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.
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.
(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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