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