Mathematics: At an airport,there are a total of 1,160 seats in the waiting areas. there are 8 separate, same size, waiting areas. how many seats are in each waiting area

At an airport,there are a total of 1,160 seats

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roseemariehunter12

10.12.2022

Mathematics

$\dfrac{y+1}{y+2}+\dfrac{y+8}{y+9}=\dfrac{y+2}{y+3}+\dfrac{y+7}{y+8}$

Write all fractions in terms of a common denominator:

$\dfrac{y+1}{y+2}=\dfrac{(y+1)(y+9)(y+3)(y+8)}{(y+2)(y+9)(y+3)(y+8)}$

$\dfrac{y+8}{y+9}=\dfrac{(y+8)^2(y+2)(y+3)}{(y+2)(y+9)(y+3)(y+8)}$

$\dfrac{y+2}{y+3}=\dfrac{(y+2)^2(y+9)(y+8)}{(y+2)(y+9)(y+3)(y+8)}$

$\dfrac{y+7}{y+8}=\dfrac{(y+7)(y+2)(y+9)(y+3)}{(y+2)(y+9)(y+3)(y+8)}$

Then move all fractions to one side and simplify the numerator:

$\dfrac{(y+1)(y+9)(y+3)(y+8)+(y+8)^2(y+2)(y+3)-(y+2)^2(y+9)(y+8)-(y+7)(y+2)(y+9)(y+3)}{(y+2)(y+9)(y+3)(y+8)}=0$

The numerator dictates when the fraction reduces to 0. The denominator can never be 0, so we know that y cannot take any of the values -2, -9, -3, nor -8.

So the equation reduces to

(y+1)(y+9)(y+3)(y+8)+(y+8)^2(y+2)(y+3)-(y+2)^2(y+9)(y+8)-(y+7)(y+2)(y+9)(y+3)=0

Expand the left side; you would end up with

-6(2y+11)=0

2y+11=0

2y=-11

$\implies\boxed{y=-\dfrac{11}2}$

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At an airport,there are a total of 1,160 seats in the waiting areas. there are 8 separate, same size, waiting areas. how many seats are in each waiting area

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