Two Statements S1 and S2 are given below with regard to two numbers followed by a Question : S1 : Their product is 21. S2 : Their sum is 10. Question : What are the two numbers? Which one of the following is correct in respect of the above Statements and the Question?
- (a) S1 alone is sufficient to answer the Question.
- (b) S2 alone is sufficient to answer the Question.
- (c) S1 and S2 together are sufficient to answer the Question, but neither S1 alone nor S2 alone is sufficient to answer the Question. ✓ UPSC's answer
- (d) S1 and S2 together are not sufficient to answer the Question.
Why the answer is (c)
• Let the two numbers be x and y. Statement S1 states that xy = 21, which has multiple integer solutions such as (1, 21), (3, 7), and their negatives, so it is not sufficient alone.
• Statement S2 states that x + y = 10, which has multiple integer solutions such as (1, 9), (2, 8), (3, 7), and (4, 6), so it is not sufficient alone.
• Combining S1 and S2, we look for a pair of numbers that multiply to 21 and add up to 10.
• The only pair of positive integers satisfying both conditions is 3 and 7, since 3 × 7 = 21 and 3 + 7 = 10.
• Since the unique solution is obtained only when both statements are used together, option (c) is correct.
Why the other options are wrong
- (a) S1 alone is sufficient to answer the Question.
- S1 alone allows multiple pairs like (1, 21) and (3, 7), so it does not uniquely determine the numbers.
- (b) S2 alone is sufficient to answer the Question.
- S2 alone allows multiple pairs like (1, 9) and (4, 6), so it does not uniquely determine the numbers.
- (d) S1 and S2 together are not sufficient to answer the Question.
- S1 and S2 together uniquely identify the numbers as 3 and 7, making them sufficient.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2020, held on 4 October 2020. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.