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## Algebra (all content)

### Unit 13: Lesson 7

Direct and inverse variation- Intro to direct & inverse variation
- Recognizing direct & inverse variation
- Recognize direct & inverse variation
- Recognizing direct & inverse variation: table
- Direct variation word problem: filling gas
- Direct variation word problem: space travel
- Inverse variation word problem: string vibration
- Proportionality constant for direct variation

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# Direct variation word problem: filling gas

Worked example: Model a context about filling gas with a direct variation equation. Created by Sal Khan and Monterey Institute for Technology and Education.

## Video transcript

We're told that the total cost
of filling up your car with gas varies directly with the
number of gallons of gasoline you are purchasing. So this first statement tells
us that if x is equal to the number of gallons purchased, and
y is equal to the cost of filling up the car, this first
statement tells us that y varies directly with the number
of gallons, with x. So that means that y is equal
to some constant, we'll just call that k, times x. This is what it means
to vary directly. If x goes up, y will go up. We don't know what the rate
is. k tells us the rate. If x goes down, y
will be down. Now, they give us more
information, and this will help us figure out what k is. If a gallon of gas costs $2.25,
how many gallons could you purchase for $18? So if x is equal to 1-- this
statement up here, a gallon of gas-- that tells us if we get 1
gallon, if x is equal to 1, then y is $2.25, right? y is what it costs. They tell us 1 gallon costs
$2.25, so you could write it right here, $2.25 is equal
to k times x, times 1. Well, I didn't even have to
write the times 1 there. It's essentially telling
us exactly what the rate is, what k is. We don't even have to
write that 1 there. k is equal to 2.25. That's what this told
us right there. So the equation, how y varies
with x, is y is equal to 2.25x, where x is the number
of gallons we purchase. y is the cost of that
purchase, so it's $2.25 a gallon. And then they ask us, how
many gallons could you purchase for $18? So $18 is going to be our total
cost. It is y cost of filling the car. So 18 is going to be
equal to 2.25x. Now if we want to solve for x,
we can divide both sides by 2.25, so let's do that. You divide 18 by 2.25,
divide 2.25x by 2.25, and what do we get? Let me scroll down
a little bit. The right-hand side, the 2.25's
cancel out, you get x. And then what is 18
divided by 2.25? So let me write this down. So first of all, I just like to
think of it as a fraction. 2.25 is the same thing-- let me
write over here-- 2.25 is equal to 2 and 1/4, which is
the same thing as 9 over 4. So 18 divided by 2.25 is equal
to 18 divided by 9 over 4, which is equal to 18 times
4 over 9, or 18 over 1 times 4 over 9. And let's see, 18 divided by 9
is 2, 9 divided by 9 is 1. That simplifies pretty
nicely into 8. So 18 divided by 2.25 is 8, so
we can buy 8 gallons for $18.