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: Work backward from the options instead of solving the quadratic fresh.
Since the seats are arranged in a square pattern first, rows equal to seats per row, the original total must be a perfect square. Among the options, $1600 = 40^2$ stands out as a clean perfect square, so let us test it.
Step 2: Set up the trial with x = 40 rows.
If there were originally $x = 40$ rows and 40 seats per row, the new arrangement has $2x = 80$ rows and $x - 16 = 24$ seats per row.
Step 3: Check the new total against the condition.
New total seats $= 80 \times 24 = 1920$. Original total $= 40 \times 40 = 1600$. The increase is \[ 1920 - 1600 = 320 \] which matches exactly what the question states.
Step 4: Confirm the answer.
Since the trial with 1600 satisfies every condition in the problem, this must be the original number of seats.
\[ \boxed{1600} \]