Direct variation
Two variables vary directly if their ratio is constant. In formulas, you have that x and y are in direct variation if
You can see that this formula can be rewritten as
by multiplying both sides by x.
So, what do we get? Since k is a fixed number, you can see that y must be a multiple of x. This situation arises everytime you have a unit "price" (it doesn't have to be necessarily money), and you multiply by a certain amount.
For example, if a candy costs $2, x is the number of candy you buy, and y is the money you spend, then x and y vary directly, because you will have
in fact, you'll spend $2 for each candy you'll buy, and so the money spent will always be twice the number of candies bought. Look at the following table:
As you can see, the ratio between the money spent and the number of candies bought will always be 2, which is our unit price. For each line, you have the ratios 2/1, 4/2, 6/3, 8/4, 10/5, and so on, and they all equal two.
Another example might be the following, to show that we don't have to deal with money. Suppose that I make 3 math exercises per day. Let x be the number of days and y be the number of exercises I make. Again, those two variables are in direct variation, because the more I'll go on, the more exercises I'll make. Moreover, the ratio between the exercises made and the days passed will always be 3:
Inverse variation
Two variables vary inversely if their product is constant. In formulas, you have that x and y are in direct variation if
You can see that this formula can be rewritten as
by dividing both sides by x.
So, this time, if x grows y gets smaller, and vice versa. Here's a couple of examples. Suppose that you want a rectangle with area 100, and you can choose the height and length of the rectangle, let's call them x and y. You know that you must satisfy
So, you can build the following rectangles:
As you can see, as the legnth increases, the height decreases. Moreover, for each line, the product between height and length is always 100. This means that the two variables vary inversely.
Another example: suppose you have to divide 20 balls in a certain number of boxes. Just like before, if you use a lot of boxes, there will be very few balls in each box. On the other hand, if you use fewer boxes, each of them will fit more balls inside. Specifically, these are your choices:
So, you can use only one box and fit all the 20 balls inside, or use 2 boxes and put 10 balls in each box, or use 4 boxes and put 5 balls in each box, and so on. Again, as you can see, as the number of boxes rises, the number of balls per box gets smaller, and their product is always 20. Again, this means that the two variables vary inversely.
A final example: suppose you have a cake. If you are alone, you can eat the whole cake for yourself! But if there are 2 people, each of them will eat 1/2 of the cake. If there are 3 people, each of them will eat 1/3 of the cake, and so on. So, as the number of people rise, the amount of cake for each person gets lower. And the product bewteen the number of people and the cake amount is always 1 (after all, you only have that one cake...). As it should be clear by now, this implies that the two variables vary inversely.
