Correct option is A
Understanding the Problem
Two children, A and B, count the number of chairs around a round table, but they start counting from different chairs.
- A’s 5th chair is B’s 9th chair, meaning B started counting 4 places ahead of A.
- B’s 3rd chair is A’s 12th chair, meaning A started counting 9 places ahead of B.
We need to determine the total number of chairs around the table.
Step 1: Express the Relationship
Since the counting is in a circular pattern, the difference in their counting positions must be consistent modulo N, where N is the total number of chairs.
A’s 5th chair is B’s 9th chair, meaning B is 4 steps ahead of A.
- Mathematically, the difference is 9 - 5 = 4.
B’s 3rd chair is A’s 12th chair, meaning A is 9 steps ahead of B.
- Mathematically, the difference is 12 - 3 = 9.
Since the counting repeats after N chairs, the total number of chairs must be the smallest number that satisfies this pattern.
Step 2: Finding the Smallest Valid N
We now find N, the total number of chairs, by determining the smallest number that is a multiple of both 4 and 9 but allows cyclic repetition.
The simplest way is to check the sum of these shifts:
N = 4 + 9 = 13, which is a valid solution.
Step 3: Verifying the Answer
For N = 13, let’s check if both conditions hold:
A’s 5th chair = B’s 9th chair
- Since B is 4 places ahead of A, after 4 more positions, B reaches A’s 5th chair, which is correct.
B’s 3rd chair = A’s 12th chair
- Since A is 9 places ahead of B, moving 9 places forward from B’s 3rd chair leads to A’s 12th chair, which is also correct.
Since N = 13 satisfies both conditions, this is the correct answer.
