This was the solution I wrote up a few months ago:
In hindsight, it's not a particularly good solution as it assumes you know what's going on. If you do, great. If not, I'll try and explain it better:
The questions tells us that x is directly proportional to z
2 whereas y is inversely proportional to z
3. Ignoring the constants of proportionality, we can write the following 2 equations:
x = z
2y = 1/z
3Now, as the image above says, we just need to try out the different options. When "try out" I mean work out what x and y would be given what they say in the answer choice, and see if they equal the same thing. Btw, because we're ignoring the constants of proportionality, replace the phrase "directly proportional" with "equals".
A - "The square of x is directly proportional to the cube of y". x
2 = z
4. The cube of y = y
3 = 1/z
9. These 2 are clearly not equal to each other, so this cannot be the answer.
B - "The square of x is inversely proportional to the cube of y". We don't even need to bother working this out, as the numbers are going to be the same as A, and we know that A can't be right.
C - "The cube of x is directly proportional to the square of y". Okay. So the cube of x = x
3 = z
6. The square of y = y
2 = 1/z
6. This is almost the right answer, but not quite.
D - "The cube of x is inversely proportional to the square of y". This is exactly the right answer. We already worked out from C that the cube of x is z
6 and the square of y is 1/z
6 and so we know that the cube of x must be inversely proportional to the square of y.