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If R and S are three-digit positive integers, what is the quotient when the sum of R and S is divided by 1000?
(1) The hundreds digit of the sum of R and S is less than the sum of the hundreds digits of R and S.
(2) When R, rounded to the nearest hundreds, is added to S, rounded to the nearest hundreds, the result is 1000.
  • a)
    Statement (1) ALONE is sufficient, but statement (2) alone is
    not sufficient to answer the question asked.
  • b)
    Statement (2) ALONE is sufficient, but statement (1) alone is
    not sufficient to answer the question asked.
  • c)
    BOTH statements (1) and (2) TOGETHER are sufficient to
    answer the question asked, but NEITHER statement ALONE
    is sufficient to answer the question asked.
  • d)
    EACH statement ALONE is sufficient to answer the question
    asked.
  • e)
    Statements (1) and (2) TOGETHER are NOT sufficient to
    answer the question asked, and additional data specific to the
    problem are needed.
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
If R and S are three-digit positive integers, what is the quotient whe...
Step 1 & 2: Understand Question and Draw Inference
  • Let R = 100a + 10b + c, where a, b and c are digits and a is non-zero (because if a is zero then R would be a 2-digit number)
  • Let S = 100d + 10e + f, where d, e and f are digits and d is non-zero (because if d is zero then S would be a 2-digit number)
 
To find: The quotient when R + S is divided by 1000
  • Now, R + S = 100(a+d) + 10(b+e) + (c+f)
  • There are 2 possibilities – either the sum R + S will also be a 3-digit number (the case of n0 carry-over at the hundreds digit) or the sum R + S will be a 4-digit number (the case of carry–over at the hundreds digit). Let’s consider both these cases one-by-one:
    • Case 1: The case of no carry-over
      • The sum R+S is a 3-digit number which is divided by 1000 (for example, think of 900 or 990 being divided by 1000)
      • In this case, the quotient when R+S is divided by 1000 is zero.
    • Case 2: The case of carry-over
      • ​The sum R+S is a 4-digit number here which is divided by 1000 (for example, think of 1800 or 1980 being divided by 1000)
      • The quotient in this case will be 1
        • If you’re wondering if the quotient can be greater than 1 in this case, the answer is, No it cannot be. The maximum value of any digit is 9. So, the maximum sum of 2 digits can be 18 (8 stays at the place of the digit and 1 gets carried over). So, the maximum carry-over that can happen when 2 digits are added, is 1.
      • So, to answer the question, we need to determine if there is a carry-over at the hundreds digit or not.
Step 3 : Analyze Statement 1 independent
The hundreds digit of the sum of R and S is less than the sum of the hundreds digits of R and S.
There are 4 possible cases for the hundreds digit of R+S:
Case A: There is no carry-over into the hundreds digit and there is no carry-over at the hundreds digit
Example: 1 2 3
+ 2 3 4
In this example, R + S = 357
So, (hundreds digit of R+S) = 3
And (hundreds digit of R) + (hundreds digit of S) = 1 + 2 = 3
In this case, (hundreds digit of R+S) = (hundreds digit of R) + (hundreds digit of S)
Case B: There is no carry-over into the hundreds digit but there is carry-over happening AT the hundreds digit
Example: 9 2 3
+ 1 3 4
In this example, R + S = 1057
So, (hundreds digit of R+S) = 0
But (hundreds digit of R) + (hundreds digit of S) = 9 + 1 = 10
So, here, (hundreds digit of R+S) < (hundreds digit of R) + (hundreds digit of S)
Case C: There is carry-over into the hundreds digit and there is no carry-over AT the hundreds digit
Example: 7 9 4
+ 1 2 2
In this example, R + S = 916
So, (hundreds digit of R + S) = 9
Whereas, (hundreds digit of R) + (hundreds digit of S) = 7 + 1 = 8
In this case, (hundreds digit of R+S) > (hundreds digit of R) + (hundreds digit of S)
Case D: There is carry-over into the hundreds digit and there is carry-over happening AT the hundreds digit
Example 1: 6 9 4
+ 3 2 1
In this example, R + S = 1015
So, (hundreds digit of R + S) = 0
Whereas, (hundreds digit of R) + (hundreds digit of S) = 6 + 3 = 9
Here, (hundreds digit of R+S) < (hundreds digit of R) + (hundreds digit of S)
Example 2: 8 9 4
+ 7 2 1
In this example, R + S = 1615
So, (hundreds digit of R + S) = 6
Whereas, (hundreds digit of R) + (hundreds digit of S) = 15
Here too, (hundreds digit of R+S) < (hundreds digit of R) + (hundreds digit of S)
Therefore, we notice that in all the cases where Statement 1 is satisfied, that is the cases where (hundreds digit of R+S) < (hundreds digit of R) + (hundreds digit of S), there is carry-over is happening at the hundreds digit. Therefore, we can be sure that Case 2 discussed in ‘Steps 1 and 2’ is applicable. So, the quotient when R + S is divided by 1000 is 1.
Since we get a unique answer from this Statement, Statement 1 is sufficient to answer the question.
Step 4 : Analyze Statement 2 independent
When the R, rounded to the nearest hundreds, is added to S, rounded to the nearest hundreds, the result is 1000.
  • R = 100a + 10b + c
    • The result of rounding R to the nearest hundreds is:
      • Either 100a
        • Happens when the digit to the right of a, that is b, is less than 5
      • Or 100(a+1)
        • Happens when b ≥ 5
  • S = 100d + 10e + f
    • The result of rounding S to the nearest hundreds is:
      • Either 100d
        • Happens when e < 5
      • Or 100(d+1)
        • Happens when e ≥ 5
  • As per Statement 2, (Result of rounding R to hundreds) + (Result of rounding S to hundreds) = 1000
    • So, the following cases are possible:
  • Case 1: 100a + 100d = 1000
  • That is, a + d = 10
  • So, in this case, carry-over will happen (in the sum R + S) into the thousands digit.
  • Therefore, the quotient in this case will be 1
 
  • Case 2: 100a + 100(d+1) = 1000
    • That is, a + d + 1 = 10
    • So, a + d = 9
    • In this case, carry-over into the thousands digit may or may not happen in the sum R + S (depending on the value of b + e)
      • For example, in the sum of 712 + 213, carry-over doesn’t happen into the thousands digit.
      • However, in the sum of 782 + 293, carry-over does happen into the thousands digit. This is because the sum of the tens digit (8 + 9) results in a carry-over into the hundreds digit first, therefore, making the sum of the hundreds digit equal to 7 + 2 + 1, that is, 10. The 0 stays at the hundreds place and the 1 gets carried over into the thousands digit.
    • So, the quotient may be 0 or 1.
 
  • Case 3: 100(a+1) + 100d = 1000
    • That is, a + d = 9
    • This is the same equation as the one obtained in Case 2 above.
    • So, here too, we may get either 0 or 1 as the quotient.
 
  • Case 4: 100(a+1) + 100(d+1) = 1000
    • That is, a + d + 2 = 10
    • So, a + d = 8
    • As we’ve discussed above, the maximum carry-over that may happen into the hundreds (or any digit for that matter) digit is 1. Even if a carried-over 1 is added to the sum of a + d, the new sum of a + d will still be 9 only. So, there will be no carry-over into the thousands digit.
    • So, the quotient in this case will be 0
Thus, we’ve seen that from Statement 2, the value of the quotient may be 0 or 1.
So, Statement 2 is not sufficient to get a unique value of the quotient.
Step 5: Analyze Both Statements Together (if needed)
Since we’ve got a unique answer in Step 3, this step is not required
Answer: Option A
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If R and S are three-digit positive integers, what is the quotient when the sum of R and S is divided by 1000?(1) The hundreds digit of the sum of R and S is less than the sum of the hundreds digits of R and S.(2) When R, rounded to the nearest hundreds, is added to S, rounded to the nearest hundreds, the result is 1000.a)Statement (1) ALONE is sufficient, but statement (2) alone isnot sufficient to answer the question asked.b)Statement (2) ALONE is sufficient, but statement (1) alone isnot sufficient to answer the question asked.c)BOTH statements (1) and (2) TOGETHER are sufficient toanswer the question asked, but NEITHER statement ALONEis sufficient to answer the question asked.d)EACH statement ALONE is sufficient to answer the questionasked.e)Statements (1) and (2) TOGETHER are NOT sufficient toanswer the question asked, and additional data specific to theproblem are needed.Correct answer is option 'A'. Can you explain this answer?
Question Description
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Can you explain this answer?, a detailed solution for If R and S are three-digit positive integers, what is the quotient when the sum of R and S is divided by 1000?(1) The hundreds digit of the sum of R and S is less than the sum of the hundreds digits of R and S.(2) When R, rounded to the nearest hundreds, is added to S, rounded to the nearest hundreds, the result is 1000.a)Statement (1) ALONE is sufficient, but statement (2) alone isnot sufficient to answer the question asked.b)Statement (2) ALONE is sufficient, but statement (1) alone isnot sufficient to answer the question asked.c)BOTH statements (1) and (2) TOGETHER are sufficient toanswer the question asked, but NEITHER statement ALONEis sufficient to answer the question asked.d)EACH statement ALONE is sufficient to answer the questionasked.e)Statements (1) and (2) TOGETHER are NOT sufficient toanswer the question asked, and additional data specific to theproblem are needed.Correct answer is option 'A'. Can you explain this answer? has been provided alongside types of If R and S are three-digit positive integers, what is the quotient when the sum of R and S is divided by 1000?(1) The hundreds digit of the sum of R and S is less than the sum of the hundreds digits of R and S.(2) When R, rounded to the nearest hundreds, is added to S, rounded to the nearest hundreds, the result is 1000.a)Statement (1) ALONE is sufficient, but statement (2) alone isnot sufficient to answer the question asked.b)Statement (2) ALONE is sufficient, but statement (1) alone isnot sufficient to answer the question asked.c)BOTH statements (1) and (2) TOGETHER are sufficient toanswer the question asked, but NEITHER statement ALONEis sufficient to answer the question asked.d)EACH statement ALONE is sufficient to answer the questionasked.e)Statements (1) and (2) TOGETHER are NOT sufficient toanswer the question asked, and additional data specific to theproblem are needed.Correct answer is option 'A'. Can you explain this answer? theory, EduRev gives you an ample number of questions to practice If R and S are three-digit positive integers, what is the quotient when the sum of R and S is divided by 1000?(1) The hundreds digit of the sum of R and S is less than the sum of the hundreds digits of R and S.(2) When R, rounded to the nearest hundreds, is added to S, rounded to the nearest hundreds, the result is 1000.a)Statement (1) ALONE is sufficient, but statement (2) alone isnot sufficient to answer the question asked.b)Statement (2) ALONE is sufficient, but statement (1) alone isnot sufficient to answer the question asked.c)BOTH statements (1) and (2) TOGETHER are sufficient toanswer the question asked, but NEITHER statement ALONEis sufficient to answer the question asked.d)EACH statement ALONE is sufficient to answer the questionasked.e)Statements (1) and (2) TOGETHER are NOT sufficient toanswer the question asked, and additional data specific to theproblem are needed.Correct answer is option 'A'. 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