Question

A school decided to award prizes to the most punctual and the most obedient student. The sum of two prizes is 150 and their products is 5600. Reprsent the given situation in the form of of quadratic equation . and find the prize money

Answer 1

Let the no.of prizes for obedient students be x

Sum of the 2 prizes is 150

the no.of prizes for punctual students be 150-x

Product of the 2 prizes is 5600

x(150-x)=5600

x^2-150x+5600=0

Quadratic formula

-(-150)+/-root(-150)^2-4*1*5600/2

150+10/2 or 150-10/2

160/2 or 140/2

Roots ={80,70}

Answer 2

The given situation represented in the form of a quadratic equation will be $x^{2} -150x+5600=0$.

Prize money:

If, for most punctual student = 80, then for most obedient student = 70.

If, for most punctual student = 70, then for most obedient student = 80.

Given,

The sum of the 2 prizes = 150,

The product of the 2 prizes = 5600.

To find,

Represent the situation in a quadratic equation form,

The prize money.

Solution,

Firstly, let the award for most punctual student be $x$, and

the award for the most obedient student is $y$.

Here, it is given that the sum of 2 prizes is 150. So,

$x+y =150$     ...(1)

Their product is 5600. So,

$xy=5600$     ...(2)

Now, from eq. (1),

$y = 150 -x$     ...(3)

Substituting this value of $y$ in equation (2), we get,

$x(150-x)=5600$

$\implies 150x-x^{2} =5600$

Rearranging,

$x^{2} -150x+5600=0$     ...(4)

So, the above equation (4) represents the given situation in the form of a quadratic equation.

Solution of the quadratic equation

So the equation formed is,

$x^{2} -150x+5600=0$

It can be solved by the factorization method as follows.

$x^{2} -(80+70)x+5600=0$

$\implies x^{2} -80x-70x+5600=0$

$\implies x(x -80)-70(x-80)=0$

$\implies (x -80)(x-70)=0$

So, $x-80 = 0$ or, $x-70= 0$

$\implies x = 80$ or, $x=70$

Now, from eq. (3), we get,

$y= 70$, when $x=80$, or

$y=80$, when $x=70$

So, the two prizes can be awarded in 2 ways, which are:

When the award for the most punctual student is 80, then the award for most obedient is 70.

And if the award for the most punctual student is 70, then the award for most obedient is 80.

Therefore, the given situation represented in the form of a quadratic equation will be $x^{2} -150x+5600=0$.

Prize money:

If, for most punctual student = 80, then for most obedient student = 70.

If, for most punctual student = 70, then for most obedient student = 80.