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Mathematics

At an annual function of a school, each student gives gift to every other student. If the number of gifts is 1980, find the number of students.

Quadratic Equations

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Answer

Let the number of students be x

If each student gives gift to every other student so each student gives gift to (x - 1) students

So, x students gives gifts to total = x(x - 1) students.

According to given,

⇒x(x−1)=1980⇒x2−x=1980⇒x2−x−1980=0⇒x2−45x+44x−1980⇒x(x−45)+44(x−45)⇒(x−45)(x+44)⇒x−45=0 or x+44=0x=45 or x=−44\Rightarrow x(x - 1) = 1980 \\[1em] \Rightarrow x^2 - x = 1980 \\[1em] \Rightarrow x^2 - x - 1980 = 0 \\[1em] \Rightarrow x^2 - 45x + 44x - 1980 \\[1em] \Rightarrow x(x - 45) + 44(x - 45) \\[1em] \Rightarrow (x - 45)(x + 44) \\[1em] \Rightarrow x - 45 = 0 \text{ or } x + 44 = 0 \\[1em] x = 45 \text{ or } x = -44

Since number of students cannot be negative hence, x ≠ -44.

Hence, the number of students are 45.

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