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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

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  • $\begingroup$ When you divide $998/2$ you should get $499$, not $449$ $\endgroup$ – Ross Millikan Mar 29 '16 at 3:12
  • $\begingroup$ @RossMillikan d'oh! $\endgroup$ – ohbrobig Mar 29 '16 at 3:24
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Your thinking is correct. You simply messed up the algebra (and the inequality sign). Fixing the result we have

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

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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:

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

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