Question

A third-grade class is told to send a valentine to every single one of their classmates. On Valentine’s Day, the class ends up with 306 valentines. How many students are in the class?

Answer 1

306 is the total cards sent by all the students in the class. Lets say x is the number of students in class, therefore each student needs to send x-1 cards (since they do not send themselves a card). So depending on your level of math, you could either take the complex way to solve it using quadratic equations, or just find the number that if multiplied by itself then subtracted, gives you 306.

`(x * x) - x = 306 \\ lets \: try \: 18 \\ (18 * 18) - 18 = 306 \\ so \: x = 18`
===============
Complex way

`(x * x) - x = 306 \\ {x}^(2) - x - 306 = 0`
So a = 1, b = -1, c=-306
using the quadratic equation:

`x = \frac{ - b + \sqrt{ {b}^(2) - 4ac } }{2a}`

`x = (1 + √(1 - (4 * 1 * - 306)) )/(2) \\ x = (1 + √(1 - ( - 1224)) )/(2) \\ x = (1 + √(1225) )/(2) \\ x = (1 + 35)/(2) \\ x = (36)/(2) \\ x = 18`

Answer 2

There are 18 students in the class.

Explanation:

Let's say there are
`x` students in the class.

Then, one student will send
`x-1` valentines.

Therefore,

Total number of valentines sent = Number of valentines sent by 1 student*Total number of students in the class.

⇒
`306=(x-1)*x`

=
`x^(2) -x=306`

=
`x^(2) -x-306=0`

=
`x^(2) -18x+17x-306=0`

=
`x(x-18)+17(x-18)=0`

=
`(x-18)(x+17)=0`

Therefore either,

`(x-18)=0` or
`(x+17)=0`

`x=18` or
`x=-17`

But the number of students cannot be a negative value, So we choose
`x=18`

Therefore there are 18 students in the class.