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There are 500 seats in Hyatt conference hall, Mumbai, placed in similar rows. After the reconstruction of the hall, the total number of seats became 10% less. The number of rows was reduced by 5 but each row contained 5 seats more than before. How many rows and how many seats in a row were there initially in the hall? 

There are 500 seats in Hyatt conference hall, Mumbai, placed in similar rows. After the reconstruction of the hall, the total number of seats became 10% less. The number of rows was reduced by 5 but each row contained 5 seats more than before. How many rows and how many seats in a row were there initially in the hall?  Correct Answer 20 rows and 25 seats

Given:

Total number of seats = 500

Decrease in total number of seats after reconstruction = 10%

Formula Used:

Total number of seats = Number of rows × Number of seats in 1 row

Calculations:

Let total number of rows be x and total number of seats in single row be y.

Before reconstruction,

xy = 500      ----(1)

After reconstruction,

Reduced number of rows = (x – 5)

Number of seats in single row = (y + 5)

Total number of seats = 90% of 500 

⇒ Total number of seats = 90/100 × 500 = 450

⇒ (x – 5) × (y + 5) = 450      ----(2)

⇒ xy + 5x – 5y -25 = 450

⇒ xy + 5x – 5y = 475

Substituting (1) in (2)

⇒ 500 + 5 × 500/y – 5y = 475

⇒ -5y + 2500/y = -25

⇒ y – 500/y = 5

⇒ y2 – 5y – 500 = 0

⇒ (y – 25)(y + 20) = 0

⇒ y = -20 or 25

∵ Total number of seats can never be negative

⇒ y = 25

Putting y in (1), we get

x × 25 = 500

⇒ x = 20

∴ The total number of rows and total number of seats in a row that were there initially in the hall is 20 and 25 respectively.

Short Trick/Topper's Approach:

Total number of seats = 500

Total number of seats after reconstruction = 90% of 500

⇒ Total number of seats after reconstruction = 90/100 × 500 = 450

Total number of seats = Number of rows × Number of seats in 1 row

On checking option (1), we get

Total number of rows = 10

Total number of seats in 1 row = 50

⇒ Total number of seats = 10 × 50 = 500

After reconstruction,

Decease in number of rows = 5 

Increase in number of seat in 1 row = 5

⇒ (10 – 5) × (50 + 5) = 5 × 55 = 275

275 ≠ 450

⇒ Option (1) is not correct

On checking option (2), we get

Total number of rows = 20

Total number of seats in 1 row = 20

⇒ Total number of seats = 20 × 20 = 400

400 ≠ 500

⇒ Option (2) is not correct

On checking option (3), we get

Total number of rows = 20

Total number of seats in 1 row = 25

⇒ Total number of seats = 20 × 25 = 500

After reconstruction,

Decease in number of rows = 5 

Increase in number of seat in 1 row = 5

⇒ (20 – 5) × (25 + 5) = 15 × 30 = 450

∵ The number of seats after reconstruction is 450

∴ Option (3) is correct. 

∴ The total number of rows and total number of seats in a row that were there initially in the hall is 20 and 25 respectively.

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