Question

$\large\underbrace{\bf{ \green{Question}}}$
Rahul and Rohan have 45 marbles together. After losing 5 marbles each, the product of the number of marbles they both have now is 124. How to find out how many marbles they had to start with.
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Hint:- Use quadratic formula.
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Explanation needed!!​

Answer 1

Answer:

Let Rahul has x marbles and Rohan has (45-x) marbles. [ Sum is 45 ]

When they both lost five marbles,

Rahul has (x-5) marbles and Rohan has (45-x-5)

=(40-x) marbles.

Now the product of number of marbles they have now is 145

$\implies \: (x - 5)(40 - x) = 124 \\ \\ \implies40x - {x}^{2} - 200 + 5x = 124 \\ \\ \implies \: 45x - {x}^{2} - 200 = 124 \\ \\ \implies \: {x}^{2} - 45x + 200 + 124 = 0$

$\implies \: {x}^{2} - 36x - 9x + 324 = 0 \\ \\ \implies \: x(x - 36) - 9(x - 36) = 0 \\ \\ \implies \: (x - 36)(x - 9) = 0 \\ \\ \implies \: x - 36 = 0 \: or \: x - 9 = 0$

$\implies \: x = </strong><strong>36</strong><strong> \: or \: x = 9$

Now,

45-x=9

x=45-9=36

or

45-x=36

x=45-36=9

So two cases are possible here.

Case I: Rahul had 9 marbles and Rohan had 36 marbles

Case II: Rahul has 36 marbles and Rohan has 9 marbles

$\rule{200pt}{2.5pt}$

Answer 2

$\large\underline{\sf{Solution-}}$

Given that,

• Rahul and Rohan have 45 marbles together.

Let assume that

• Rahul have x marbles

So,

• Rohan have 45 - x marbles.

According to statement, After losing 5 marbles each, the product of the number of marbles they both have now is 124.

After losing 5 marbles,

• Rahul have x - 5 marbles

• Rahul have 45 - x - 5 = 40 - x marbles

Now,

$\rm \: (x - 5)(40 - x) = 124$

$\rm \: 40x - {x}^{2} - 200 + 5x= 124$

$\rm \: 45x - {x}^{2} - 200= 124$

$\rm \: {x}^{2} - 45x + 324 = 0$

$\rm \: {x}^{2} - 9x - 36x + 324 = 0$

$\rm \: x(x - 9) - 36(x - 9) = 0$

$\rm \: (x - 9)(x - 36) = 0$

$\rm\implies \:x = 9 \: \: or \: \: x = 36$

So, it means either

Rahul have 9 marbles

and

Rohan have 36 marbles

OR

Rahul have 36 marbles

and

Rohan have 9 marbles

$\rule{190pt}{2pt}$

Additional Information :-

Nature of roots :-

Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.

• If Discriminant, D > 0, then roots of the equation are real and unequal.

• If Discriminant, D = 0, then roots of the equation are real and equal.

• If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.

Where,

• Discriminant, D = b² - 4ac