Question

For Christmas, each member of a class

Answer 1

Answer:

number of pupils in the class = 32

Step-by-step explanation:

Let n be the number of students. So n-1 cards will be sent by each student.

n(n-1) =192

n² - n =192

n² - n - 192 = 0

n² - 32n + 31n - (31*32) = 0

n(n - 32) + 31 (n-32) = 0

(n-32)(n+31) = 0

n - 32 = 0    or n + 31 = 0

n = 32  or n = -31 is not possible because no. of students cannot be negative

n= 32

Answer 2

Answer:

There are 32 pupils in the class

Step-by-step explanation:

Let's say there are N pupils in the class. Then each pupil must send N-1 cards - because it would make no sense to send one to themselves! So each of the N pupils send N-1 cards, which becomes 992 cards in total. In equation form, this is

$N(N-1)=992\\N^2-N-992=0$

This is a second degree polynomial, which has the solutions

$N=\frac{-b\pm \sqrt{b^2-4\cdot a \cdot c}}{2a}$

where $a=1, b=-1, \text{and }c=-992$

If we insert these numbers in the equation,

$N=\frac{-(-1)\pm \sqrt{1^2-4*1*(-992)}}{2*1}\\ = \frac{1\pm \sqrt{1+4*992}}{2}\\= \frac{1 \pm 63}{2}$

If we choose the solution with the minus sign, we get

N=-31

but this makes no sense! There can't be a negative number of pupils in the class!

So we choose the solution with the plus sign,

$N=\frac{1+63}{2}\\ =\frac{64}{2}\\ =32$

So there are 32 pupils in the class