# Ex.4.2 Q2 Quadratic Equations Solutions - NCERT Maths Class 10

## Question

Solve the problems given in example \(1.\)

(i) John and Jivanti had \(45\) marbles. Both of them lost \(5\) marbles each and the product of the no. 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 ₹\(750.\) We would like to find out the number of toys produced on that day.

## Text Solution

**(i)**

**What is known?**

(i) John and Jivanti together had \(45\) marbles

(ii) Both of them lost \(5\) marbles each.

(iii) Product of number of marbles they now have is \(124.\)

**What is Unknown?**

Number of marbles John and Jivanti started with (each).

**Reasoning:**

Let the number of marbles that John had be \(x.\)

The number of marbles Jivanti had will be (Total marbles MINUS the Number of marbles John had) \(= 45-x\)

(i) Both of them lost \(5\) marbles each:

John \(= x-5\)

Jivanti \(=45 - x - 5 = 40 - x\)

(ii) Product of current number of marbles \(=124\)

\[\left( {x - 5} \right)\left( {40 - x} \right) = 124\]

**Steps:**

\[\begin{align}\left( {x - 5} \right)\left( {40 - x} \right) &= 124\\40x - {x^2} - 200 + 5x &= 124\\

- {x^2} + 45x - 200 - 124 &= 0\\ - {x^2} + 45x - 324 &= 0\\{x^2} - 45x + 324 &= 0\\{x^2} - 36x - 9x + 324 &= 0\\x\left( {x - 36} \right) - 9\left( {x - 36} \right) &= 0\\\left( {x - 36} \right)\left( {x - 9} \right) &= 0\\x - 36 = 0 &\qquad x - 9 = 0\\x = 36 &\qquad x = 9\end{align}\]

John and Jivanti started with \(36\) and \(9\) marbles.

**(ii)**

**What is known?**

(i) Cost of each day is \(55\, –\) (number of toys produced in a day) rupees.

(ii) On a particular day the total cost of production was \(\rm{Rs.}\,750.\)

**What is Unknown?**

Number of toys produced on that day.

**Reasoning:**

Let the number of toys produced in a day is \(x.\)

(i) Cost of each toy \(= (55 - x)\) rupees

(ii) Total cost of production \(=\) Cost of each toy \(\times\) Total number of toys

\[\begin{align}(55 - x)(x) &= 750\end{align}\]

**Steps:**

\[\begin{align}(55 - x)(x) &= 750\\55x - x &= 750\\55x - x - 750 &= 0\\

x - 55x + 750 &= 0\\x - 25x-30x + 750 &= 0\\x(x - 25)-30\left( {x - 25} \right) &= 0\\

(x - 25)(x - 30) &= 0\\x-25 = 0 &\qquad x-30 = 0\\x = 25 &\qquad x = 30\end{align}\]

Number of toys produced on that day is \(25\) or \(30.\)