Squaring any Number close to 50

08:21 Squaring numbers 4 Comments

You likely know how to square any 2-digit numbers and how to square any number near the base 100 mentally by now. Here is a trick to square numbers closer to 50 ranging from 30 – 70 (ranges vary at capacity) at extremely fast speed.

In order to find the square of any number closer to 50, all we need to do is,

1. Find out how much more or less than 50 is that number i.e. the distance of that number from 50.
2. Add that distance to 25 if the number to be squared is above 50, or subtract the distance from 25 if the number is below 50.
3. For the last 2 digits, find the square of that distance. If it is a single digit answer, make sure to put an extra ‘0’ before it to make it a 2 place. If it is a 3-digit answer, carry forward the left digit. 

That’s it.

Let’s illustrate with example,

57² =?

1)  57 is 7 more than 50.

2)  Add 7 to 25. 25 plus 7 is 32.

3)  For the last 2 digits, find the square of that distance i.e 7. Square of 7 is 49. So, put 49.

So, 57² =3249

53² =?

1)   53 is 3 more than 50.

2)   25 plus 3 is 28.

3)   Square of 3 is 9. So, put 09.

So, 53² =2809

See the pattern?

So, 62² =?

1)  62 is 12 more than 50.

2)  Add 12 to 25 to get 37.

3)  Square of 12 is 144. Put 44 to the last 2 places and carry forward 1 to 7. 7 plus 1 is 8.

So the answer is 3844.

Now what if the numbers of which you are trying to square is less than 50? This is the same technique. Just subtract the distance from 25.

 Let’s say 47² =?

1)   47 is 3 less than 50.

2)   Subtract this distance from 25. 25 minus 3 is 22.

3)   Square of the distance i.e. 3 is 9. So, put 09.

 So, 47² =2209

42² =?

1)   42 is 8 less than 50.

2)   25 minus 8 is 17.

3)   Square of 8 is 64.

So, 42² =1764

37² =?

1)   37 is 13 less than 50.

2)   25 minus 13 is 12.

3)   Square of 13 is 169. Put 69 and carry forward 1 to 2. 2 plus 1 is 3

           So, 37² = 1369

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Multiplying Numbers Close to base 100 (part-1)

08:38 Multiplication 4 Comments

You must have known the fastest and easiest method of multiplying any 2-digit numbers together in just one line by now.

Here's the smarter way of multiplying numbers close to base 100.

1. Find the difference between both of the numbers and 100.
2. Multiply these two values together and write it down. Make sure the answer takes up 2 place values. For a single digit result add a ‘0’ before it and for a 3-digit result, carry the left digit.
3. Subtract the difference found in step-1 of one of the numbers with the remaining number. You can subtract either way; you will always get the same answer. Write the result.

Suppose you want to multiply 88 by 98.

With ‘Vertically and Cross-wise’ you can give the answer immediately.

Both 88 and 98 are close to 100.
88 is 12 below 100 and 98 is 2 below 100.

You can imagine the sum set out like this:

            

Multiplying vertically both the differences (12 and 2) from 100 results in 24.

And, 86 comes from subtracting crosswise: 88 - 2 = 86 (or 98 - 12 = 86: you can subtract either way, you will always get the same answer).

So, 88 x 98 = 8624

This is so easy it is just mental arithmetic.

Multiplication can also be carried out starting from the left, which can be better because we write and pronounce numbers from left to right. In this case, remember to carry, if any.

Example, 96 by 92

96 is 4 below the base and 92 is 8 below.

We can cross-subtract either way: 96-8=88 or 92-4=88. This is the first part of the answer and multiplying the "differences" vertically 4x8=32 gives the second part of the answer.

                       
                  

Multiplying numbers just over 100

This works equally well for numbers above the base. Here we add the differences.

• 103 x 104 =?
• The answer is in two parts: 107 and 12,
  107 is just 103 + 4 (or 104 + 3), and 12 is just 3 x 4.
• So, 103 x 104 = 10,712
• Similarly 107 x 106 = 11,342

107 + 6 = 113 and 7 x 6 = 42

And, 105 x 111 = 11,655 

For 205x211; just double the first part of the answer, because 200 is 2x100.

Hence, 205x211=43,255

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Dividing any number By 9, 90, 900 and so on

16:33 Division 9 Comments

The fastest way to divide any number mentally by 9 is simply to reduce a complex division to a very simple addition. The technique follows the below steps:

Step-1: Copy the first (left) digit from the dividend as it is and you have the first answer digit.

Step-2: Add it to the next digit of the dividend and you have the 2^nd answer digit. Add this number to the next adjacent digit of the dividend and continue the step with the other digits. Write these numbers down as answer digit except the last set of digit(s) to get partial Quotient. Remember to add any carried numbers.

Step-3: If last set of digits is = or >9; divide it by 9 and get the balance Quotient and final Remainder.

Step-4: Add the partial and balance quotient to get the final quotient.

Here are some examples to help illustrate the method:

221013 ÷ 9 =?

Start putting the first (left) digit of the dividend as it is which is 2, and you have the first answer digit. ->2

The next one would be this answer digit plus the next digit of the dividend. So, 2 plus 2 is 4. ->2-4

The next answer digit would be this 4 plus 1 that is 5. ->2-4-5

Now, using the same logic, the next answer digit would be 5 plus 0, which is again 5. ->2-4-5-5

Next one would be 5 plus 1 that is 6. ->2-4-5-5-6

And then the last step would be 6 plus 3 that is 9. But do not write down 9, because this would be the remainder, if there is a remainder. In this case, there would be none, because 9 still goes 1 times in 9. This 1 would be added to the last findings i.e. 6 resulting 7. ->2-4-5-5-(6+1)

So, the answer would be 2-4-5-5-(6+1) that is 24557.

See the pattern?

So, 32142 ÷ 9 =?

The answer would be the first digit that is 3, ->3

3 plus the next digit 2 that is 5, ->3-5

5 plus the next digit 1 that is 6, ->3-5-6

6 plus the next digit 4 that is 10, now 10 is a 2 digit number so carry '1' (to be added with 6 making 7) and put 0 here. ->3-5-(6+1)-0

And 10 plus the last digit 2 would be 12, remember not to put 12 here. Now, 9 goes how many times in 12? Nine 1 times is 9. So, this 1 would be added to previous finding '0' and 3 is the remainder. ->3-5-(6+1)-(0+1)

So, the answer would be 3-5-(6+1)-(0+1) that is 3571 with 3 reminder.

8346425 ÷ 9 =?

The answer would be the first digit 8, ->8

8 plus 3 that is 11, put 1 and add 1 to the previous finding ->(8+1)-1

11 plus 4 that is 15, write 5 and add 1 to the previous finding ->(8+1)-(1+1)-5

15 plus 6 that is 21, write 1 and add 2 to previous finding ->(8+1)-(1+1)-(5+2)-1

21 plus 4 that is 25, write 5 and add 2 to previous finding ->(8+1)-(1+1)-(5+2)-(1+2)-5

25 plus 2 that is 27, write 7 and add 2 to the previous finding ->(8+1)-(1+1)-(5+2)-(1+2)-(5+2)-7

27 plus 5 that is 32. Now as it is the last digit, don’t write 27 down. See, 9 goes how many times in 32? Nine 3 times is 27 with 5 remainder. So, this 3 would be added to last finding that is 7 and 5 is the remainder. ->(8+1)-(1+1)-(5+2)-(1+2)-(5+2)-(7+3)

So, the answer would be ->(8+1)-(1+1)-(5+2)-(1+2)-(5+2)-(7+3)

that is 92737-10, Now again don’t write this 10 straightaway. 

Write ‘o’ and carry ‘1’ to add to previous finding that is 7 to get 8.

Therefore, the final answer is 927380 with a remainder of 5.

43967 ÷ 9 =?

Partial Quotient => 4, 7, 16, 22

=> 4, 7, (16+2), 2

=> 4, 7, 18, 2

=> 4, (7+1), 8, 2

=> 4, 8, 8, 2

Balance Quotient = 29/9 = 3, Remainder 2

Final Quotient = 4882 + 3 = 4885

Answer: Quotient 4885 Remainder 2

This way you can reduce a complex division to a very simple addition. And everyone is really very good at adding numbers.

For dividing by 90, shift the decimal point to 1 digit left and for dividing by 900, move the decimal point to 2 digits left after diving by 9 or following the above steps.

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Extracting Square Roots Mentally

11:01 Extracting Roots 19 Comments

Faster way to extract the square root of any perfect square
 
Finding the square root of a number is the inverse operation of squaring that number. Remember, the square of a number is that number times itself.
Square of n = n²
Square of 5 = 5² = 5 x 5 = 25
Therefore, the square root of 25 is 5
 

The perfect squares are the squares of the whole numbers such as 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 so on and so forth.

Before learning the procedure, it is wise that the performer memorizes the squares of the numbers 1-10 which is very elementary:
                                

12 =   1 22 =   4 32 =   9 42 = 16 52 = 25

62 =   36 72 =   49 82 =   64 92 =   81 102 = 100

To extract the square root of any perfect square follow:

Step-1: Look at the magnitude of the “hundreds number” (the numbers preceding the last two digits) and find the largest square that is equal to or less than the number. This is the 1^st part of the answer.

Step-2: Now, look at the last (unit’s) digit of the number. If the number ends in a:

0 -> then the ending digit of the answer is a 0

1 -> then the ending digit of the answer is 1 or 9.

4 -> then the ending digit of the answer is 2 or 8.

5 -> then the ending digit of the answer is a 5.

6 -> then the ending digit of the answer is 4 or 6.

9 -> then the ending digit of the answer is 3 or 7.

To determine the right answer from 2 possible answers (other than 0 and 5), mentally multiply the findings in step-1 with its next higher number. If the left extremities (the numbers preceding the last two digits) are greater than the product, the right digit would be the greater option (9,8,7,6) and if left extremities are less than the product, the right digit would be the smaller option (1,2,3,4).

Let us illustrate the trick with some examples:

Extracting square root of 784 (√784)

1.  Look at the magnitude of the “hundreds number” (the numbers preceding the last two digits) which is 7. Now, 2²=4 and 3²=9. So, the highest square in 7 is 2 which is the 1^st part of the answer.
2. Now, look at the last digit of the number which is 4. We know if the number ends in a 4 then the ending digit of the answer would be 2 or 8.

Now, 2 (findings in step-1) times its next higher number which is 3 is (2×3=) 6. The left extremity which is 7 is greater than 6. Therefore, the right digit of the answer must be the greater option which is 8. 

So, our final answer is 28.

Let’s go for another example: √3969 (square root of 3969)

1. The magnitude of the “hundreds number” is 39. Now, 6²=36 and 7²=49. So, the highest square in 39 is 6.
2. Looking at the last digit of the number which is 9; we know if the number ends in a 9 then the last digit of the answer would be 3 or 7.

Now, 6 (findings in step-1) times its next higher number 7 is (6×7=) 42. And 39 (the left extremities) is less than 42. Therefore, the right digit of the answer must be the smaller option i.e. 3. 

So, our final answer is 63.

So, square root of 5476 (√5476) =?

1. The numbers preceding the last two digits is 54; the highest square in it is 7.
2. The last digit of the number is 6 so; the ending digit of the answer would be 4 or 6.

Now, 7 times its next higher number (8) is 56. Since 54 is less than 56, the right digit of the answer must be the smaller option i.e. 4. 

So, our final answer is 74.

Square root of 13689 (√13689) =?

1.  Focusing 136; the highest square in it is 11 (since, 11² = 121 and 12² = 144).
2. The last digit of the number is 9 so; the ending digit of the answer would be 3 or 7.

11 times its next higher number (12) is 132 and 136 is greater than 132, so the right digit of the answer would be 7. 

So, the final answer is 117.

Square root of 15376 (√15376) =?

1. The highest square in 153 is 12 (12² = 144 and 13² = 169).
2. The last digit of the number 6 makes the ending digit of the answer a possibility of 4 or 6.

12 times its next higher number (13) is 156. Since 153 is less than 156, the right digit of the answer must be 4 giving the final answer 124. 

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