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

Hint: 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 $$x={\displaystyle \frac{-b\pm \sqrt{b^2-4ac}}{2a}}$$.
So, we get the roots as, $$x={\displaystyle \frac{-(-45)\pm \sqrt{2025-4(1)(324)}}{2}}$$
$$\Rightarrow x={\displaystyle \frac{45\pm \sqrt{729}}{2}}={\displaystyle \frac{45\pm 27}{2}}$$
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.
$$\Rightarrow x={\displaystyle \frac{-(-55)\pm \sqrt{3025-4(1)(-750)}}{2}}$$
$$\Rightarrow x={\displaystyle \frac{55\pm 77.62}{2}}$$
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.