 = 5x6   3x7

53 x 57 = 3021

61 x 69 = ?
61 x 69 = 6x7   1x9

61 x 69 = 4209

26 x 24 = ?
26 x 24 = 2x3   6x4

26 x 24 = 624

35 x 35 = ?
35 x 35 = 3x4   5x5

35 x 35 = 1225

#Matamata

Computing Percent Savings

We love to get discounts whenever we buy something. Know first hand how much you are about to save for a given amount of percent off sale with these mental tricks.

Example, a shirt with a price of 300 Php with a given % sale.

For 10%, moving 1 decimal point, 300 becomes 30.
10% of 300 = 30 Php.

For 5%, that's half of 10% savings.
5% of 300 = 30/2 = 15 Php.

For 15%, add the 10% and 5% savings.
15 % of 300 = 30 + 15 = 45 Php.

For 20%, that's twice of 10% savings.
20% of 300 = 30x2 = 60 Php.

For 25%, add 20% and 5% savings.
25% of 300 = 60 + 15 = 75 Php.
Or take 1/4 of the price. To make it simple, divide by 2 twice.
25% of 300 = 300/2 = 150/2 = 75 Php.

For 30%, that's 3 times of 10%.
30% of 300 = 30x3 = 90 Php.

For 35%, add 30% and 5% savings.
35% of 300 = 90 + 15 = 105 Php.

For 40%, multiply 10% savings with 2 twice.
40% of 300 = 30x2 = 60x2 = 120 Php.

For 45%, add 40% and 5% savings.
45% of 300 = 120 + 15 = 135 Php.

For 50%, that's half of the price.
50% of 300 = 300/2 = 150 Php.

More examples:

10% of 985 = 98.5
15% of 280 = 28 +14 = 42
20% of 740 = 74x2 = 148
25% of 460 =  460/2 =230/2 =115
30% of 1500 = 150x3 = 450
35% of 600 = 60x3 + 30 =210
40% of 240 = 24x2 = 48x2 = 96
45% of 240 = 96 + 12 = 108
50% of 649 = 649/2 = 324.5

#Matamata

Checking Exact Change

"Count your money before you leave." A friendly reminder when you are paying at the cashier or in vendor's stand. Check the exact change by mastering "All from 9, Last from 10."

Example:
You hand 1000 Php bill for payment of goods worth 648 Php. How much is the exact change? Know the exact change without the aid of calculators.

Just deduct all the digits from 9 and the last non-zero digit from 10.
1000 - 648 = ?
9 - 6 = 3
9 - 4 = 5
10 - 8 = 2
1000 - 648 = 352

Or you can think of it this way:
1000 - 648 = ?
   9  9  10
 - 6  4  8  
   3  5  2   answer

Or simply think of a number that will complete the digits into 9 9 10.
 1000 - 468 = ?
 6 4 8 
 3 5 2 answer

More examples:
 100 -78 = ?
 7 8 
 2 2 answer

 1000 - 385 = ?
 3 8 5 
 6 1 5 answer

 1000 - 740 = ?
   9 10 0           (10 for last non-zero)
- 7  4  0  
  2  6  0   answer

1000 - 407 = ?
   9  9 10
- 4  0  7   
  5  9  3  answer

#Matamata

Subtracting From Left to Right

Subtraction can be done from left to right. You can even start answering while the question is still ongoing.

Example 1

Step 1
  7,896
- 5,2--   (question ongoing)
  2,6--   (answer ongoing)

Step 2
  7,896
- 5,28-   (question ongoing)
  2,61-   (answer ongoing)

Step 3
  7,896
- 5,281   (question done)
  2,615   (answer also done)

Example 2
  8,652
- 5,478   
  3,2(-2)(-6)
  3,174 answer

The answer is done by subtracting digit to digit.
8 - 5 = 3
6 - 4 = 2
5 - 7 = (-2)
2 - 8 = (-6)

3,2(-2)(-6)
From here, apply "All from 9, Last from 10" and subtract 1 from the preceding positive number. Bar above the number indicates negative.
      _ _
3,2 2 6 = 3, 2-1, 9-2, 10-6
3,1 7 4 answer

Example 3
   134
 -  78   
     _ _
  1 4 4   = 1-1, 9-4, 10-4
     5 6 answer

Example 4
  52,485
- 38,347  
   _        _
2 6, 1 4 2   = 2-1, 10-6, 1, 4-1, 10-2
1 4, 1 3 8  answer

Example 5
  654,321
- 345,678   
      _  _ _ _
3 1 1, 3 5 7  =  3, 1-1, 9-1, 9-3, 9-5, 10-7
3 0 8, 6 4 3 answer

"All from 9, last from 10" is applied for negative numbers because these are simply subtracted from base number with zeros.
       _ _
3, 2 2 6  = 3,200 - 26 = 3,174
   _ _
1 4 4  = 100 - 44 = 56
   _        _
2 6, 1 4 2  = 20-6, 1, 40-2 = 141,388
      _  _ _ _
3 1 1, 3 5 7  =  310,000 - 1, 357 = 308,643

#Matamata

Square of Any 2-digit Number

The square of any 2-digit number can be done easily by this trick.

Square the tens digit and the ones digit and put it on both sides. Double the product of the digits and put it in the middle. If the results have no carry over, these become the digits of the answer.

Below are examples of no carry over.

13 x 13 = ?
1*1   2*1*3   3*3
  1       6        9
13 x 13 = 169

12 x 12 = ?
1*1   2*1*2   2*2
  1       4        4
12 x 12 = 144

11 x 11 = ?
1*1   2*1*1   1*1
  1       2        1
11 x 11 = 121

21 x 21 = ?
2*2   2*2*1   1*1
 4        4        1
21 x 21 = 441

41 x 41 = ?
4*4   2*4*1   1*1
 16       8        1
41 x 41 = 1681

70 x 70 = ?
7*7   2*7*0   0*0
 49       0        0
70 x 70 = 4900

When the results are 2-digit numbers, carry the tens digit to the preceding number.
Below are examples with carry over.

62 x 62 = ?
6*6     2*6*2   2*2
 36        24       4
 36+2    4         4
 38        4         4
62 x 62 = 3844

18 x 18 = ?
1*1   2*1*8     8*8
 1         16      64
 1+1     6+6     4
 2         12       4
2+1       2        4
3           2        4
18 x 18 = 324

15 x 15 = ?
1*1     2*1*5    5*5
 1         10       25
 1+1      0+2      5
 2          2          5
15 x 15 = 225

It seems long to write because I am showing here the step by step process. But this trick takes only seconds in the mind.

#Matamata

Square Roots of Perfect Squares

Square root is the reverse for square. If you memorized the squares of 1 to 10, you can easily answer the square roots of 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100. How about the square roots of perfect squares greater than 100? Image below shows the relation between square and square root.

The method is done by reversing the process of squaring 2-digit numbers. Let's take an example from the image, square root of 169. Segregate the digits of 169:
1      6      9

First, think of a number that when squared will equal to 1 .
1*1 = 1
So, first digit of the answer is 1.
Next, double this number then divide from the next digit 6.
6/(2*1) = 3
So, last digit of the answer is 3.
Square root of 169 = + - 13  (positive and negative 13)

To check if perfect square, take the last digit 3 then square it.
3*3 = 9 This equals to the last digit of the problem 169. Therefore, perfect square.

More examples:

Square root of 1681 = ?
16     8          1
4*4   8/(2*4)
4     1
Square root of 1681 = + - 41

Square root of 441 = ?
4         4             1
2*2     4/(2*2)
2       1
Square root of 441 = + - 21

Square root of 3600 = ?
36       0      0
6*6      0
6       0
Square root of 3600 = + - 60

Square root of 1225 = ?
12        2         5

Here, 12 is not a perfect square. We will use a number that if squared will be near to 12, but not over it. If 3x3 = 9, less than 12. If 4x4 = 16, more than 12. Therefore, use 3, then there will be remainder of 12 - 3*3 = 3. Put this remainder beside the next digit 2 and become 32.

32/(2*3) = 5 with remainder 2

Square root of 1225 = + - 35

Rewriting the solution:

Square root of 1225 = ?
12        2                      5
n*n      R2/(2n)             r5  R = 12 - 3*3 = 3
3*3    32/(2*3)              r5
3        5 with r = 2       25
Square root of 1225 = + - 35
5*5 = 25 = r5   Therefore, perfect square.

Square root of 888 = ?
888 is not a perfect square. There is no perfect square ending with 2, 3, 7 and 8. This can be check on the squares of 1 to 10.

These are long to write but quick on the mind.

#Matamata

Square of Numbers Ending with 5

To find the square of any two digit number ending with 5, multiply the first digit by itself plus 1 then put 25 after it.

Example

35 x 35 = ?

First digit is 3 multiplied with 3+1 equal to 4.

3x4=12 , then put 25 after it.

35 x 35 = 1225

More examples

65 x 65 = ?

6x7 = 42

65 x 65 = 4225

75 x 75 = ?

7x8 = 56

75 x 75 = 5625

#Matamata

Cube Roots of Perfect Cubes

Calculate cube roots of perfect cubes in seconds.

Preparation:

1. Memorize the cubes of 1 to 10 shown below.

1³  =  1

2³  =  8

3³  =  27

4³  =  64

5³  = 125

6³  = 216

7³  = 343

8³  = 512

9³  = 729

10³ = 1000

2. From the above cubes of 1 to 10, remember the following properties.

If the last digit of the perfect cube = 1, 4, 5, 6, 9 & 0, the last digit of cube root is equal to that number.

If the last digit of the perfect cube = 2, 3, 7 & 8, the last digit of the cube root = 8, 7, 3 & 2 respectively, their  10’s complement.

It’s very easy to remember.

1  ->  1  (Same numbers)

8  ->  2 (8+2 = 10)

7  ->  3 (7+3 = 10)

4  ->  4 (Same numbers)

5  ->  5 (Same numbers)

6  ->  6 (Same numbers)

3  ->   7 (3+7 = 10)

2  ->   8  (2+8 = 10)

9  ->   9 (Same numbers)

0  ->   0 (Same numbers)

Take note also that

8 -> 2 and 2 ->  8 

7 -> 3 and 3-> 7

Example 1:  Find Cube Root of 4913

Step 1

Split the digits, group of 3 digits.

      4        913

Step 2

Take the first group which is 4 .

Find out a maximum cube that is not greater than 4. Use 1³ = 1.

Hence the first digit of the answer  = 1. 

Step 3

Take the last group which is 913. 

The last digit of 913 is 3.

Remember point 2, If the last digit of the perfect cube = 3, the last digit of the cube root = 7

Hence the last digit of the cube root  = 7

That is, the answer = 17

More examples:

Cube root of 804357 ?       

804           357

9³ = 729          7

9                     3

Answer: 93

Cube root of 9261 ?

9              261

2³ = 8              1

2                     1

Answer: 21

Cube root of  32768 ?

32               768

3³ = 27            8

3                     2

Answer: 32

Cube root of 97336 ?

97            336

4³ = 64            6

4                     6

Answer: 46

Cube root of 216000 ?

216          000

6³ = 216          0

6                     0

Answer: 60

It is as simple as that, all you need to do is memorize the cubes of 1 to 10.

#Matamata

Memorizing Tip

Memorizing tip for students:

During school days, studying is not hard as it is. Do what you want to do in your free time. At home, eat dinner, watch TV shows, play games and have fun. But before going to sleep, read your school notes without memorizing what you read. Be it Math, Science or History, just read and do not memorize. Every school day, new lessons, new notes, new chapter to read. Read from the very start until the present. Do it every night.

Eventually, you will be amazed that at time of your exams, you will remember what you read and even have memorized it already, without the intention of memorizing it in the first place. This technique will make you always prepared whenever there is an exam. You don't even have to study the night before exam. You become always ready for it. You will not be afraid anymore to be called and answer your teacher's questions about your lessons.

Just read, don't think or analyze, don't pressure your mind. Just read, read from start to end, every night. Feed your mind.


#Matamata

Simple Product of Big Numbers

Some people find it hard to multiply big numbers. There is a trick to do it simpler. This is applicable for numbers near a base. A base can be 10, 100, 1000 or higher.

Example:
98 x 94 = ?

This question can be answered in seconds. The answer is 9212.

Both 98 and 94 are numbers near 100. First, take their differences from 100.
100 - 98 = 2
100 - 94 = 6
Then, subtract one difference from the other number.
98 - 6 = 92
94 - 2 = 92
It will always have the same answer, so choose which one is easier for you. This is the first two digits of the final answer. The other two digits is the product of the differences 2 and 6.
2 x 6 = 12
Putting 92 & 12 together, we have 9212 as the answer.

 -2    -6
98 x 94 = ?
98 - 6 = 92 
2  x  6 = 12
98 x 94 = 9,212

More examples:

98 x 87 = ?

-2   -13
98 x 87 = ?
87 - 2 = 85
2 x 13 = 26
98 x 87 = 8,526

998 x 986 = ?

  -2    -14
998 x 986 = ?
986 - 2 = 984
2  x  14 = 028   (base is 1000, 3 zeros means 3 digits for answer)
998 x 986 = 984,028

7 x 8 = ? 

-3    -2
7  x  8 = ?
7  -  2 = 5
3  x  2 = 6   (base is 10, 1 zero means 1 digit for answer)
7  x 8 = 56

For numbers above the base, the trick becomes easier. Instead of subtracting the differences, add.

    3       5
103 x 105 = ?
103  +  5 = 108
3 x 5 = 15   (base is 100, 2 zeros means 2 digits for answer)
103 x 105 = 10,815

    7       8
107 x 108 = ?
107 + 8 = 115
7 x 8 = 56   (base is 100, 2 zeros means 2 digits for answer)
107 x 108 = 11,556

  3      5
13 x 15 = ?
13 + 5 = 18
3 x 5 = 15   (base is 10, 1 zero means 1 digit for answer. Therefore add 1 to 18, 18+1=19)
13 x 15 = 195

For numbers that one is below the base and the other one is above the base, subtract the base to these numbers and consider the positive and negative differences.

Example:
96 x 108 = ?

 -4     +8
96 x 108 = ?
96 + 8 = 104
108 - 4 = 104 (same answer)
(-4) x 8 = (-32) 
96 x 108 = 10,4(-32)
96 x 108 = 10,400 - 32
96 x 108 = 10,368

More examples:

98 x 114 = ?

 -2   +14
98 x 114 = ?
114 - 2 = 112
98 + 14 = 112 (same answer)
(-2) x 14 = (-28) 
98 x 114 = 11,2(-28)
98 x 114 = 11,200 - 28
98 x 114 = 11,172

102 x  97 = ?

    2    -3
102 x 97 = ?
102 -  3 = 99
97 + 2 = 99 (same answer)
2 x (-3) = (-06)   (base is 100, 2 zeros means 2 digit for answer.)
102 x 97 = 99(-06)
102 x 97 = 9,900 - 06
102 x 97 = 9,894

These are long when written, but just quick in the mind.

#Matamata

Completion by Addition and Subtraction

Adding 400 and 68 is as easy as

400 + 68 = 468 .

How about adding 598 and 78 ?

Add 2 to 598 to make it whole 600, making the addition simpler. Since 2 more is added, subtract 2 from the sum to get the final answer.

598 + 78 = ?

600 + 78 - 2 = 676

More examples:

44 + 19 = ?

44 + 20 - 1 = 63

97 + 86 = ?

100 + 86 - 3 =183

994 + 678 = ?

1000 + 678 - 6 = 1672

Same goes with subtraction. Subtracting 20 from 46 is as easy as

46 - 20 = 26 .

How about subtracting 18 from 54 ?

Add 2 to 18 to make it 20. Since 2 more is subtracted, add back 2 to get the final answer.

54 - 18 = ?

54 - 20 + 2 = 36

More examples:

444 - 198 = ?

444 - 200 + 2 = 246

55 - 29 = ?

55 - 30 + 1 = 26

875 - 496 = ?

875 - 500 + 4 = 379

#Matamata

Squares of Any Number

Case 1: Below a base number.

Base number can be 10, 100, 1000 or higher.

997^2 = ?

1000 - 997 = 3

997^2 = 997 - 3 | 3^2

997^2 = 994 | 009  ••

997^2 = 994009

•• Number of zeros of base number indicates number of digits for the last group of the answer. That is why 3^2 = 009 .

Other examples:

96^2 = ?

100 - 96 = 4

96^2 = 96 - 4 | 4^2

96^2 = 92 | 16

96^2 = 9216

9987^2 = ?

10000 - 9987 = 13

9987^2 = 9987 - 13 | 13^2

9987^2 = 9974 | 0169 ••

9987^2 = 99740169

Case 2: Above the base number.

13^2 = ?

13 - 10 = 3

13^2 = 13 + 3 | 3^2 •••

13^2 = 16 | 9

13^2 = 169

••• If a number is below the base, subtract. If a number is above the base, add. That is why 13 + 3 and not 13 - 3.

Other examples:

108^2 = ?

108 - 100 = 8

108^2 = 108 + 8 | 8^2

108^2 = 116 | 64

108^2 = 11664

1012^2 = ?

1012 - 1000 = 12

1012^2 = 1012 + 12 | 12^2

1012^2 = 1024 | 144

1012^2 = 1024144

Case 3: Near a multiple of base number.

204^2 = ?

Base number is 2*100. Follow the same procedure with case 1 and case 2, but this time, first group of the answer will be multiplied by base factor (2).

204^2 = ?

204 - 200 = 4

204^2 = 2*(204+4) | 4^2

204^2 = 416 | 16

204^2 = 41616

Other examples:

196^2 = ?

200 - 196 = 4

196^2 = 2*(196 - 4) | 4^2

196^2 = 2*192 | 16

196^2 = 384 | 16

196^2 = 38416

6080^2 = ?

6080 - 6000 = 80

6080^2 = 6*(6080+80) | 80^2

6080^2 = 6*6160 | 6400

6080^2 = 36960 + 6 | 400 ••••

6080^2 = 36966400

•••• Number of zeros of base number indicates number of digits for the last group of the answer. Carry over the excess digits to the first group ( 36960+6).

#Matamata

Times 11 and + 10% Charge

1. Any number multiplied by 11

15 x 11 = ?

Take 1 as first digit and 5 as last digit of the answer, then add the digits for the middle number.

15 x 11 = 1   1+5   5

15 x 11 = 1     6     5

15 x 11 = 1 6 5

For more than 2 digits:

1726 x 11 = ?

Take 1 as first digit and 6 as the last digit of the answer. Then add every two adjacent digits for the middle numbers.

1726 x 11 = ?

1726 x 11 = 1    1+7   7+2   2+6     6

1726 x 11 = 1      8       9        8       6

1726 x 11 = 18986

2. Cost + 10% charge

For anything that will cost you an additional 10% charge, same procedure as above but take the last digit as decimal. Because adding 10% is same as multiplying 110% equal to 1.1 in decimal form.

15 + 10% charge = ?

15 x 1.1 = 1   1+5   .5

15 x 1.1 = 1      6    .5

15 x 1.1 = 16.5

1726 x 1.1 = ?

1726 x 1.1 = 1    1+7   7+2   2+6    .6

1726 x 1.1 = 1      8       9        8      .6

1726 x 1.1 = 1898.6

3. When sums of digits are greater than 9, take only the ones digit and add 1 to the preceding number.

472 x 11 = ?

472 x 11 = 4    4+7    7+2    2

472 x 11 = 4     11*     9       2

472 x 11 = 5*     1       9       2

472 x 11 = 5192

164 x 1.1 = ?

164 x 1.1 = 1   1+6    6+4   .4

164 x 1.1 = 1      7      10    .4

164 x 1.1 = 1      8       0     .4

164 x 1.1 = 180.4

#Matamata

The Average

The product of any two numbers is equal to the square of their average minus the square of their difference from average.

46 x 54 = ?

46 x 54 = 50^2 - 4^2

46 x 54 = 2500 - 16

46 x 54 = 2484

Average = (46+54)/2 = 50

Difference from average:

50-46 = 4 or 54-50 = 4

Or think of it this way:

46 is 4 below 50

54 is 4 above 50

These numbers are equally 4 below and above 50. Therefore, their average is 50 and their difference from the average is 4.

More examples:

59 x 61 = ?

59 x 61 = 60^2 - 1^2

59 x 61 = 3600 - 1

59 x 61 = 3599

73 x 87 = ?

73 x 87 = 80^2 - 7^2

73 x 87 = 6400 - 49

73 x 87 = 6351

115 x 125 = ?

115 x 125 = 120^2 - 5^2

115 x 125 = 14400 - 25

115 x 125 = 14375

394 x 406 = ?

394 x 406 = 400^2 - 6^2

394 x 406 = 160000 - 36

394 x 406 = 159964

6989 x 7011 = ?

6989 x 7011 = 7000^2 - 11^2

6989 x 7011 = 49000000 - 121

6989 x 7011 = 48999879

#Matamata

Square of Any Number of Digits

Square of Any Number of Digits

ab^2 = a*(ab+b) | b^2

•Where a & b represents digits, 'ab' does not mean a times b.

•b can be more than one digit.

•Number of zeros of base determines number of digits of b^2, excess digit will be added to the preceding number.

Base 10:

12^2 = 1*(12+2) | 2^2

         = 14 | 4 = 144

Base 20:

21^2 = 2*(21+1) | 1^2

         = 44 | 1 = 441

Base 30:

31^2 = 3*(31+1) | 1^2

         = 96 | 1 = 961

Base 40:

42^2 = 4*(42+2) | 2^2

         = 176 | 4 = 1764

and so on...

Base 100:

112^2 = 1*(112+12) | 12^2

           = 124 | 144

           = 124+1 | 44

           = 12544

Base 200:

213^2 = 2*(213+13) | 13^2

           = 2*(226) | 169

           = 452+1 | 69

           = 45369

Base 300:

308^2 = 3*(308+8) | 8^2

           = 3*(316) | 64

           = 94864

and so on...

Base 1 000 000 000:

1000001012^2 = ?

= 1*(1000001012+1012) | 1012^2

= 1000002024 | 1024144

= 1000002024001024144

#Matamata

Day of the Week of Any Date

Day of the Week of Any Date

     On what day of the week were you born? On what day of the week we'll be celebrating New Year 2015?

     Be a walking calendar or perform this trick as magic to amaze anyone. =)

Step 1: Assign these numbers to the days of the week:

0 = Sunday        (NONEday)

1 = Monday       (ONEday)

2 = Tuesday       (TWOsday)

3 = Wednesday (3 fingers looks W)

4 = Thursday     (FOURsday)

5 = Friday          (FIVEday)

6 = Saturday      (SIXturday)

Step 2: Remember these codes for the month (mcode).

Jan. = 6  (5 for leap year)

Feb. = 2  (1 for leap year)

Mar. = 2

Apr.  = 5

May  = 0

Jun.  = 3

Jul.   = 5

Aug. = 1

Sep. = 4

Oct.  = 6

Nov. = 2

Dec. = 4

     That is 622 503 514 624. Read it 10 times to paste in your memory. =) Month codes will be explained later.

Step 3: Year code (ycode) for 2015 is 4. Year codes will be explained later.

Step 4: Formula.

     Day = mcode + date + ycode

Example 1: January 1, 2015

                  (New Year's Day)

Day = mcode + date + ycode

Day = 6 + 1 + 4 = 11

       = 11 - 7 = 4 ***

Day = 4 = Thursday

January 1, 2015 is Thursday!

***Always reduce numbers by subtracting multiples of 7 (i.e. 7, 14, 21 & 28).

Example 2: February 14, 2015

14 - 14 = 0

Day = mcode + date + ycode

Day = 2 + 0 + 4  ***

Day = 6 = Saturday

February 14, 2015 is Saturday!

     Using year code of 4 and had memorized the month codes, you will now know what day of the week an any date in 2015 is!

     Now, you want to check your birthdate? Or any date in the past or future?

Just wait. =)

     Before we cook something, we must have to prepare the ingredients first. So, how is the formula and codes were derived?

Days:

     For any given month, all the multiple of 7 days fall on the same day of the week. If day 7 is Sunday, day 14, 21 & 28 are also Sundays. Thus, day 11 is the same with day 4  a Thursday (11-7=4). So, always reduce numbers by deducting multiples of 7 and still get the same result. In this way, you will be adding numbers not greater than 6 mentally.

Month codes (mcode):

     Start with January as 6. Next month adjustment is due to extra days from the exact 4 weeks (4x7=28). January has 31 days (31-28=3), so February dates will jump 3 days from January. (i.e. If January 1 is Thursday, February 1 will be Sunday)

Feb mcode = Jan mcode + 3

Feb mcode = 6 + 3 = 9 (- 7) = 2 ***

      February has 28 days, that's exactly 4 weeks (4x7=28), so dates in March will fall on the same days with February. (i.e. 28-28=0)

Mar mcode = Feb mcode + 0

Mar mcode = 2 + 0 = 2

And so on...

     That's how the month codes are generated. But you can easily just memorize 622 503 514 624, with exemption of 5&1 (instead of 6&2) for Jan. & Feb. respectively on leap years.

Year codes (ycode):

     A normal year consist of exactly 52 weeks (52x7=364) plus 1 day, to make 365 days. This 1 extra day makes a given date in a given month fall one day later in the next normal year. ( e.g. If January 1, 2014 is Wednesday,  January 1, 2015 will be Thursday).

     So, the year code will have plus 1 day adjustment in the next normal years.

Year codes (ycode):

     2000 = 0

     2001 = 1

     2002 = 2

     2003 = 3

     However, on leap years (366 days), everything must jump ahead 2 days after Feb. 29th. By that, the year code for 2004 (leap year) is not 4 but 5. And there is 5 & 1 mcode adjustment for Jan. and Feb respectively. The days jump by 2. Leap years happen every 4 years. Leap, jump, leapfrog, jump-start, ...hmmm whatever =)

     Let's have a walk to the year codes:

2000 = 0  (leap year)

2001 = 1

2002 = 2

2003 = 3

2004 = 5  (leap year, jump 2)

2005 = 6

2006 = 0  (6+1=7, 7-7=0)

2007 = 1

2008 = 3  (leap year, jump 2)

2009 = 4

2010 = 5

2011 = 6

2012 = 1  (leap year, 6+2=8, 8-7=1)

2013 = 2

2014 = 3

2015 = 4

2016 = 6  (leap year, jump 2)

2017 = 0  (6+1=7, 7-7=0)

2018 = 1

2019 = 2

2020 = 4 (leap year, jump 2)

2021 = 5

2022 = 6

2023 = 0  (6+1=7, 7-7=0)

2024 = 2 (leap year, jump 2)

2025 = 3

2026 = 4

2027 = 5

And so on...

     Notice the codes for 2012 and 2014, take the last 2 digits and divide by 12 and this goes until 2096.

2000 = 0

2012 = 1 = 12/12

2024 = 2 = 24/12

2036 = 3 = 36/12

2048 = 4 = 48/12

2060 = 5 = 60/12

2072 = 6 = 72/12

2084 = 0 = 7 = 84/12 (drop 7)

2096 = 1 = 8 = 96/12 (drop 7)

     Without leap years, year code pattern will repeat every 7 years. But because of the effect of leap years every 4 years, year code pattern will repeat in every 28 years.

     Meaning to say, year 2000, 2028, 2056 & 2084 have the same year codes which is 0. Example for year 2030, take 2030 minus 28 = 2002. And you know year code for 2002 is 2.

     This will work for any year. For 1900 - 1999 just add 1 for adjustment.

Example:

     Your birthday was on June 15, 1982, what day of the week is it? We know year code of 84 = 84/12 = 7 = 0. Moving backwards to 82:

leap year 84 to 83 is 7-2 = 5

Reg. year 83 to 82 is 5-1 = 4

ycode for 82 is 4 then add 1 for 1900-1999 adjustment

1982 ycode = 4+1 = 5

June 15, 1982

mcode = 3 (from 622 50[3] 514 624)

date = 15 - 14 = 1 ***

ycode = 5

Day = mcode + date + ycode

       = 3 + 1 + 5 = 9

       = 9 - 7 = 2

Day = 2 = Tuesday

June 15, 1982 is Tuesday!

     Below are the adjustments for year codes:

1600 to 1699 = add 0

1700 to 1799 = add 5

1800 to 1899 = add 3

1900 to 1999 = add 1

2000 to 2099 = add 0

2100 to 2199 = add 5

2200 to 2299 = add 3

2300 to 2399 = add 1

And so on... (0-5-3-1 pattern)

     One important note here for January and February in the years ending in 00: If a year ends in 00, apply leap year adjustment (5&1 month code) only if it is divisible by 400. The years 1600, 2000, and 2400 are divisible by 400 while the years 1700, 1800, 1900 & 2100 are not.

Tip: When performing magic trick to anyone, ask first the year. So you will have time to find the year code. Say 2014, year code of 2012 is 12/12 = 1, add 2 years, 1+2 = 3. When you already have the year code in your mind, ask for the complete date.

Say Dec. 25, 2014,

Dec mcode = 4

Date = 25 - 21 = 4 ***

2014 ycode = 3

Day = mcode + date + ycode

Day = 4 + 4 + 3 (- 7) = 4 = Thursday

Dec. 25, 2014 is... (drum rolls...) ...Thursday!!!

#Matamata #Math #Tricks & #Shortcuts

Make Times 5 Easier

Multiplying any number with 5 can be done easily by adding digit "0" after it, then divide it by 2.

Example:
You are buying 5 pieces of something worth 245 Php. How much is the total cost?

245 x 5 = ?

Add "0" to 245, then divide it by 2.
245 x 5 = 2450/2  = 1225 Php


More examples:

14 x 5 = ? 
14 x 5 = 140/2 = 70

160 x 5 = ?
160 x 5 = 1600/2 = 800

887 x 5 = ?
887 x 5 = 8870/2 = 4435

1268 x 5 = ?
1268 x 5 = 12680/2 = 6340

#Matamata #Math #Tricks & #Shortcuts

If one is in ratio the other one is zero

If one is in ratio the other one is zero.

Problem: Find the value of x and y.

2x + 4y = 8          (eq.1)

6x + 8y = 24        (eq.2)

Notice 2x & 6x and also 8 & 24, they are in ratio, therefore y = 0 .

2x:6x and 8:24 = 1:3 ratio

Solution:

y = 0

2x = 8

x = 4

Press the buzzer or raise your hand.

"Sir! x = 4 and y = 0 !"

Here is why or how:

Multiplying equation 1 by 3.

(2x + 4y = 8) x 3

6x + 12y = 24       (eq.3)

Then, subtract equation 2 from equation 3.

    6x + 12y = 24    (eq.3)

(-) 6x +   8y = 24    (eq.2)

=  0   +   4y = 0

                y = 0

When one is zero, you can solve for the other one with just one equation.

#Matamata #Math #Tricks & #Shortcuts