This is what I did
6 = 1/5
6 x the numerator (4) = 24
24 + 6 = 30
Next problem
12 = 2/5
12 ÷ 2 = 6
6 x 3 = 18
18 + 12 = 30
Aside from the fact that your notation is so abysmal as to be almost indecipherable (and because of that, if I were grading this exercise, I'd not grade you very favorably), your methodology and answers are correct for both problems. Let's step through both step-by-step, clean up any errors, and see if we can't get a better feel for what's going on...
Clearly this is a bit of nonsense. As we all know, 6 is
not the same number as 1/5, so we cannot say they are equal. Instead, what I'm sure you mean (though you absolutely need to specify this as part of writing your thought process and answers in a clear manner) is that 6 feet is 1/5
of the length of the stick.
6 x the numerator (4) = 24
And here we have even more nonsense. You reference "the numerator (4)" but you've not shown a fraction which has a numerator of 4. Instead, it would be more appropriate and accurate to write something like, "Since 6 feet is 1/5 of the length of the stick, it follows that 6 x 4 = 24 feet is 4/5 of the length of the stick."
Although this statement is technically true, it isn't clear how this bit of arithmetic relates to the problem at hand. Here I would say, "4/5 of the length of the stick plus 1/5 of the length of the stick is 5/5, or the total length, of the stick. I previously found that 1/5 of the length was 6 feet and 4/5 of the length was 24 feet, so the total length must be 24+ 6 = 30 feet."
And similarly for the second problem...
See the comments for the previous statement of this sort.
Okay, so here you've noted that since 12 feet is 2/5 of the length of the pole, 6 feet is 1/5 of the length of the pole. That's a good, accurate deduction, though again I would write this out clearly, using English words and phrases if needed.
In the same manner, you've now deduced that 3/5 of the length of the pole is 18 feet. Very good.
As with the last problem, you've noted that 5/5 = 3/5 + 2/5, so the total length of the pole must be 18 + 12 = 30 feet.
Now, since your methodology checks out at every step along the way, the only possible conclusion is that the answer key which told you 30 feet is not the correct answer for problem 2 is wrong. Unfortunately, you should get used to this, as it is all too common for answers keys to be wrong.
