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>> Julie Harland: Hi.
This is Julie Harland,
and I'm Your Math Gal.
Please visit my Web site
at yourmathgal.com
where you can search for any
of my videos organized
by topic.
All right.
We're going
to do this puzzle problem.
In a triangle,
the length
of the sides are
consecutive integers.
The perimeter is 11 more
than twice the shortest side.
What is the perimeter?
Hmm. First of all,
you have to know what
consecutive integers are.
So consecutive integers are
numbers like 5, 6, 7,
one after the other.
Or you might have 20, 21, 22.
That would be three
consecutive integers, etc. So,
if you have one number,
the next consecutive integer
is one more than that.
So, if the first one is n,
whatever that number is,
you would have to add one
to it to get the next number.
And you would have to add 1
to that to get the
next number.
And you'd have to add 1
to that to get the next
number, etc. So,
in other words,
this goes on and on.
If the first number would be
10, another one --
the next number would be 10
plus 1, which would be 11.
This would be 10 plus 2,
which would be 12.
This would be 10 plus 3;
it would be 13,
etc. All right.
So we have to know that.
That's what consecutive
integers means.
So we're going
to define our sides using
that idea, the first, second,
and third side.
So, if the first one's n,
all right,
so we don't know what it is.
What would the next one be?
It would be n plus 1.
And the other,
the third side would be n
plus 2.
So this is the shortest side,
right?
The smallest number is the
shortest side in this case.
And we're going
to assume these are positive
numbers as well.
Next part.
The perimeter is 11 more
than twice the shortest side.
Okay. Perimeter means you add
the three sides.
So I'm -- if I --
in other words,
I want to add my first side.
This is not the really
mathematical part,
but it's a way
of thinking of it.
The first side plus the second
side plus the third side,
that's the perimeter.
And it says that that's going
to be the same thing as --
this is, you know, kind of --
I don't like to use equal
signs this way,
but it says that's basically
going to be the same thing.
Has 11 more
than twice the shortest side.
So I'm going
to take twice the shortest
side which, in this case,
would be the first side.
And I'm going to add 11 to it.
Okay. So this is a way
of thinking about it.
So let's actually write
down what those pieces are.
The first side is n. The
second side is n plus 1.
And the third side is n
plus 2.
Some people like to put each
of these in parentheses.
That's okay,
but then you would take them
off at the next step.
And that's going
to be the same thing
as 2 times the first side.
Well, what's the first side?
It's n. So it's 2 times n,
right, plus 11.
So that's 11 more
than twice the shortest side.
So now we have an equation.
So to solve this equation
let's simplify the left-hand
side by combining like terms.
Remember, n plus n plus n,
you've got three n. Remember
you add the coefficients,
1 plus 1 plus 1.
3n plus 3 is equal
to 2n plus 11.
So we can subtract 2n
from both sides,
so we get the variables
on the left-hand side.
And I'm going to subtract 3
from both sides
so that the constants are
on the right.
So what I'm doing is I'm
adding a negative 2
and minus 3 to both sides.
You could think
of it that way.
So what I have on the left --
oh, how convenient.
I just get n equals 8.
So I found n. Now,
what did n stand for?
It stood for the
shortest side.
And notice this problem had
nothing about feet or inches,
so it was sort
of a puzzle problem.
We weren't given any units,
so it was just asking
in numbers.
So in this case we don't have
to write feet or inches
because the problem didn't
give us any units.
So those would be our numbers,
8, 9, and 10.
Those are three consecutive
integers, right?
So, if n is 8,
the next number would be 9.
The next number would be 10.
And, now, what's the question?
So I've got three
consecutive integers.
I did that part right.
But the other thing it said
was what?
All right.
So, if we read this again,
it said that the perimeter is
11 more than twice the
shortest side.
Let's make sure that's true.
Well, what would the perimeter
of the triangle be
if we had 8, 9, and 10?
We had 8, 9, and 10.
That adds up to 27.
All right.
So we're wondering
if that's really true
about the perimeter.
So the question is --
is this: Is 27, right,
the perimeter, is that 11 more
than twice the shortest side?
Well, the shortest side is 8,
so what's twice the
shortest side?
16. And if I add 11, I get 27.
So that makes sense.
All right.
So what we've got here is
that the sides are 8, 9,
and 10; and the perimeter is
27, and all this makes sense
given our original
problem here.
And so the question is:
What is the perimeter?
So that's all we have
to do is write 27;
or we could actually write it
out in words,
which is how I usually do it.
I would write,
"The perimeter is 27."
[ Silence ]
Please visit my Web site
at yourmathgal.com
where you can view all
of my videos,
which are organized by topic.