Writing equations for relationships between quantities | 6th grade | Khan Academy

We're told Ahmad is going to walk 20 kilometers for a charity fundraiser. In the first part of this question, they say to write an equation that represents how many hours ( t ) the walk will take if Ahmad walks at a constant rate of ( r ) kilometers per hour. Pause the video and see if you can have a go at that.

All right, now let's work through this together. You might be familiar with the notion that distance is equal to rate times time. For example, if you were to walk at a rate of five kilometers per hour for two hours, you would say five times two. Five kilometers per hour times two hours would give you 10 kilometers.

Now, in this situation, they've given us the number of kilometers, or the distance. In this situation, ( d ) is equal to 20. So, 20 is going to be equal to our rate, which we are told is going to be ( r ) kilometers per hour, times our time, which is ( t ) hours.

Now they're writing, they're asking us for an equation that represents how many hours the walk will take if Ahmad walks at a constant rate of ( r ). The way that it's phrased, it sounds like they want us to solve for ( t ), where ( t ) is going to be equal to some expression here that deals with ( r ) and probably some other things. So, if we put in any ( r ) here, then we can get the time.

If we know what the rate is, if you put that in here, because it's already solved for ( t ), we'll be able to solve for that time. You could think of ( r ) as the independent variable that you could try different values out for and that ( t ) is the dependent variable; it's the thing that we have solved for.

So, let's do that! Let's rewrite this expression here by solving for ( t ). I could do it right over here:

If I have ( 20 ) is equal to ( rt ), if I want to solve for ( t ), what can I do? Well, I could divide both sides by ( r ). If I do that on the right-hand side, then I'm just left with a ( t ) here because ( r ) divided by ( r ) is just 1, and on the left, I have ( \frac{20}{r} ).

So, I have ( t ) is equal to ( \frac{20}{r} ), and we're done. This will tell us how many hours Ahmad will take to walk based on the rate. You give me a rate, I'm just going to divide 20 by that and I'm going to give you ( t ).

You might say, why is this useful? Well, this is useful because now that we have it written this way, any time someone gives an ( r ) to you, you just take 20 divided by that and it essentially is already going to solve for what your time is; how far, how long Ahmad's going to have to walk.

Question two: How many hours will the walk take if Ahmad walks at a constant rate of six kilometers per hour?

Well, here is an example of that, where they are giving us the actual rate and they want the time. So, we just take the 6 and replace it in for ( r ). So we get ( t ) is equal to ( \frac{20}{6} ), which is ( 3 ) and a third hours, which would be the same thing as ( 3 ) hours and ( 20 ) minutes, depending on how you want to view it.

Let's do another example here. So, here we're told, at the end of each day, a restaurant makes soup with whatever amount of vegetable stock is unused that day. Let me center this a little bit. The soup recipe calls for ( 400 ) milliliters of water for every ( 500 ) milliliters of vegetable stock. Write an equation that represents how much water the restaurant should use, and we'll use the variable ( w ) with any amount of vegetable stock ( b ).

All right, and then we'll do part two right after that. So, let's look at this: ( 400 ) milliliters of water for every ( 500 ) milliliters of vegetable stock.

To get my head around this, I like just to think about—let's put a little table here. So, you could say amount of water, let me write it this way, water and vegetable stock. Vegetable stock:

So, for every ( 500 ) milliliters of vegetable stock, say you had ( 500 ) milliliters of vegetable stock—and I won't write the milliliters—then you're going to have ( 400 ) milliliters of water.

If you had a thousand here, which is two times that, well, you're going to have twice as much water, which is going to be ( 800 ). So you can see this relationship that's forming. No matter what the vegetable stock is, if you essentially take ( \frac{4}{5} ) of that, that is the amount of water you take.

So, if you had only ( 5 ) milliliters of vegetable stock, you take ( \frac{4}{5} ), you get the amount of water. So, another way to think about it is the water that you need to use is going to be ( \frac{4}{5} ) of the amount of vegetable stock that you are going to be using.

And so actually, we just did part one; we wrote an equation that represents how much water the restaurant should use with any amount of vegetable stock. The way that they phrased it, we're solving for ( w ) given some ( v ) that you might have.

Since we're solving for ( w ) here, we would consider ( w ) the dependent variable, and ( v ) is the independent variable. You can give me different ( v )'s, and then I can put that into this little equation here and I can solve for ( w ).

So, we've done the first part. If there are ( 800 ) milliliters of unused vegetable stock, how much water should the restaurant use to make soup?

Well, we can just take this ( 800 ) and substitute it in for ( v ) to figure that out. In this situation, the amount of water to use is ( \frac{4}{5} ) times ( 800 ).

And that's going to be, let's see, ( 800 ) divided by ( 5 ), ( 100 ) divided by ( 5 ) is ( 20 ). And so, you're going to have ( 8 ) of those, so it's ( 160 ). So, ( 160 ) times ( 4 ) is equal to ( 640 ) milliliters, and we are done.