Chapter 12.6, Problem 13P

To determine

To find: number of students in the club.

Answer to Problem 13P

The number of students in the club is $18$ .

Explanation of Solution

Given information:

Total cost for the calculator was $\$72$ and if two more students have been added then the cost each student have to contribute was $50$ cents less.

Formula:

The quadratic equation $\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ .

Calculation:

Consider that the student in the club was $x$ .

Total cost for the calculator was $\$72$ .

Then the cost for $1$ student will be $\frac{72}{x}$ .

Since as mentioned above that if two more students have been added then the cost for each student have to contribute was $50$ cents less.

Therefore total students is $\left(x+2\right)$ .

Each have to contribute $50$ cents less.

Therefore $\$\left(\frac{72}{x+2}+\frac{1}{2}\right)$ have to contribute. $\left(\text{here 50 cents less implies half dollar}\right)$ .

Therefore $\frac{72}{x+2}+\frac{1}{2}=\frac{72}{x}$ .

Therefore $\frac{72}{x+2}+\frac{1}{2}=\frac{72}{x}\Rightarrow \frac{72}{x}-\frac{72}{x+2}=\frac{1}{2}\Rightarrow \frac{72\left(x+2\right)-72\left(x\right)}{x\left(x+2\right)}=\frac{1}{2}\Rightarrow 288=x^2+2x\Rightarrow x^2+2x+288=0.$

Recall the quadratic equation $\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ where $a=1,b=2,c=-288$ .

Therefore

  $\begin{array}{l}\frac{-2\pm \sqrt{{\left(2\right)}^2+4\times 1\times 288}}{2}\Rightarrow x=\frac{-2\pm \sqrt{4+1152}}{2}\Rightarrow x=\frac{-2\pm \sqrt{1156}}{2}\\ \Rightarrow x=\frac{-2\pm 34}{2}\Rightarrow x=\frac{32}{2},\frac{-36}{2}\Rightarrow x=16,-18.\end{array}$

Therefore total number of student in the club was $\left(x+2\right)=16+2=18$ .

Here we reject $x=-18$ as the student cannot be in negative.