Question

Solve the following:
(i) 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. We would like to find out how many marbles they have 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 Rs. 750. We would like to find out the number of toys produced on that day.

Answer 1

Hint: From the question, it is clear that we have to transform the conditions into mathematical equations. So, we will assume the asked value as x and then we will try to form the equation according to the situation given. We will realise that it is a quadratic equation that we get. Then we will use Shridhara Charya’s formula, that is, \[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\] to find the roots of the quadratic equation or we can say the possible values of x. Now, the first part, we can take the number of marbles that John has as x and the number of toys manufactured on that day as x for the second part.

Complete step-by-step answer:
(i) In this question, we have been asked to find the number of marbles John and Jivanti had when they started. So, to solve this question, let us consider the number of marble John had when they started as x. So, we can say Jivanti had (45 – x) marbles because they had a total of 45 marbles together.
Now, we have been given that both of them lost 5 marbles. So, we can say that John will be left with x – 5 marbles whereas Jivanti will be left with (45 – x – 5) = 40 – x marbles. Also, we have been given that after they both cost 5 marbles, the product of the number of marbles they now have is 124. So, we can say,
\[\left( x-5 \right)\left( 40-x \right)=124\]
Now, we will simplify it to get the value of x. So, we get,
\[40x-200-{{x}^{2}}+5x=124\]
\[\Rightarrow 45x-{{x}^{2}}-200-124=0\]
\[\Rightarrow {{x}^{2}}-45x+324=0\]
Now, we know that for the quadratic equation, \[a{{x}^{2}}+bx+c=0\] roots are given by \[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}.\]
So, for a = 1, b = – 45 and c = 324, we get,
\[x=\dfrac{-\left( -45 \right)\pm \sqrt{{{\left( -45 \right)}^{2}}-4\left( 1 \right)\left( 324 \right)}}{2\left( 1 \right)}\]
\[\Rightarrow x=\dfrac{45\pm \sqrt{{{2025}^{2}}-1296}}{2}\]
\[\Rightarrow x=\dfrac{45\pm \sqrt{729}}{2}\]
\[\Rightarrow x=\dfrac{45\pm 27}{2}\]
\[\Rightarrow x=\dfrac{45+27}{2};x=\dfrac{45-27}{2}\]
\[\Rightarrow x=\dfrac{72}{2};x=\dfrac{18}{2}\]
\[\Rightarrow x=36;x=9\]
Hence, we get the value of x as 36 and 9. So, we get the value of 45 – x for x = 36 as 9 and for x = 9, we get 45 – x as 36.
Hence, we can say if John has 36 marbles then Jivanti has 9 marbles and if John has 9 marbles, then Jivanti has 36 marbles.
(ii) In this question, we have been asked to find the number of toys manufactured that day. To solve this question, we will consider the number of toys manufactured on that day as x. So, we can say that the cost of each toy will be (55 – x) because we have been given that the cost of each top is 55 minus the number of toys produced that day.
So, we can say that the total cost of toys produced on that day is Rs. 750. So, we can say,
\[x\left( 55-x \right)=750\]
Now, we will simplify it to get the value of x. So, we get,
\[55x-{{x}^{2}}-750=0\]
\[\Rightarrow {{x}^{2}}-55x+750=0\]
And we know that for \[a{{x}^{2}}+bx+c=0,\] roots are given by \[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}.\] So, for a = 1, b = – 55 and c = 750, we get,
\[\Rightarrow x=\dfrac{-\left( -55 \right)\pm \sqrt{{{\left( -55 \right)}^{2}}-4\left( 1 \right)\left( 750 \right)}}{2\left( 1 \right)}\]
\[\Rightarrow x=\dfrac{55\pm \sqrt{3025-3000}}{2}\]
\[\Rightarrow x=\dfrac{55\pm \sqrt{25}}{2}\]
\[\Rightarrow x=\dfrac{55\pm 5}{2}\]
\[\Rightarrow x=\dfrac{55+5}{2};x=\dfrac{55-5}{2}\]
\[\Rightarrow x=\dfrac{60}{2};x=\dfrac{50}{2}\]
\[\Rightarrow x=30;x=25\]
Hence, we can say that the total number of toys produced or that day is either 30 or 25.

Note: While solving this question, the possible mistake one can make is either a mistake in understanding the question or calculation mistake. For first part, they may assume two variables x and y as number of marbles with Jivanti and John. Then they will get stuck as forming equations might get confusing. Also, we can solve this question by solving the quadratic equation by middle term splitting. But that could be lengthy and complicated. So, it is better to use Shridhara Charya’s formula to get the answer.