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TIPS AND TRICKS FOR SPEEDY CALCULATIONS – MODULE I – SQUARES
In this module we deal with finding squares of two digit and three digit numbers and some special cases arising out of these forms. This section is not only important in itself but forms the very basis of more tedious and calculations of higher difficulties. Mastery over these techniques of finding squares of two and three digit numbers shall form a strong foundation for more techniques to come such as finding the square roots of various numbers.
- Square of a number between 26 and 74
This is a simple method of finding square of any number between 26 to 74 without using calculator.
To apply this method you should know squares of 1 to 25 by heart. You can refer to this table to learn the same.
|
Number |
Square |
Number |
Square |
|
1 |
1 |
13 |
169 |
|
2 |
4 |
14 |
196 |
|
3 |
9 |
15 |
225 |
|
4 |
16 |
16 |
256 |
|
5 |
25 |
17 |
289 |
|
6 |
36 |
18 |
324 |
|
7 |
49 |
19 |
361 |
|
8 |
64 |
20 |
400 |
|
9 |
81 |
21 |
441 |
|
10 |
100 |
22 |
484 |
|
11 |
121 |
23 |
529 |
|
12 |
144 |
24 |
576 |
|
* |
* |
25 |
625 |
For finding square of any number between 26 to 75
Step 1. Find the difference between 50 and the number you want to square.
Scenario 1: If the number to be squared is greater than 50
Step 2. Add that many 100s to 2500 (which is the square of 50)
Step 3. Then add the square of the difference to the result of step 1
Scenario -2: If the number is less than 50
Step 2. Subtract that many 100s to 2500.
Step 3. Then add the square of the difference to the result of step 1
Example
Find out the Square of 67.
Step 1. Difference of 67 and 50 = 67-50 = 17
Step 2. This number is greater than 50. So add 1700 to 2500 = 4200
Step 3. Add square of 17 to step 2.
Answer = 4200+ 289 = 4489
Alternative method of calculating the square of a number:
Since, 67-50 = 17
67^2
We will be getting answer in 2 parts; see below – right hand side gives you tens and units digit. Left hand side gives you the remaining digits.
= 25 + 17 | (17)^2 ( | denotes separation )
= 42 | 289 (17^2 is 289. The 2 shown in subtext will be carried over and added to left hand side)
= 4489
- Square of any two-digit number ending with 1
Let us take a 2 digit number in its generic form. Any two digit number whose unit digit is 1, say a1 can be expressed as 10a+1, where a is the digit in ten’s place
Square of a1= a2 | 2xa | 1
Here, ‘|’ is used as separator.
That means for the left most part of the answer, a is squared, hence first part will be a2. The middle part will be twice of a and the last or the right most part will always be 1.
Let us see a few examples.
(21)2= 2 squared | 2 . 2 |1 = 441
(31)2= 3 squared | 2 . 3 |1 = 961
(41)2= 4 squared | 2 . 4 |1 = 1681
(51)2= 5 squared | 2 . 5 |1 = 2601 (Here the square of 5 is 25 but since the product of 2.5 is 10 we write down 0 and add 1 to 25). Similarly,
(91)2= 9 squared | 2 . 9 |1 = 8281
- Square of any 3 digit number ending with 1
Now let us try to extend the above shortcut method to 3 digit numbers as well. Let us straight away start with an example 131.
Like earlier separate the given number in 2 halves, left hand side will have digits other than 1 and right hand side will have 1 as usual.
Hence, the answer is
(131)2= 13 squared | 2 x13 | 1 =
= 169 | 26 | 1
= 17161
Let’s take another example of squaring a three digit number ending in 1.
261 = 26 squared | 2×26 | 1
= 676 | 52 | 1
= 68121
- Square of any two digit number ending with 5
Let us take a 2 digit number in generic form, say the number is a5 (=10a+5), where a is the digit in ten’s place
Square of a5= a x (a+1) | 25
That means a is multiplied by the next higher number, i.e. (a+1). Now let’s take example of a real number ending in 5, say 45.
452 = Left hand side of the answer will be 4 multiplied by its successor i.e. 5 and the right hand side part will always be 25 for squares of numbers of which the unit’s digit is 5.
Giving the answer a x (a+1) | 25 ( | stands for concatenation} i.e. 4 x (4+1) | 25 = 4 x 5 | 25 = 2025
- Square of any 3 digit number ending with 5
This is an extrapolation of the above method. The only difference being that here you have to multiply two digit numbers with each other. This has been explained with the help of following examples;
1252 = 12 x 13 | 25 = 15625
5052 = 50 x 51 | 25 = 255025
- Square of any two digit numbers ending with 9
Finding the square of 39
Firstly add 1 to the number. The number now ends in zero and is easy to square.
40^2 = (4*4*10*10) = 1600. This is our subtotal.
In the next step, add 40 plus 39 (the number we squared plus the number we want to square)
40 + 39 = 79
Subtract 79 from 1600 to get an answer of 1521. To easily do such subtractions, subtract 80 (1 more than 79) from 1600 to get 1520 and then add 1 to get the answer as 1521.
1600 – 79 = 1521 is the square of 39.
- Square of any three digit numbers ending with 9
Finding the square of 159
159^2 =
159 + 1 = 160
160^2 = 25600
160 + 159 = 319
25600 – 319 = 25600 – 320 + 1
25600 – 319 = 25281
Another Example,
449^2
450^2 = 202500 (use the shortcut Squaring number ending in 5 to calculate 45^2 and then put double zeroes in front of the answer)
450+449 = 899
202500 – 899 = 202500 – 1000 + 101 = 201601
- Square of any two digit number
Let us take first example of squaring 32;
Step 1. In finding the last two digits of the answer, we shall find the square of the last digit of the number. Square the right-digit digit, which is 2 in this case. Hence we get 04
_ _04
Step 2. We shall now need to use the cross product. This is what we get when we multiply the two digits of the given number together. Multiply the two digits of the number together and double it: 3 times 2 is 6, doubled is 12: We write 12 as 2 and carry over 1 to the next step.
_ 24
Step 3: In finding the first two digits of the answer we shall still need to square the first digit of the number. That means we square the left hand figure of the number. Here square of 3 will be 9. Add 1 which is carried over from last step. Hence we get 9 + 1 =10
1024
Let us take second example of squaring 64;
Square of 64 = Square of 6 | double of cross product of both given digits 4 & 6| square of 4
Square of 64 = 36 |2x6x4 | 16 = 36 | 48 | 16
Collapsing the numbers
= 36 | 48 + 1 | 6
= 36 + 4 | 9 | 6
= 40 | 9 | 6
Hence the answer is 4096.
Let us take third example of squaring 83;
- Square each digit individually, making sure that you get a two-digit number for each square. If the number is low and its square gives you only one digit, use 0 as a placeholder.
8² = 64 and 3² = 9 = 09, giving you 6409
- Now, multiply the two digits and double your answer, adding a 0 to the end.
8 X 3 = 24; 24 X 2 = 48, giving you 480
- Simply add your answers from steps one and two.
6409 + 480 = 6889, or 83²
- Square of any three digit number
To understand and appreciate squaring of 3 digit numbers you should be well versed with shortcut of squaring any 2 digit number. Let us start Learning this with the help of an example.
Let us find the square of 384
Step 1: To begin with ignore the 3 of 384. You are left with only 84, a two digit number. Using the method of squaring 2 digit numbers, find the square of 84. We get the answer as
Square of 8 | twice of 8 X 4 | square of 4
64 64 16
7056 (consolidating the result obtained above)
Step 2: This step is new and different from what we’ve learned in the previous post for squaring 2 digit numbers. Watch carefully.
We have to multiply the first and last digits of our original number and double it. Essentially, that is multiplying together 3 and 4 and then doubling it. Hence we get 24.
Add this number directly to the two left hand digits of our number obtained from the first step.
7056
Add 24 to 70. 70+24=94. So 7056 gets converted to 9456.
Step 3: In the first step we left out the first digit of our number and squared the last two digits. Now we will forget about the unit’s digit 4 and square the first two digits i.e. 38 as before just omitting to square the last digit 8.
Square 38 as a regular 2 digit number, except that you omit the 8 squared.
Square of 3 | twice of 3 X 8
9 | 48
Step 4: Consolidating this with the result obtained in step 2,
9 | 48 | 9456
14 7 456
Hence the answer is 147456.