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>> Hi, this is Julie Harland
and I'm your Math Gal.
Please visit my website
at yourmathgal.com
where you can search for any
of my videos organized
by topic.
I'm going to show you a cool,
fast and easy way
to square numbers ending in 5
such as 25 squared
or 35 squared, et cetera,
and then at the end
of the video
if you know a little bit
of [inaudible] I'm going
to show you why this
method works.
First of all let's just do a
little review
if you have numbers like this,
10, 20, 30,
40 or 50 squared remember this
means 10 times 10,
which is 100,
and actually the fast way
of doing that is you do 1
times 1, which is a 1 and then
since you're squaring this
number you put
in 2 0s after it.
So, for 20 squared it means 20
times 20 and that's really 2
times 2 which is 4
and then you put the 2 0s
after it because it's really
times another 10 times
another 10.
So, the fast way to do this is
to do, well,
30 squared is just 3 times 3
and since you're squaring this
number you put in 2 0s;
40 squared you'll be squaring
the 4, which is 16 and put
in the extra 0s.
So you can see how this works.
So, if I asked you what was 80
squared, what would that be?
Well, you would just do 8
times 8 and add 0s.
This applies
if I were cubing this number
like if I did 10 cubed you
would do 1 cubed and put
in 3 0s, if this was 20 cubed,
you'd do 2 cubed put in 3 0s,
et cetera,
but we're just working
on squaring things today.
All right so now here's
the question.
What would it be
if I had 15 squared?
What is the answer to that
or 25 squared?
So, in other words,
numbers ending
in 5 that's what we're going
to figure out today.
So first thing I want you
to think about is an estimate.
If 10 squared is 100
and 20 squared is 400,
hopefully it seems reasonable
to you that 15 squared is
somewhere between 100 and 400.
Okay? Same thing for 25.
That's in between 20 squared
and 30 squared.
So, it's going to be somewhere
between 400 and 900.
I'm going to write
down what the answer is
and then I'll show you how I
came up with these in my head.
Fifteen squared
if you just get
out a calculator and square it
or multiply it
out you get 225;
25 squared is 625,
35 squared is 1225.
Forty-five squared let's just
do that one is 2025.
So, before going
on how I did this I want you
to notice a pattern.
The pattern is they all end
in 25.
So, that's kind of cool.
So you can guess
if I was going
to do 55 squared this would
also end in 25 and, in fact,
it will end in 25.
How do we get this first part
before the 25?
So how do I get the 2 or the 6
or the 12 or the 20
or whatever this is going
to be?
Here's the interesting thing.
If you're doing 10 squared,
so you take the number 1
and multiply it
by the next digit so 1 times 2
so I get 1 times 2 that's how
I get that 2.
Think of it as in
between 1 squared
and 2 squared, you know 2 is
in between that, right?
Okay. So, how
about this one the digit,
so I'm looking
at this first digit 2,
this is really the digit I was
looking at,
it's going to be 2 times
when you're counting the next
number is 3,
that's how I got the 6.
What about this one, 3?
How would I get that 12?
I do 3 times its next number,
3 times 4.
That's the pattern.
So, do you see how I got the
45 squared was 2025?
It's really just 4,
write that digit,
times the next number
which is 5,
that's the first two digits
and then they all end in 25.
So, here we go what's it going
to be for 55 squared?
What will that be?
It ends in 25
and you take the first digit
here, which is 5
and multiply it
by the next number
on the sequence
when you're counting is 6
so that is 55 squared.
So let's see
if you can do these
on your own now.
Fifteen squared, 25 squared,
35 squared, 45 squared,
55 squared, et cetera.
Let's just do
up to 75 squared for now.
All right so what's
the pattern?
They all end in 25
and how do we get the first
part before the 25?
Well, if the first digit is 1,
you do 1 times 2 which is 2
and then it ends in 25.
All right.
Put the video on pause and see
if you can get the rest
of these on your own.
All right let's do 25 squared.
So, 2 times the next digit is
3 so 2 times 3 is 6
and that always ends in 25.
Thirty-five squared;
3 times the next digit is 4,
3 times 4 is 12 and then 25.
Forty-five squared;
4 times the next digit is 5
so 4 times 5 is 20,
ends in 25.
I hope you're doing as well
as I'm doing.
Five, 5 times 6 is 30,
ends in 25.
Isn't this cool how fast we
could do this?
Sixty-five squared.
Six times 7 is 42 ends in 25.
Seventy-five,
7 times 8 is 56 ends
in 25 that's all there is
to it.
Now, let's take a little bit
bigger number
like 115 squared,
you can do this
with any number still works.
You have to do 11 times 12.
Well, there's a little trick
for multiplying by 11s
and you might want to look
at my video on that,
11 times 12
but if you don't know
that you go
over here what's 11 times 12
is 132 and ends in 25.
Now you may be saying to me,
wait, I still have
to do multiplication.
Well, what's easier?
Multiplying 115 times 115 long
hand or just doing 12 times 11
and putting a 25 at the end?
I vote for this way 11 times
12 is 132, tack on the 25.
So that's how you can do
any number.
In fact, let's do this one.
How about 205 squared what
would that be?
Well, I'd have
to do 20 times 21.
That's not too bad;
21 times 20 I always write it
like that,
you know there's an extra 0
at the end I just think
of 21 times 2 it's 42
so I've got 420 and tack a 25
at the end
and that's all there is to it
so that's a fast, easy,
fun way to square any number
ending in 5.
Why does this work?
Let's say you have a number
ending in 5 like 85.
Remember when you see the
number 85 the 8 is
in the 10s place and the 5 is
in the 1s or units place.
So, 85 the actual value
of 85 is really 8 times 10
plus 5.
That's what it really
stands for.
Let's take another number.
What about if I had 115?
It's true we have the 100s,
10s, and 1000s
but it's also true
that I could write
that as 11 times 10 plus 5.
So, if you have a number
ending in 5,
you could always write it the
number in front
of the 5 could be multiplied
by 10 and then you add 5;
that's the actual value.
So here you do the 11 times 10
for this first one we had 8
times 10.
So let's take any number
in that form.
Say we have something X5.
So in other words, 35,
you know, 115 whatever X,
the value of this number is
really X times 10.
I'm going to write it
as 10X plus 5.
So, for instance,
if this was 35,
it would be 10 times 3 plus 5.
So the question is what is 10X
times 5 squared?
Squaring something means
to multiply it by itself.
You have to know some algebra
to follow this,
beginning algebra,
and then you can do the
foil method.
So we do 10X times 10X,
which is 100X squared.
Then the outer terms 10X times
5 is 50X and the inner term is
also 50X that always happens
when you're squaring a
binomial the two middle terms
are the same plus 5 times 5
is 25.
So, I could add these 2
like terms in the middle
and get 100X.
All right now what I'm going
to do is group just the first
two terms together.
I'm going to take
out the common factor
which is 100X.
So, I have 100X times X plus
1, plus 25.
So this proves it.
What you really have here is
X, that's the number before
the 5 up here
in our number X5,
times the next digit whatever
that must be,
and then it says times 100.
Well, remember the way we
write X times X plus 1.
We wrote it whatever it ended
up being and then we just
wrote the number 25
after it it's
because when you multiply
by 100 it moves this 2 places
to the left.
So, if this was,
so this is what it looks
like the number X times X plus
1 whatever number you get
and you just wrote 25 after it
that really is X times X plus
1 times 100 with a 25
at the end.
So, if you have 35 squared,
what we really did was 3 times
4 times 100 plus 25,
which looks exactly like 1225
because this is 1200,
25 so that's the reason
that it's always going to work
to take the number in front
of 5, multiply
by the next consecutive
number, and then tack on 25
at the end.
[ Silence ]
Please visit my website
at yourmathgal.com
where you can view all
of my videos
which are organized by topic.