If R and S are three-digit positive integers what is the quotient when

If R and S are three-digit positive integers, what is the quotient when the sum of R and S is divided by 1000?

To find the quotient when dividing by 1,000 we need to know the thousands digit of the sum. Notice that since we are adding two three-digit numbers, their sum is either a three-digit number or a four-digit number, so the quotient upon division by 1,000 is either 0 or 1. For example:

If R = 150 and y = 350, then R + S = 500 and 500 divided by 1,000 gives the quotient of 0.
If R = 750 and y = 350, then R + S = 1,100 and 1,100 divided by 1,000 gives the quotient of 1.

(1) The hundreds digit of the sum of R and S is less than the sum of the hundreds digits of R and S. The only way it could happen is if there is carry over 1 to the thousands digit after adding the hundreds digits of R and S. Thus, R + S is a four-digit number, which means that the quotient upon division by 1,000 is 1. Sufficient.

For example:
750
350
___
1,100

As you can see above, the sum of the hundreds digit of the sum (1,100) of R and S (1) is less than the sum of the hundreds digits of R and S (7 + 3 = 10): 1 < 10. Hence the carry over 1 from hundreds to thousands, making the sum a four-digit number.


(2) When R, rounded to the nearest hundreds, is added to S, rounded to the nearest hundreds, the result is 1000. If R = S = 500, then rounded to the nearest hundreds both R and S are still 500, so in this case R + S = 1,000 and the quotient upon division by 1,000 is 1. But if R = S = 499, then rounded to the nearest hundreds both R and S are 500, so in this case R + S = 998 and the quotient upon division by 1,000 is 0. Not sufficient.

Answer: A.

Hope it's clear.