Example 1: Y is directly proportional to x. When x = 5, y = 8. What does y equal when x = 9?
First, we set up our general equation. Because y is directly proportional to x, we have:
y = cx
where c is the constant of proportionality. In other words, when x goes up, y goes up, and when x goes down, y goes down.
The next thing we do is plug our values for x and y into the equation so we can solve for c:
8 = (c)(5)
Solving for c, we get c = 8/5 = 1.6 and we plug this into our equation:
y = 1.6x
Now, we can plug x = 9 into the equation to find out what y equals:
y = (1.6)(9)
y = 14.4
So, our answer is 14.4
Example 2: Y is directly proportional to the square of x. When x = 2, y = 32. What does y equal when x = 5?
This time, our general equation is slightly more complicated because x is squared:
y = cx2
Like before, we solve for our constant:
32 = (c)(22)
32 = (c)(4)
We get c = 8:
y = 8x2
Solving for y when x = 5, we get y = (8)(52) = (8)(25) = 200
Example 3: Y is inversely proportional to x. When x = 2, y = 8. What does y equal when x = 24?
This time, because y is inversely proportional to x, our general equation is different:
xy = c
so when x goes up, y goes down, and vise versa. But, other than that, we solve these kinds of problems the same way as direct proportion problems. Solving for the constant, we get:
(2)(8) = c
So c = 16 and our equation is now:
xy = 16
Solving for y when x = 24 we get y = 16/24 = 2/3
Example 4: Y is inversely proportional to the square root of x. When x = 36, y = 2. What does y equal when x = 64?
As before, we set up our equation:
eq001
Since the square root of 36 is 6, it is easy to solve for c:
(6)(2) = c
We get c = 12 and our equation is now:
eq002
Solving for y when x = 64 we get 8y = 12 or y = 12/8 = 1.5 because the square root of 64 is 8.
Step-by-step explanation:
Example 1: Y is directly proportional to x. When x = 5, y = 8. What does y equal when x = 9?
First, we set up our general equation. Because y is directly proportional to x, we have:
y = cx
where c is the constant of proportionality. In other words, when x goes up, y goes up, and when x goes down, y goes down.
The next thing we do is plug our values for x and y into the equation so we can solve for c:
8 = (c)(5)
Solving for c, we get c = 8/5 = 1.6 and we plug this into our equation:
y = 1.6x
Now, we can plug x = 9 into the equation to find out what y equals:
y = (1.6)(9)
y = 14.4
So, our answer is 14.4
Example 2: Y is directly proportional to the square of x. When x = 2, y = 32. What does y equal when x = 5?
This time, our general equation is slightly more complicated because x is squared:
y = cx2
Like before, we solve for our constant:
32 = (c)(22)
32 = (c)(4)
We get c = 8:
y = 8x2
Solving for y when x = 5, we get y = (8)(52) = (8)(25) = 200
Example 3: Y is inversely proportional to x. When x = 2, y = 8. What does y equal when x = 24?
This time, because y is inversely proportional to x, our general equation is different:
xy = c
so when x goes up, y goes down, and vise versa. But, other than that, we solve these kinds of problems the same way as direct proportion problems. Solving for the constant, we get:
(2)(8) = c
So c = 16 and our equation is now:
xy = 16
Solving for y when x = 24 we get y = 16/24 = 2/3
Example 4: Y is inversely proportional to the square root of x. When x = 36, y = 2. What does y equal when x = 64?
As before, we set up our equation:
eq001
Since the square root of 36 is 6, it is easy to solve for c:
(6)(2) = c
We get c = 12 and our equation is now:
eq002
Solving for y when x = 64 we get 8y = 12 or y = 12/8 = 1.5 because the square root of 64 is 8.
