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? ...?
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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 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