Find the two-digit number — which statements suffice? I. The difference between the number and its digit-reversed form is 27. II. The difference between the two digits is 3. III. The units digit is less than the tens digit by 3.
Verbal Reasoning
Data Sufficiency
Difficulty: Medium
Choose an option
Answer
Correct Answer: None of these
Explanation
Introduction / Context:We must decide which statements allow us to identify a unique two-digit number. Represent the number as 10a + b with digits a (tens) and b (units).
Given Data / Assumptions:
- I: |(10a + b) − (10b + a)| = 27 ⇒ 9 * |a − b| = 27 ⇒ |a − b| = 3.
- II: |a − b| = 3.
- III: b = a − 3 (units less than tens by 3).
Concept / Approach:Combine conditions and see if a unique (a, b) pair emerges. With |a − b| = 3 and 1 ≤ a ≤ 9, 0 ≤ b ≤ 9, many numbers satisfy the constraints.
Step-by-Step Solution:
From I: |a − b| = 3.From II: |a − b| = 3 (duplicate of I).From III: b = a − 3, which is one orientation of the absolute difference.Numbers satisfying III include 41, 52, 63, 74, 85, 96 (all valid two-digit numbers).Even with all three, multiple solutions exist; hence no unique number can be determined.Verification / Alternative check:Test any candidate from the set; each meets I, II, III and yields the same 27 difference from its reverse. Non-uniqueness persists.
Why Other Options Are Wrong:
- Only I and II: Merely restate |a − b| = 3 ⇒ many solutions.
- Only I and III: Still produces a family of solutions with a − b = 3.
- Only I and either II or III: Same issue; no uniqueness.
- All I, II and III: Still not unique.
Common Pitfalls:Assuming the difference with reverse fixes the number; it fixes only the digit gap.
Final Answer:None of these