# Represent the following situations mathematically.

Question.

Represent the following situations mathematically.

(i) John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. We would like to find out how many marbles they had to start with.

(ii) A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produced in a day. On a particular day, the total cost of production was Rs.750. We would like to find out the number of toys produced on that day.

Solution:

(i) Let the number of marbles John had be x.

Then the number of marbles Jivanti had

= 45 – x (Why?).

The number of marbles left with John, when he lost 5 marbles = x – 5.

The number of marbles left with Jivanti, when she lost 5 marbles = 45 – x – 5 = 40 – x

Therefore, the is product $=(x-5)(40-x)$

$=40 x-x^{2}-200+5 x=-x^{2}+45 x-200$

So, $-x^{2}+45 x-200=124$

(Given that product = 124)

i.e., $-x^{2}+45 x-324=0$

i.e., $x^{2}-45 x+324=0$

Therefore, the number of marbles John had, satisfies the quadratic equation

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

which is the required representation of the problem mathematically.

(ii) Let the number of toys produced be x.

$\therefore$ Cost of production of each toy $=\operatorname{Rs}(55-x)$

It is given that, total cost of production of the toys = Rs 750

$\therefore \quad x(55-x)=750$

Therefore, $x^{2}-55 x+750=0$

which is the required representation of the problem mathematically.

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