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# Solve 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.

(i) Let the number of marbles John has be $$x$$.
Thus, the number of marbles Jivanti has will be $$45 - x$$.

After losing 5 marbles each, number of marbles left with John will be $$x - 5$$

and with Jivanti will be $$45 - x - 5$$
= $$40 - x$$

Since the product of their final number of marbles is given to be 124,

$$\Rightarrow (x - 5)(40 - x) = 124$$
$$\Rightarrow 40x - x^2 - 200 + 5x = 124$$
$$\Rightarrow -x^2 + 45x - 324 = 0$$
$$\Rightarrow x^2 - 45x + 324 = 0$$

(Splitting -45x as -9x - 36x )
$$\Rightarrow x^2 - 9x - 36x + 324 = 0$$
$$\Rightarrow x(x - 9) - 36(x - 9) = 0$$
$$\Rightarrow (x - 36)(x - 9) = 0$$

The roots of this equation are the values of x for which $$(x - 36)(x - 9) = 0$$

which are,
$$x - 36 = 0$$ or $$x - 9 = 0$$
Thus, $$x = 36$$ or $$x = 9$$

So, if the number of marbles with John is 36, the number of marbles with Jivanti will be 45 - 36 = 9.

If the number of marbles with John is 9, the number of marbles with Jivanti will be 45 - 9 = 36.

(ii) Let the number of toys produced in a day be $$x$$

Thus, the cost of production of each toy will be $$55 - x$$ Rs.

Since the total production cost is given to be 750 Rs,
$$\Rightarrow x(55 - x) = 750$$
$$\Rightarrow 55x - x^2 - 750 = 0$$
$$\Rightarrow x^2 - 55x + 750 = 0$$

(Splitting -55x as -30x - 25x )
$$\Rightarrow x^2 - 30x - 25x + 750 = 0$$
$$\Rightarrow x(x - 30) - 25(x - 30) = 0$$
$$\Rightarrow (x - 25)(x - 30) = 0$$

The roots of this equation are the values of x for which $$(x - 25)(x - 30) = 0$$

which are,
$$x - 25 = 0$$ or $$x - 30 = 0$$
Thus, $$x = 25$$ or $$x = 30$$

So, if the number of toys produced in a day is 25, the cost of production of each toy will be 55 - 25 = 30Rs.

If the number of toys produced in a day is 30, the cost of production of each toy will be 55 - 30 = 25Rs.

Thus, the number of toys produced on that day are either 25 or 30.