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(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.

Answer :

(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

\(\Rightarrow x^2 - 30x - 25x + 750 = 0\)

\(\Rightarrow x(x - 30) - 25(x - 30) = 0\)

\(\Rightarrow (x - 25)(x - 30) = 0\)

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.

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