Question

the soccer club has planned a trip to a tournament. the cost of the van is $210. when 3 students who are not members of the club join the trip, the transportation cost per person drops by $5.83. how many soccer club members are going to the tournament?
a. 9
b. -12, 9
c. -12
d. 12

Explanation:

Step1: Set up the equation

Let the number of soccer - club members be \(x\). The cost per person initially is \(\frac{210}{x}\). After 3 non - members join, the number of people is \(x + 3\), and the cost per person is \(\frac{210}{x+3}\). We know that \(\frac{210}{x}-\frac{210}{x + 3}=5.83\approx\frac{35}{6}\).
Cross - multiply: \(210(x + 3)-210x=\frac{35}{6}x(x + 3)\).
Simplify the left - hand side: \(210x+630 - 210x=\frac{35}{6}(x^{2}+3x)\).
So, \(630=\frac{35}{6}(x^{2}+3x)\).
Multiply both sides by \(\frac{6}{35}\): \(x^{2}+3x - 108=0\).

Step2: Solve the quadratic equation

For a quadratic equation \(ax^{2}+bx + c = 0\) (here \(a = 1\), \(b = 3\), \(c=-108\)), use the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).
\(x=\frac{-3\pm\sqrt{9+432}}{2}=\frac{-3\pm\sqrt{441}}{2}=\frac{-3\pm21}{2}\).
We get two solutions: \(x_{1}=\frac{-3 + 21}{2}=9\) and \(x_{2}=\frac{-3-21}{2}=-12\).
Since the number of people cannot be negative, we discard \(x=-12\).

Answer:

A. 9