Question

An audience of 450 people is seated in an auditorium. Each row contains the same number of seats and each seat in the auditorium is occupied. With three fewer seats per row, and five extra rows, the same audience could still be seated, occupying all seats. How many rows does the auditorium have? ...?

Answer 1

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SR = 450 
S = 450/R 

(S - 3)(R + 5) = 450 
(450/R - 3)(R + 5) = 450 
(450 - 3R)(R + 5) = 450R 
450R + 2250 - 3R^2 - 15R = 450R 
3R^2 + 15R - 2250 = 0 
R^2 + 5R - 750 = 0 
R^2 + 30R - 25R - 750 = 0 
R(R + 30) - 25(R + 30) = 0 
(R + 30)(R - 25) = 0 
R ∈ {-30,25} 

Only positive numbers make sense in this context, therefore R = 25.

Answer 2

Answer with Step-by-step explanation:

Let there be r rows and s seats in every row.

An audience of 450 people is seated in an auditorium and each seat is occupied.

i.e. rs=450

or s=450/r

With three fewer seats per row, and five extra rows, the same audience could still be seated, occupying all seats.

i.e. (s-3)(r+5)=450

s(r+5)-3(r+5)=450

rs+5s-3r-15=450

450+5s-3r-15=450

Subtracting both sides by 450,we get

5s-3r-15=0

i.e. 5s-3r=15

5(450/r)-3r=15

Dividing both sides by 3 and multiplying by r, we get

750-r²=5r

r²+5r-750=0

r² + 30r - 25r - 750 = 0  

r(r + 30) - 25(r + 30) = 0  

(r + 30)(r - 25) = 0

either r+30=0 or r-25=0

either r= -30 or r=25

r can't be negative

Hence, number of rows in auditorium are:

25