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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.