Question

In an auditorium, the sloping floor allows the seats to be arranged to give a clear view of the stage. The seats are arranged in such a way that the number of rows is equal to the number of seats in each row. When the number of rows are doubled and the number of seats in each row is reduced by 16, then the total number of seats increases by 320. Based on the above information, the total number of seats in the original arrangement is :

• A. 400
• B. 800
• C. 1600
• D. 3200

Hint:

Since the original arrangement has a square configuration, the original total number of seats must be a perfect square.
Looking at the options, both \(400\) (which is \(20^2\)) and \(1600\) (which is \(40^2\)) are perfect squares.
Test \(x = 40\):
Original seats = \(1600\).
New arrangement: rows = \(80\), seats/row = \(24\).
New total = \(80 \times 24 = 1920\).
Difference: \(1920 - 1600 = 320\), which matches the problem description. This verification takes less than 15 seconds.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This question is a word problem that can be formulated and solved using Quadratic Equations.
We are given an initial arrangement where the number of rows equals the number of seats per row.
We are also given a secondary arrangement with modified rows and seats, and we need to determine the original total number of seats.

Step 2: Key Formula or Approach:
Let the original number of rows be \(x\).
Since the number of seats in each row equals the number of rows, the seats per row is also \(x\).
The total number of seats in the original arrangement is:
\[ \text{Original Seats} = x^2 \]
In the modified arrangement:
\[ \text{New Rows} = 2x \]
\[ \text{New Seats per Row} = x - 16 \]
The new total number of seats is:
\[ \text{New Seats} = 2x(x - 16) \]

Step 3: Detailed Explanation:
According to the problem, the new number of seats is \(320\) more than the original number of seats:
\[ 2x(x - 16) = x^2 + 320 \]
Expand the left-hand side:
\[ 2x^2 - 32x = x^2 + 320 \]
Move all terms to one side to form a standard quadratic equation:
\[ 2x^2 - x^2 - 32x - 320 = 0 \]
\[ x^2 - 32x - 320 = 0 \]
Now, solve this quadratic equation by splitting the middle term. We look for two numbers whose product is \(-320\) and whose sum is \(-32\).
These numbers are \(-40\) and \(+8\):
\[ x^2 - 40x + 8x - 320 = 0 \]
Factor by grouping:
\[ x(x - 40) + 8(x - 40) = 0 \]
\[ (x - 40)(x + 8) = 0 \]
This gives two possible values for \(x\):
\[ x = 40 \quad \text{or} \quad x = -8 \]
Since the number of rows cannot be negative, we discard \(x = -8\).
Thus, \(x = 40\).
The original total number of seats is:
\[ \text{Original Seats} = x^2 = 40^2 = 1600 \]

Step 4: Final Answer:
The total number of seats in the original arrangement is \(1600\).