How many positive integers between 100 and 999 inclusive are odd?

Question

I found the answer to this in a pdf online but don't understand their method: Every 2nd number is odd. 1000 div 2 − 100 div 2 = 500 − 50 = 450

The method I thought I could use didn't work either. If someone can explain to me why my logic doesn't make sense I would appreciate it. I simply created an inequality as such:

100 ≤ 2k+1 ≥ 999

and then solved for k

99 ≤ 2k ≥ 998

49.5 ≤ k ≥ 449.5

since its integers only:

50 ≤ k ≥ 449

and then the numbers in this range would be (449-50)+1=400

Answer 1

Your thinking is correct. You simply messed up the algebra (and the inequality sign). Fixing the result we have

$$100 \leq 2k+1 \leq 999$$ $$99 \leq 2k \leq 998$$ $$ \frac{99}{2} \leq k \leq 499$$

Answer 2

1000 - 100 = 900

You can include 1000 since it's not an odd number, and wouldn't count.

So there are 900 numbers in total, and exactly half of them will be odd:

$\frac{900}{2} = 450$