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### Course: 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.

## Want to join the conversation?

- why do we always have to set X?(15 votes)
- "x" is the horizontal line in a cartesian plane. "x" and "y" are the most commonly used variables, though "x" is prefered more. Hope that helped :)(3 votes)

- when point p moves at a speed of x cm per second and the time taken to reach point b after starting point a is y seconds, express the relation between x and y as an equation.

y= blank/x how do find the constant of variation.(5 votes)- Jed,

There are lots of letters in your question points p (which is moving) and points a and b as well as x cm/second speed and y seconds.

Thats a lot of letters, but to answer the question you need one more letter.

Let "d" be the distance point p has traveled after y seconds.

The cm/second * seconds = cm * (seconds/seconds). The seconds over seconds cancel out giving you an answer in cm. (words can cancel just like numbers)

If you multiply x cm/second * y seconds you get xy cm as your answer.

And this answer would be equal to the distance (d) in cm that point p has traveled after y seconds.

so d cm = x cm/second * y seconds. so

d=xy which can be converted to

d/x = y or

y=d/x which you had as y=blank/x

The blank is the distance that point p has traveled.

I hope that helps.(8 votes)

- Why is the constant called k? Why not c?(6 votes)
- It abbreviates the German word “konstante” which means… you guessed it: constant. Also, the letter 'c' is used for many other tasks and is usually not available.(2 votes)

- Is the constant of variation the same as slope?(5 votes)
- Couldn't you just get the answer to this question faster by dividing 18 by 2.25 and just skipping all the "Oh so Y is this and X is this now time to solve"?(3 votes)
- Definitely! But Sal wants us to understand the process of actually solving this problem, since in real life, you come upon much more complex problems.(2 votes)

- can a direct variation be vertical on the x-axis(2 votes)
- I see the quick answer without much calculation, i.e. 18/2.25 = 8, Now I tried to think more structured as the solution suggested at the video, sorry, but this time I don't get where does the 9/4 fraction come from? Checking the tutorials here at Khan Academy have been really helpful to clear my doubts in math problems. Appreciate your answer. Many thanks to the team!(2 votes)
- The 9/4 fraction comes from the conversion of 2.25 into a fraction. When it is converted, it becomes 2 1/4 and when it becomes an improper fraction, it becomes 9/4 (because 2*4=8, then 8+1=9)(1 vote)

- Hi my name is Grace and I have a quiz on this on wednesday and I really need help! I am very confused and don't know what to do.Can you give me some help??(2 votes)
- I don't get how to do direct variation. I need help with the graph parts in it. Can anyone explain it to me? Thanks(2 votes)
- will you do my home work plz ?(1 vote)
- You aren't supposed to ask homework questions on here. If you post, I usually tell you how to do the problem, but won't give you the answer. That's the point of homework - to learn how to get the answer yourself.(2 votes)

## 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.