Word Problem Solving Plan 1

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Word Problem Solving Plan 1

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Let's do some word problems. It says this year
you got a 5% raise.
If your new salary is $45,000, what was your
salary before the raise?
So let's say my old salary is x.
x is equal to old salary.
And I got a 5% raise off of that salary.
So x plus 5% of x, plus 5% times x-- that's my raise
right there-- is going to be equal to my new salary.
Well my new salary, they tell us what it is.
It's $45,000.
And let's see if we can algebraically manipulate this
a little bit.
5% is the same thing as 0.05, 5%.
Per cent literally means per 100.
So this is the same thing as 5 over 100.
This is 5%, per 100, which is the same thing as 0.05.
So here we could say this is the same thing is 1x plus--
let me write that x a little bit nicer-- 1x plus 0.05x is
equal to 45,000.
1 of something plus 0.05 of that same something.
You can add the coefficients.
So you add those two.
You're going to get 1.05x is equal to 45,000.
And then to solve for x, you divide both sides by 1.05.
These will just cancel out.
And we're going to get x is equal to 45,000 over 1.05.
And let's figure out what that is.
So 45-- one, two, three-- $45,000 divided by 1.05 is
equal to $42,857.
I'll say that again, $42,857.
This is how much your annual income was last year.
Let's make sure that that makes sense.
Let's see if that works out.
So if we check it, let's say we start with this.
Let's take 5% of that.
So times 0.05 is equal to-- so that's the amount of raise you
had-- plus your old salary, plus 42,857.
And, of course, we round it off a little bit,
so it won't be exact.
And it equals almost exactly $45,000.
We're like 14 cents below that.
So $45,000.
So this answer checks out.
Let's do this next one.
It costs $250-- let me write this down-- it costs $250 to
carpet a room that is 14 by 18.
How much does it cost to carpet a room that is 9 by 10?
And the units here are feet.
Well in case you have never carpeted a room, you normally
carpet a room by the amount of area the room is, or the
square footage of the room.
So what we could do is use this information up here to
figure out how much it costs per square foot.
So in this situation we had 14 by 18, so how many
square feet is that?
So I'll multiply it out.
18 times 14.
4 times 8 is 32.
1 times 4 is 4, plus 3 is 7.
You have a 0.
1 times 8 is 8.
1 times 1 is 1.
You have a 2.
You have a 15.
And then you have a 2.
So it's 252 square feet.
So 14 times 18, this is 252 square feet.
Which is kind of a strange number.
It's almost exactly $1.00 per square foot, but not exactly
equal to $1.00 per square foot.
So if you want to know per square foot, they charged us
$250 for 252 square feet.
So it's going to be a little bit less than
$1.00 per square foot.
Let me confirm my math.
So you have 18 times 14 is equal to 252.
And then so they're charging us $250 for 252 square feet.
So it's like 99.2 cents per square foot.
So if we write this as a function, they're going to
charge us as a function of, let's say the square footage,
or let me say as a function of the area where a is the area
of our floor.
They're going to charge us-- what was that number I just
had-- 99.2 cents, 0.992 times our area in square footage.
This is an expression of how much they're
going to charge us.
So here, what's our area?
We have 9 by 10.
So this is our area is equal to 90 square feet.
So the cost of 90 square feet of carpeting for 90 square
feet is going to be a little bit less than $90.00.
We know this is a little bit less than
$1.00 per square foot.
We can actually do it.
It's 0.992 times 90.
So let's multiply this times 90.
It's equal to like $89.29.
If I were to have written this problem, I would have made
this number 250 so that it would have been exactly $1.00
per square foot, and so we wouldn't have had to break out
the calculator.
But anyway, I didn't write this problem.
Problem 6.
It costs $12.00 to get into the San Diego County Fair, and
$1.50 per ride.
If Rena spent $24.00 in total, how may rides did she go on?
So no matter what, it's going to cost you $12.00,
plus $1.50 per ride.
Plus $1.50 times the number of rides you go on.
This is how much it's going to cost for you to spend your day
at the San Diego County Fair.
I could write this as a function.
I could write cost as a function of rides is $12.00
plus $1.50 times the number of rides.
So we know that Rena spent $24.00 in total.
So what was the number of rides she went on?
Well we could solve for this.
We could just look at this part of the
equation right there.
So what do we get?
We get 12 plus-- I'll just write this as--
1.5r is equal to 24.
Let's subtract 12 from both sides of the equation.
So that becomes-- so I'll write it here-- negative 12
plus negative 12.
Subtract the 12 from both sides.
You get 1.5r is equal to 12.
And divide both sides by 1.5.
You get r is equal to-- what's 12 divided by 1.5-- that is 8.
1.5 times 8 is 12.
1-1/2 times 8.
One 8 is 8, and 1/2 of an 8 is 4, that's 12.
So she went on 8 rides.
And let's verify.
Let's verify by looking at what's the cost of 8 rides?
It's going to be $12.00 plus 1.5 times the 8 rides.
So that's going to be $12.00 plus, well
1.5 times 8 is $12.00.
So this is going to be equal to $24.00.
So we have verified our problem.
All right.
Let's do one more.
The sum of angles in a triangle is 180 degrees.
That's good to know in general.
You'll see that many times in geometry.
Let me write this down.
So let me draw some random triangle right there.
So that is my triangle.
If the second angle is twice the size of the first angle.
So let me make my first angle.
Let's say that the angle is x.
That's my first angle.
The second angle is twice the size of the first angle.
So let's say this is the second angle, that's 2x.
And the third angle is three times the size
of the first angle.
So the third angle is three times this one, so that's 3x.
What are the measures of the angles in the triangle?
Well they tell us that they all have to
add up to 180 degrees.
So we get x plus 2x plus 3x has to be
equal to 180 degrees.
And I could even write this as 1x.
Just remember that there's a coefficient there.
So 1 of something plus 2 of something plus 3 of something.
That's going to be 6 of the somethings.
1 plus 2 plus 3.
So 6x is equal to 180 degrees.
Divide both sides by 6.
And you get x is equal to 180 divided by 6 is 30.
So this angle right here is 30.
This angle right here is two times that, right?
It's 2x.
So this angle right here is 60.
And this angle right here is three times that.
That's equal to 90.
So let's see if this meets our requirements.
This is a 30, 60, 90.
They definitely add up to 180.
30 plus 60 is 90, plus 90 is 180.
And then if we look at the requirements, the second angle
is twice the size of the first. Right, 60 is twice 30.
And the third angle is three times the size of the first.
90 is 3 times 30.
So it all worked out.