If 500 is the multiple of 100 that is closest to x and 400 is the mult

I see your point.

"x rounded to the nearest hundred is 500" means \(450\leq{x}<550\) (inequality you've written) BUT it IS NOT the same as "500 is the multiple of 100 that is closest to x", which should be \(450<x<550\) as 450 is equidistant from 400 and 500 and we cannot say that 450 rounded to nearest multiple of 100 is 500, it's 500 OR 400. That's why endpoints must be excluded.

If 500 is the multiple of 100 that is closest to X and 400 is the multiple of 100 closest to Y, then which multiple of 100 closest to X + Y ?

"500 is the multiple of 100 closest to X" --> \(450<x<550\);
"400 is the multiple of 100 closest to Y" --> \(350<y<450\).

(1) x<500 --> \(450<x<500\) --> add this inequality to inequality with y --> \(800<x+y<950\). If x+y=810 then closest multiple of 100 is 800 BUT if x+y=860 then closest multiple of 100 is 900. Not sufficient.

(2) y<400 --> \(350<y<400\) --> add this inequality to inequality with x --> \(800<x+y<950\). The same here: if x+y=810 then closest multiple of 100 is 800 BUT if x+y=860 then closest multiple of 100 is 900. Not sufficient.

(1)+(2) Sum \(450<x<500\) and \(350<x<400\) --> \(800<x+y<900\) --> and again if x+y=810 then closest multiple of 100 is 800 BUT if x+y=860 then closest multiple of 100 is 900. Not sufficient.

Answer: E.

Hope it's clear.