Thhe members of a soccer team are to share the $540 cost of a banquet. When 5 students are unable to attend the cost per student was $1.50 greater. How many students attended the banquet?
To solve this problem, we can use algebra. Let's assume that the number of students who initially attended the banquet is "x".
Since the cost per student when all students attend is not given, let's represent it as "y".
According to the given information, when 5 students are unable to attend, the cost per student increases by $1.50. So, the new cost per student becomes "y + $1.50".
Now, let's set up an equation to solve for the number of students who initially attended the banquet:
x * y = (x - 5) * (y + $1.50)
Expanding this equation, we get:
xy = xy + $1.50x - 5y - 7.50
By canceling out the xy terms on both sides, the equation simplifies to:
$1.50x = 5y + $7.50
Now, we need to use the fact that the total cost of the banquet is $540. Since each student pays the cost per student, we have another equation:
x * y = $540
Now, we have a system of two equations:
$1.50x = 5y + $7.50 ...(Equation 1)
xy = $540 ...(Equation 2)
To solve this system of equations, we can substitute Equation 2 into Equation 1:
$1.50x = 5($540/x) + $7.50
Multiplying through by x to eliminate the fraction, we get:
$1.50x^2 = $2700 + $7.50x
Rearranging the equation, we have:
$1.50x^2 - $7.50x - $2700 = 0
Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. Upon solving, we find that x ≈ 25.37 or x ≈ -11.37.
Since the number of students cannot be negative, we can conclude that approximately 25 students initially attended the banquet.