Q2:

Solve the problems given below.

Represent the following situations mathematically:(i) John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, andthe product of the number of marbles they now have is 124. We would like to findout how many marbles they had to start with.(ii) A cottage industry produces a certain number of toys in a day. The cost ofproduction of each toy (in rupees) was found to be 55 minus the number of toysproduced in a day. On a particular day, the total cost of production wasRs.750. We would like to find out the number of toys produced on that day.

Given:

(i) Total marbles with John and Jivanti = $45$. After losing $5$ marbles each, the product of the remaining marbles is $124$.

(ii) Let $x$ be the number of toys produced. The cost of production per toy = $55 - x$. Total cost of production = $Rs. 750$.

To Find:

(i) The initial number of marbles John and Jivanti had.

(ii) The number of toys produced on that day.

Part (i): Solving for Marbles

Step 1: Define variables.

Let the number of marbles John had be $x$.

Since the total number of marbles is $45$, the number of marbles Jivanti had is $(45 - x)$.

Step 2: Formulate the equation based on the condition.

After losing $5$ marbles each:

John's remaining marbles = $x - 5$

Jivanti's remaining marbles = $(45 - x) - 5 = 40 - x$

The product of these is $124$:

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

Step 3: Simplify the quadratic equation.

$40x - x^2 - 200 + 5x = 124$

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

$x^2 - 45x + 324 = 0$ [Rearranging terms to standard form $ax^2 + bx + c = 0$]

Step 4: Solve by factorization.

We need two numbers that multiply to $324$ and add to $-45$. These are $-36$ and $-9$.

$x^2 - 36x - 9x + 324 = 0$

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

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

Therefore, $x = 36$ or $x = 9$.

If John had $36$ marbles, Jivanti had $9$. If John had $9$ marbles, Jivanti had $36$.

Final Answer (i): John and Jivanti started with 36 and 9 marbles respectively.

Part (ii): Solving for Toys

Step 1: Define variables.

Let the number of toys produced be $x$.

Cost of production per toy = $(55 - x)$

Step 2: Formulate the equation.

Total cost = (Number of toys) $\times$ (Cost per toy)

$x(55 - x) = 750$

Step 3: Simplify the quadratic equation.

$55x - x^2 = 750$

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

Step 4: Solve by factorization.

We need two numbers that multiply to $750$ and add to $-55$. These are $-25$ and $-30$.

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

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

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

Therefore, $x = 30$ or $x = 25$.

Final Answer (ii): The number of toys produced on that day was either 25 or 30.