Question

Represent the following situations mathematically.
(1) 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. Find the number of marbles they had initially.
(2) A cottage industry produces a certain number of toys in a day. The cost of production of each toy was found to be 55 minus the number of toys produced that day. On a particular day, the total cost of production was 750. Find the number of toys produced on that day.

Answer 1

We will use the concepts of variables and linear equations in two variables. We will look at some solving techniques of these equations.
A variable is an algebraic term, a value which changes its value according to situations and conditions.

Complete step by step answer:
(1) Let the number of marbles with John be $x$.
Let the number of marbles with Jivanti be $y$.
The total sum of marbles they have together is 45.
$\Rightarrow x + y = 45$ ------(1)
After losing 5 marbles each, the remaining marbles count will be, $x - 5$ for John and $y - 5$ for Jivanti.
And the product of marbles they now have is 124.
So, we can write it as,
$(x - 5)(y - 5) = 124$ ------(2)
On multiplication, we get, $xy - 5x - 5y + 25 = 124$
$\Rightarrow xy - 5(x + y) = 99$
From equation (1), we know, the value of $x + y$
$\Rightarrow xy - 5(45) = 99$
$\Rightarrow xy = 99 + 225 = 324$
From equation (1), we know, $y = 45 - x$
$\Rightarrow x(45 - x) = 324$
$\Rightarrow {x^2} - 45x + 324 = 0$
To solve this problem, we will use a quadratic formula. For an equation $a{x^2} + bx + c = 0$, the roots are $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$.
So, we get the roots as, $x = \dfrac{{ - ( - 45) \pm \sqrt {2025 - 4(1)(324)} }}{2}$
$\Rightarrow x = \dfrac{{45 \pm \sqrt {729} }}{2} = \dfrac{{45 \pm 27}}{2}$
So, we get $x = 36{\text{ or 9}}$
From equation (1), $y = 45 - x$
$\Rightarrow y = 45 - 36 = 9$
Or
$\Rightarrow y = 45 - 9 = 36$
So, the number of marbles with John and Jivanti are 36 and 9 respectively.
OR
The number of marbles with John and Jivanti are 9 and 36 respectively.
(2)
Let the number of toys produced in a day be $x$.
Cost of each toy is $x - 55$.
Total cost of production on a day is the product of the number of toys produced and cost of each toy.
On a particular day, it was 750.
So, we can write it as,
$\Rightarrow x(x - 55) = 750$
$\Rightarrow {x^2} - 55x - 750 = 0$
We got a quadratic equation and we will solve this by quadratic formula.
$\Rightarrow x = \dfrac{{ - ( - 55) \pm \sqrt {3025 - 4(1)( - 750)} }}{2}$
$\Rightarrow x = \dfrac{{55 \pm 77.62}}{2}$
So, the roots that we will get are, $x = 66.31{\text{ or }} - 11.31$
As total number of toys can not be in decimal, we can conclude that $x = 66$
So, the total number of toys produced on that day is 66.

Note: Make a note that the count of things or cost of an object or length, breadth, height, area and volume of a 2-dimensional or 3-Dimensional object are always positive. You won’t get a negative value for these units of measurements.
The term ${b^2} - 4ac$ is called determinant. So, if the determinant is less than zero, then the roots are imaginary.