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x of Lixin's classmates plan to buy her a birthday gift. The cost of the birthday gift is $36 (i) Given that 5 of Lixin's classmate decided not to contribute to the birthday gift, the rest of her classmates will have to pay an additional $0.24 to make up for the difference. Write down an expression, in terms of x, for the amount which each of the remaining classmates will now have to pay. (ii) Form an equation in x and show that it reduces to x square- 5x - 750 = 0. (i) Find the number of classmates Lixin has.

Sagot :

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

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