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 \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\].
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.