I'm having a little trouble solving inverse variation. I have four problems - 2 are solved using proportions, 2 are not --- why can they not all be solved using proportions?

1. y varies inversely with x. If y = 40 when x = 16, find x when y = -5
If I use a proportion for that, I get -2, however, the answer is -128.

They solved like this: (40)(16) = -5x
x = -128

2. y varies inversely with x. If y = 7 when x = -4, find y when x = 5. If I use a proportion for that, I get -8.75, and the answer is -5.6

They solved: 7x(-4) = 5y
y = -5.6

3. y varies directly with x. If y = 15 when x = -18, find y when x = 1.6. I did a proportion and got the correct answer of -1.33

4. y varies directly with x. If y = 75 when x = 25, find x when y = 25. I did a proportion and got 8.333

Could you please explain #1 and #2 and why those can't be solved using proportions, and how they solved them?

direct ... y = k x ... y1 / x1 = y2 / x2

... y1 / y2 = x1 / x2

indirect ... x y = k
... x1 y1 = x2 y2 ... x1 / x2 = y2 / y1
... a proportion will work
... but it's different than direct
... it is "inverted"

Sure! I can explain why problems #1 and #2 cannot be solved using proportions and how they were solved.

In inverse variation, two variables are said to vary inversely with each other if their product remains constant. Mathematically, for inverse variation, we have the equation:

y = k/x

where y and x are the variables, and k is the constant of variation.

Let's start with problem #1: y varies inversely with x. If y = 40 when x = 16, and we need to find x when y = -5.

If we try to solve this using a proportion, we would set up the equation:

40/16 = -5/x

If we solve this proportion, we get x = -2. However, this is incorrect. The correct approach is to use the equation for inverse variation, which is:

(40)(16) = -5x

We multiply y and x together because they are inversely related. Then, we can solve for x:

640 = -5x
x = -128

For problem #2: y varies inversely with x. If y = 7 when x = -4, and we need to find y when x = 5.

Again, if we try to solve this using a proportion, we would set up the equation:

7/-4 = y/5

Solving this proportion gives us y = -8.75. But this is incorrect. To solve correctly, we use the inverse variation equation:

7(-4) = 5y

Multiplying y and x together, we get:

-28 = 5y
y = -5.6

In both cases, using proportions does not work because inverse variation does not follow a direct proportionality relationship (where y = kx). Instead, it follows an inverse relationship (y = k/x). To solve inverse variation problems correctly, you need to use the appropriate equation, multiplying or dividing the variables as necessary to isolate the unknown.