Math Calculations: Simplified --- Part 2


This is part 2 of the Math Calculations: Simplified blog.
Welcome back! This time I’m going to share tricks about counting squares such as 412 or 522. There are 2 simple methods that I am going to explain. Here is the first one:

METHOD  1

For numbers that begin with 1, the formula is simple:  (N-1)²+ (2N-1).  For example, let’s calculate 51². The equation becomes (50)² + (101). It is now easy to figure out the solution, which is 2601. This method makes it much easier to do these types of calculations rather than using the traditional method of carry-multiplying.

Next, we have the 2 digit numbers that begin with 9. We can figure this out by using pretty much the opposite of the method used to calculate the square of one digit numbers. The formula: (N+1)² - (2N+1). Let’s use 29² as an example. We put this into the equation, which makes it (30)² - (59). That leads to a solution of 841. This method also helps with these types of calculations.

Another thing to account for is that similar methods to the ones stated above will help you find the squares of 2 digit numbers ending with 4 and 6, except instead of using  (a number that ends with 0)², you use (a number that ends with 5)². For digits that end with 4, you can use the exact same formula as the one used for 2 digit numbers that end with 9. For the ones that end with 6, you can use the formula for 2 digit numbers that begin with 1.

Last but not least, we need to also find an easy way to calculate the square of 2 digit numbers that start with 2, 3, 7, and 8. I will list the equations below.
2 and 7: (N-2)² + (4N-4)
3 and 8: (N+2)² - (4N+4)







METHOD  2

Now I’m going to share the second tricks. For example let’s do 412.

All you have to master is to count 40x40, which I have shared in my previous blog. 
So, 40 x 40 = 1600

Then, to count 41 x 41, just add 40 (previous number) and 41 (current number)!

41 x 41 = (40x40) + 40 + 41            = 1600 + 40 + 41 = 1681

Furthermore, 42 x 42 = (41x41) + 41 + 42               = 1681 + 41 + 42 = 1764

And, 43 x 43 = (42x42) + 42 + 43                                 = 1764 + 42 + 43 = 1849

And so on…..
See how easy that is? A little practice and you’ll be an expert in no time!


SQUARE ROOT

And now that you are (hopefully) an expert at squaring 2 digit numbers, now let’s look into the method for square rooting 2 digit numbers. We will make this a mixed fraction. First you will want to take the perfect square number just before it. That will be the ones place. 

Next we take the difference of your number and the perfect square just before it. That will be the numerator of the mixed fraction. Finally, we will take the number in the ones place and double it to become the denominator. That is how you square root a 2 digit number with an average accuracy of 1 decimal place. When you are in more advanced math problems, you will indeed need more than one decimal place, but they usually allow you to use calculators for that.

Here is an example. Let’s try to calculate √67. First we take the perfect square just before it, which is 64. √64 is equal to 8, so now our mixed fraction is 8 ?/?. Next, we take the difference between our number (67) and the perfect square just before it (64), which is 3. Now that we know 3 is the numerator, we have 8 3/?. Now we simple double the number in the ones place (8) to get 16. Our fraction is now 8 3/16, which is 8.1875. This is quite accurate as the real answer is about 8.18535277187.

Thanks for reading! ~Nat



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