Question

There are rooms numbered from 1 to 99 in an apartment. A mathematician notices that the sum of all room numbers before his room is equal to the sum of all room numbers after his room. Find his room number.

• A. 20
• B. 35
• C. 49
• D. 50

Hint:

This type of problem can be solved quickly with an alternative setup. Let (S_{before} = S_{after} = K). The total sum is (S_N = S_{before} + x + S_{after} = K + x + K = 2K + x). Substituting (K = S_{x-1}), we get (S_N = 2S_{x-1} + x). [ frac{N(N+1)}{2} = 2 frac{(x-1)x}{2} + x = x^2 - x + x = x^2 ] This gives the relation (x^2 = frac{N(N+1)}{2}). For an integer solution for (x), (frac{N(N+1)}{2}) must be a perfect square. For (N=49), (frac{49 times 50}{2} = 1225 = 35^2). This confirms (x=35).

Answer 1

The Correct Option is B

Step 1: Understanding the Question: Let the mathematician's room number be (x). The total number of rooms is (N=99).
The condition is: (Sum of room numbers from 1 to (x-1)) = (Sum of room numbers from (x+1) to 99).

Step 2: Key Formula or Approach:
The sum of the first (n) natural numbers is given by the formula:
[ S_n = frac{n(n+1)}{2} ] Let's analyze the problem statement. The question as stated with (N=99) does not yield an integer solution, which suggests a likely typo in the question, as this is a common problem type that usually has a clean solution. A very similar question (Q-28) in the same paper uses 49 houses. Let's solve the problem assuming (N=49), which matches one of the options. We will point out the issue with (N=99).

Step 3: Detailed Explanation (Assuming N=49):
Let the total number of rooms be (N=49). Let the mathematician's room be (x).
Sum of numbers before room (x): (S_{x-1} = frac{(x-1)x}{2}).
Sum of numbers after room (x): This can be calculated as the total sum minus the sum up to room (x).
Sum after (x) = (S_{49} - S_x).
Total sum (S_{49} = frac{49(49+1)}{2} = frac{49 times 50}{2} = 1225).
Sum up to (x), (S_x = frac{x(x+1)}{2}).
Now, we set the two sums equal:
[ S_{x-1} = S_{49} - S_x ] [ frac{(x-1)x}{2} = 1225 - frac{x(x+1)}{2} ] Multiply the entire equation by 2 to clear the denominators:
[ x(x-1) = 2450 - x(x+1) ] [ x^2 - x = 2450 - x^2 - x ] Add (x^2) and (x) to both sides:
[ 2x^2 = 2450 ] [ x^2 = frac{2450}{2} = 1225 ] [ x = sqrt{1225} = 35 ] The room number is 35. This matches option (B).
Analysis of the original problem (N=99):
If we use (N=99), the equation becomes:
[ S_{x-1} = S_{99} - S_x ] [ S_{99} = frac{99(100)}{2} = 4950 ] [ frac{(x-1)x}{2} = 4950 - frac{x(x+1)}{2} ] [ x^2 - x = 9900 - x^2 - x ] [ 2x^2 = 9900 implies x^2 = 4950 ] (x = sqrt{4950} approx 70.35). This is not an integer, so no such room number exists for (N=99). Given that this is a memory-based paper and option (B) 35 is a valid answer for (N=49), it is highly probable that the number of rooms was intended to be 49.

Step 4: Final Answer: Assuming the number of rooms was 49 (due to a likely typo), the mathematician's room number is 35.