arithmetic properties problem

Is it E ?

Given: x < y
x = n7m
y = p5r
-------------
x + y = abcd

Note that the only thing you can tell is that the hundreds digits (plus any carryover) must sum to 10 or greater. We know that there is a 1 carried over from the tens column, so n+p+1>9 or n+p>8

Let's look at each individually

I. The units digit of x + y is greater than the units digit of either x or y.

This DOES NOT need to be true. m+r based on my convention can be any numbers (5 and 1 or 7 and 8). I. is asking you if you NEED the values to result in carrying over a value into the next column.

Think about this. There is nothing you need to do with the singles column that will make a difference to us at this point. We know that the tens column already adds to 10 or greater and is carrying a digit over. If the singles column doesn't bring a 1 over, then it doesn't matter. The only way you would start caring is if the tens column otherwise added to 9. Then carrying a 1 would overflow the tens column. Since this is not the case, the singles digit is pretty much irrelevant (unless for some reason it was necessary to show that y>x).

II. The tens digit of x + y equals 2.

This DOES NOT need to be true. As already explained in I, there could be a carryover from the singles column. if m+r<10, then the tens (7+5) results in a 2. However, if there is m+r >9, then the tens is 7+5+1 = 13.
There still is no evidence to what is in the singles column.

III. The hundreds digit of y is at least 5.

First of all, I already do not believe that I or II has to be true, and there is no "NONE" option. So III has to be true or I have missed something earlier.

As explained in the "given" section, the hundreds digits MUST total to 10 or above. The minimum requirements for this is that the hundreds are 4 and 5, plus a carryover from the tens digits, or they are 5 and 5.

At this point, we know that the hundreds digit of 5 MUST be at least 5. The hundreds digit of x must be at least 4 since the tens column did bring over a 1.

Therefore, the answer is III. Only (B)

III. The hundreds digit of y is at least 5.

First of all, I already do not believe that I or II has to be true, and there is no "NONE" option. So III has to be true or I have missed something earlier.

I am with you till here

As explained in the "given" section, the hundreds digits MUST total to 10 or above. The minimum requirements for this is that the hundreds are 4 and 5, plus a carryover from the tens digits, or they are 5 and 5.

They dont have to be (4,5) or (5,5). we can still satisfy the condition of X+y being 4 digits with out any carry over with (4,6) or (6,4) (3,7) or (7,3) and so on For example 476,653 or 676,453

At this point, we know that the hundreds digit of 5 MUST be at least 5. The hundreds digit of x must be at least 4 since the tens column did bring over a 1.

I would say either x or y has to have hundred's digit greater than 5 but we can't say that Y has hundered's digit greater than 5

Therefore, the answer is III. Only (B)[/i]

only III is true

x = 476
y = 558

since x+y>999, y has to be >500

My bad, I missed the X < Y condition and hence I was ending up with different pairs.