Answer :
Lixin has x number of classmates and plan to buy her a birthday gift worth $36.
So basically, they need to contribute 36/x dollars each in order to buy the gift.
But 5 of them will not contribute and it cost them to add $0.24 each
So from 36/x dollars, it become (36/x + 0.24) dollars each
Multiply this individual contribution by the number of her classmates which was reduced by 5. (x - 5)
The equation will be :
[tex]\begin{gathered} \text{contribution}\times\text{ number of classmates}=36 \\ (\frac{36}{x}+0.24)(x-5)=36 \end{gathered}[/tex]Simplify the equation :
[tex]\begin{gathered} (\frac{36}{x}+0.24)(x-5)=36 \\ \frac{36}{x}(x-5)+0.24(x-5)=36 \\ (36-\frac{180}{x})+(0.24x-1.2)=36 \end{gathered}[/tex]Multiply both sides by x to remove the denominator.
[tex]\begin{gathered} x(36-\frac{180}{x})+x(0.24x-1.2)=36x \\ 36x-180+0.24x^2-1.2x=36x \end{gathered}[/tex]Rearrange and combine like terms :
[tex]\begin{gathered} 0.24x^2+36x-1.2x-36x-180=0 \\ 0.24x^2-1.2x-180=0 \end{gathered}[/tex]Divide both sides by 0.24 to make the coefficient of x^2 become 1 :
[tex]\begin{gathered} \frac{0.24x^2}{0.24}-\frac{1.2x}{0.24}-\frac{180}{0.24}=0 \\ x^2-5x-750=0 \end{gathered}[/tex]Solve by factoring completely :
[tex]\begin{gathered} x^2-5x-750=0 \\ (x-30)(x+25)=0 \end{gathered}[/tex]Equate both factors to 0, then solve for x :
x - 30 = 0
x = 30
x + 25 = 0
x = -25
Neglect negative value since there is no negative value for the number of persons.
The answer is x = 30
To summarize :
i Expression in terms of x :
[tex](\frac{36}{x}+0.24)(x-5)=36[/tex]ii Equation in x that reduces to x^2 - 5x - 750 = 0
[tex]\begin{gathered} 0.24x^2-1.2x-180=0 \\ \Rightarrow x^2-5x-750=0 \end{gathered}[/tex]iii Number of classmates Lixin has.
x = 30
