Squaring up 25 !

After squaring up numbers ending with 5, me and my daughter have moved onto squaring numbers ending with 25. We proceed exactly the same way we did with 5, but with a larger number. That is all.

We will begin with same equation of (a + b)2= a2 + 2ab + b2.

Here b = 25.

Now the equation gets simplified like this:  

a2 + (2 x a x 25) + 625

That is further simplified into: 

 a2 + (50a) + 625. 

Now we will leave out 625 to be added at the end.

a2 + (50a) is further simplified as:

a (a + 50)

Here 'a' is any number that ends with 00, like 100, 200, 300 and so on. We have 625 at the end of the number. 

a (a + 50) is a round figure that will end with at least '000' because we are multiplying 100 x 150 or 200 x 250 and so on. Leaving out the 'three zeros' we will further simplify the equation a (a + 50).

We will take the two zeros [00] from 'a' and arrive at a simpler number 'n'. We will remove the third zero [0] from the [a + 50]. We remove these three zeros [000] because we have 625 at the end of the answer and hence we are simplifying the multiplication leaving out the '0's.

That means: 150 becomes 15, 250 becomes 25, 350 becomes 35 and so on.

So what we have is following simplified equation:

n (n5) - *n5* here is NOT a product here, but we just write n and place 5 next to it to derive the number. So if 'n' is 7, n5 is 75.

Now the n (n5) product is placed before 625 and we have the whole square.

Like: n25 whole squared = n(n5) followed by 625 

Some examples: 

For Whole square of 125: 1 x 15 = 15. Whole square: 15625

For Whole square of 225: 2 x 25 = 50. Whole square: 50625

For Whole square of 425: 4 x 45 = 180. Whole square: 180625

For Whole square of 725: 7 x 75 = 525. Whole square: 525625

For Whole square of 1025: 10 x 105 = 1050. Whole square: 1050625

For Whole square of 1325: 13 x 135 = 1755. Whole square: 1755625

This is quicker, accurate and time saving way of doing Arithmetic for school kids. This again can be worked out without a paper and pencil, if we know our simple multiplication well.

 

Dr. Punned-it

Solving the 9th Wonder !

Multiplying any number ending with 9 is a big 'scare crow' for kids in primary school. It becomes virtually a non-issue if we approach it as a process of addition and subtraction rather than multiplication. Once again unfortunately, people shut their minds even before we count from 1 - 9.

My daughter used to do the same till I made her use a method that has eradicated the fear of multiplication from her mind. She now enjoys doing it in the mind and she is always correct. But she still hasn't exorcised the 'Ghost' from her mind and wants to check if her answer is correct. She will be confident with time.

So what do we do ?

Simple. Let us take the simplest of numbers ending from 9, that is 19. Most kids would know the multiplication table by heart. But is that necessary ? I have never found that necessary. I approach it as adding 19 as many times as I am supposed to multiply. So 38, 57, 76 and there goes the table.

Often kids get stuck on reaching 7, 8 and 9. That is where the realization comes that it actually is easier to add and delete than do a table. I do it this way.

19 is 1 less than 20. 

We all can have multiples of 20 easily because 20 is nothing but 2 x 10. 

So just check this method. 

I do 20 x 9 = 180. 

When it comes to 19 x 9, that means it actually is one 9 less than 20 x 9. 

So just delete 9 from 180. 

That means 19 x 9 = 171. 

That is it !

Let us move to larger numbers.

Let us take 89 x 7. 

What I do is 90 x 7 = 630. 

We know 89 x 7 is one less times 7.

So just delete 7 and that means 623.

Let us take another example of 99 x 8. 

100 x 8 = 800. 

Minus 8 means the answer is 792. 

The riddle solved, in the mind. 

No paper, no pencil and in no time. And of course no tension !

Dr. Punned-it

High Fives to Mathematics !

Continuing with my Maths posts, let me deal with one more easy way to Maths. We all love 'Five', may be because we have 'five' fingers and 'five' toes in each of our limbs; well mostly. But '5' is a lot more useful in Mathematics. It is a wonderful number that makes Maths lot more easier, fun and even entertaining.

I asked my daughter, "Do you know the easy way to obtain whole squares of numbers ending with 5 ? Like say, 25, 35, 45 or even 115 and so on ?" I knew she doesn't and hence went on to explain. This isn't anything new again, but somehow not many Maths teachers follow this and I don't know why !

Let me first show how we do it and then explain how it actually happens. We have to remember all numbers ending with 5 will end with 25 when squared. The 25 will remain constant. We just have to look at the first number. Let us say 'u' and multiply that with (u+1) and the product has to be written before 25 and we have a whole square. 

u x (u + 1) before 25

For example: 

Let us take 25: u here is 2. Hence: 2 X 3 = 6. The whole square of 25 is 625.

Let us take 35: u here is 3. Hence: 3 x 4 = 12. The whole square of 35 is 1225.

Let us take 45: u here is 4. Hence: 4 x 5 = 20. The whole square of 45 is 2025.

Let us take 55: u here is 5. Hence: 5 x 6 = 30. The whole square of 55 is 3025.

This will go on and on. Now let us see how this actually happens. Once we know that, we will never have a problem with 'Five' !

-> Let us begin with 55. What is the easiest way to square a big number ? 

-> We have already dealt with that in (a + b)2= a2 + 2ab + b2.

-> When we deal with any number ending from 5, we know the 'b' in the equation and that is 5. 

-> So the equation gets simplified as follows.

* a2 + (2 x a x 5) + 25.

-> That is further simplified into

* a2 + 10 x a + 25

-> Now what are we left with ?

* a2 + 10a + 25 

Now let us keep the 25 away for the time being because we are going to add it at the end. Thus we have

* a2 + 10a

This is further simplified into

* a (a +10)

-> In reality 'a' is a round figure here like 10, 20, 30, 40, 50, so on to 90, 100, 110 and more. Multiplying a round figure with a round figure [a x (a + 10)] always yields a multiple of 100 with 00 at the end. We just have to ignore those zeros for the sake of ease and speed. Thus the 00 goes with 25 as just 25.

So we have to remove the '0's from [a] and [a + 10] to get the smaller simpler numbers.

-> Thus we have: 

*a / 10 and [a + 10] / 10

Thus the equation becomes: 

* (a / 10) x (a + 10) / 10 = (a / 10) x (a / 10 + 1)

-> [a / 10] here is the simplified small number after removing the 'Zero'. 

So we will name it [u] for convenience. So the equation becomes simpler:

* u x (u + 1) with the 00 going with 25. 

Now we just put the product of u x (u + 1) before 25 and we have the whole square.

Let us take 145: u here is 14. Hence: 14 x 15 = 210. The whole square of 145 is 21025 !

Simply put, what are we doing ?

Take the number before 5 and multiply that with the one more than the same number and put the product before 25. We are home !

Hope I have made it simpler for those who like to make things easier in life; especially Mathematics !

Dr. Punned-it

Multiply from Left to Right: Eco-friendly Mathematics !

After assisting my daughter to "Love Maths" I have decided to indulge in Mathematics because that has always been a favorite subject. If this sharing helps people, that should be wonderful. I am mostly dealing with simple Mathematics here, because I believe in beginning from the foundation. If this endeavor proves to be useful, I shall dive deeper into this in future with a series of short articles.

Yesterday Piya was struggling with 'Pi' diagrams. I don't understand why the texts and the so called guides [I would call them misguides] make it so convoluted and so tough. A simple act of multiplication for a Class 6 or 7 or 8 student is made so complicated by the texts.

I have a problem with doing addition and multiplication from right to left. It is cumbersome, time consuming and downright unintelligent. Addition and Multiplication should be done left to right; just like we write and most of this world writes.

How do we go about it ? In reality, we don't even need a paper and pencil to do multiplication of two digit numbers and three digit numbers. Honestly, this isn't any new or revolutionary idea. It just is that most of our education simply doesn't encourage using the mind or the brain. 

Kids are encouraged and even forced to learn all things by heart. We used to call that 'mugging' in the Medical College. There can't be a better word to describe this system. We are 'mugging' the thinking ability of our kids. Let me explain how we go about it. 

For a beginning, let us take a multiplication of 137 X 25. 

Our books teach us to multiply 137 X 5

And then 137 X 2 and then add the second number one place to the left to get the answer. 

We have to begin by multiplying 7 X 5

And move with the product to left and add it to 3 X 5

And then move further left and add the next product to 1 X 5. 

Then we multiply 137 X 2 similarly and add the product one place to the left. 

Most of the kids get confused here.

Now let us do it from left to right. 

For this, we use the simple formula of,

A X B = [u + v + w] X B = [u X B + v X B + w X B]. 

Here [u + v + w = A]

Let us make it simpler. 137 = [100 + 30 + 7]. 

So what do we have ? 

[100 + 30 + 7] X 25.

Now look at the answer: 2500 + 750 + 175 = 3425. 

This splitting, multiplying and adding is all done in the mind once we develop the practice. 

Right to left Maths can never be done in the mind so easily.

If we teach our kids to learn multiplication tables with proper technique, Maths becomes a cakewalk. There is no point in mugging up tables. We have to help them to understand multiplication table is nothing but adding the same number as many times as the second number suggests. By doing away with papers, we can save a lot of trees too. Thus Mathematics becomes Eco-friendly !

Dr. Punned-it