## Solving Proportions Problems

### Algebra: Rates and Proportions Help

Step 1: Put together the general equation.
Step 2: Solve for the constant of proportionality.
Step 3: Plugging the constant into the equation, solve for the unknown variable.

Let's solve a few problems to see how this works, shall we?

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

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: 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: Solving for y when x = 64 we get 8y = 12 or y = 12/8 = 1.5 because the square root of 64 is 8. 