Question

Solve:
(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 1

$\left(i\right)$ Let John have $x$ marbles, so Jayanti will have $(45-x)$ marbles.

After losing $5$ marbles,

John will have $(x-5)$ marbles

Jivanti will have $(45-x-5)=(40-x)$ marbles

As per the given condition,

$(x-5)(40-x)=124$

$\Rightarrow -(x-5)(x-40)=124$

$\Rightarrow -x^2+45x-200=124$

$\Rightarrow x^2-45x+324=0$

$\Rightarrow x^2-36x-9x-324=0$

$\Rightarrow x(x-36)-9(x-36)=0$

$\Rightarrow (x-36)(x-9)=0$

$\Rightarrow x-36=0$ or $x-9=0$

$\Rightarrow x=36$ or $x=9$

So, if John has $36$ marbles, Jivanti will have $9$ marbles.

If John has $9$ marbles, Jivanti will have $36$ marbles.

$\left(ii\right)$ Let the total number of toys produced in a day be $x$.

Cost of production of one toy $=Rs.(55-x)$

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

$\Rightarrow x^2-55x+750=0$

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

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

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

$\Rightarrow x-25=0$ or $x-30=0$

$\Rightarrow x=25$ or $x=30$

So, total number of toys produced in a day are $25$ or $30$.