squares tricks
Easy tips to find the square of numbers ending in 5:
When you have any number ending in 5, the square of the number can be found easily using this simple trick. Add 5 to the number and also subtract 5 from the same. Now multiply the two numbers obtained after this. And to it add the square of the number you added and subtracted. To explain it in a simpler way,
For example: the number concerned in 45. To find the square of 45, first add 5 to 45;
45+5= 50
Now subtract 5 from 45
45-5=40
Now multiply 50 and 40
40*50 = 2000
To it add the square of the number that was added and subtracted i.e. 5
5*5=25
2000+25 = 2025
Therefore the square of 45 is 2025. This method is faster than a calculator at times. You just have to get a hang of doing it. And you can solve problems without using calculators!!!!
To make it easier, here’s another example
For example we need to find the square of 125
125+5= 130
125-5= 120
120*130 = 15600
5*5=25
15600+25= 15625
Now punch the number 125 into your calculator and check the result. Try it yourself on a few numbers. It’s a much faster and reliable technique.
Let us try another example for practice;
Let us find out the square of 185
For this we will have to first subtract 5 from the number,
185-5 =180.
Now add 5 to 185
5 +185=190.
Now multiply the two answers obtained
180*190=34200.
Square the digit that is added and subtracted.
5*5=25
Add the square to the answer obtained in the third step to get the final answer;
Hence the final answer is 34200+25 = 34225.
Deriving squares:
To derive the square of any number simply take the base of the nearest power of 10 available, check the difference between the number and this power of 10 and subtract the difference from the number to get the left hand side digits of your answer. Then square the difference to get the right hand side digits.
Let us try and understand this technique using an example,
For example we need to find the square of 9
The nearest power of 10 is 10. Difference between 9 and 10 is 1. Now 9-1 =8. So the first digit of the answer is 8. For the last digit square the difference i.e. 1 square=1. Therefore the square of 9 is 81.
Similarly if it the number is greater than the nearest power of 10 we take the surplus and add it to the actual number instead of subtracting as we did earlier, in order to calculate the answer. For example the square of 14, the nearest power of 10 is 10. So the surplus is 14-10 =4. Now add 4 to 14, which equals to 18. For the last digit we square 4, which is 16. Of which the 1 is carried over and added to the first two digits as usual is 1 is carried over and added to 18. Hence the square of 14 is 196.
Let us find the square of 98
The nearest power of 10 available is 100.
The difference between the number and the nearest power is 100-98 = 2
Subtracting the difference from the number;
98-2=96. These are the left hand side digit so of the answer.
Square the difference to get the right hand side of the answer.
2*2=4
Hence the square of 98 is 9604.
[ Since the number in the question is a two digit number , the square has to be a four digit number hence we write the right hand side digits as 04 as opposed to 4]
Now let us try another problem where the number is greater than the concerned power of 10. For example let us try and find the square of 105
Now the difference between this number and the concerned power of 10 is;
105-100 = 5.
Since the number is greater than the power of 10 we shall add the difference rather than subtracting it from the number,
Hence 105 + 5 = 110.
This forms the left hand side of the answer.
To determine the right hand side of the answer we will have to square the difference.
Hence 5*5=25
Hence the square of 105 is 11025.
How to cube a number that lies close to a power of 10:
For example let us consider the cube of 14
In the first step we will subtract the nearest power of 10 (in this case it is 10) from 14.
14-10=4
Now multiply this answer by 2 and add the original number to it
4*2+14= 8+14=22
Subtract the closest power of 10 from this answer 22-10=12.
Following this we shall multiply the answer from the previous step with the answer obtained in the first step.
Hence 12*4=48
Now we shall cube the answer obtained in the first step, hence 4^3 = 64.
Since we are working with a two digit number we shall two zero’s to the answer obtained in the second step and one zero to the answer obtained in the fourth step. Now add the answers obtained in step five, seven and eight to obtain the actual answer.
Hence the answer is 2200+480+64= 2744.
Let us take another example, to find the cube of 15
Let start,
The nearest power of 10 is 10.
15-10=5
5*2+15=25
25-10=15
5*15=75
5^3=125
Now adding the answers we get 2500+ 750+125= 3375
To understand it better let us try another example,
The cube of 13
The nearest power of 10 is 10
13-10=3
3*2+13=19
19-10=9
3*9=27
3^3= 27
Now adding the answers we get 1900+270+27= 2197