Okay, let's see. I need to figure out what fraction of the remaining books on the three shelves are math books after two books are taken off each shelf. Hmm, let's start by understanding the problem step by step.

First, the bookcase has 3 shelves with a total of 24 books. The top shelf has 8 mystery books, the middle shelf has 10 math books, and the bottom shelf has 6 science books. Then, two books are taken off each shelf. I need to find the fraction of math books out of the remaining books.

Wait, so initially, each shelf has a certain number of books. Let me confirm the numbers: top shelf is 8, middle is 10, bottom is 6. Let's add those up to make sure they total 24. 8 + 10 is 18, plus 6 is 24. Yep, that's correct. Total books are 24, which matches the problem statement.

Now, two books are taken off each shelf. So, each shelf loses 2 books. Let me calculate how many books remain on each shelf after this removal.

Starting with the top shelf: originally 8 mystery books. Subtract 2, so 8 - 2 = 6 books left. The type of book (mystery) isn't specified to change, so we assume the remaining books are still mystery unless specified otherwise. But since the problem doesn't mention any changes in the types, I can assume the categories remain the same. So, top shelf still has mystery books, but now 6 of them.

Middle shelf: originally 10 math books. Subtract 2, so 10 - 2 = 8 math books left. Therefore, the middle shelf now has 8 math books.

Bottom shelf: originally 6 science books. Subtract 2, so 6 - 2 = 4 science books left. So, bottom shelf has 4 science books.

Now, let's check the total number of remaining books. Top shelf has 6, middle has 8, bottom has 4. 6 + 8 + 4 = 18 books in total. Let me verify that: 24 total minus 2 books taken off each of the 3 shelves. Each shelf loses 2, so total books removed are 2*3 = 6. Therefore, 24 - 6 = 18. Correct.

Now, the question is asking for the fraction of the remaining books that are math books. The math books remaining are 8 (from the middle shelf). So, the fraction is 8 (math books) over 18 (total books). Let me write that as a fraction: 8/18.

But we can simplify that fraction. Both numerator and denominator are divisible by 2. 8 divided by 2 is 4, and 18 divided by 2 is 9. So, 4/9. Is that the simplest form? Yes, since 4 and 9 have no common divisors other than 1.

Wait, let me double-check my steps to make sure I didn't make a mistake.

Original counts:
- Top shelf: 8 mystery
- Middle shelf: 10 math
- Bottom shelf: 6 science

After removing 2 from each shelf:
- Top: 6 mystery
- Middle: 8 math
- Bottom: 4 science

Total remaining: 6 + 8 + 4 = 18

Math books remaining: 8

Fraction: 8/18 = 4/9. That seems right.

Hmm, seems straightforward. Let me check if there's any alternative way to approach this problem that might lead to a different answer, just to be thorough.

Alternatively, we could consider each shelf's contribution to the total math books and total books. But since the problem is straightforward with each shelf losing 2 books, and the types remaining the same, the initial approach seems solid.

Another way: total math books initially are 10. After removing 2, they become 8. Total books initially 24, after removing 6, 18. So 8/18. Yep, same result.

I think that's correct. The fraction is 4/9.
Therefore, the fraction of the remaining books that are math books is $\boxed{\frac{4}{9}}$.
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