Question

A cottage industry produces a certain number of potteries in a day. It was found that the cost of production of each article was Rs. 5 more than twice the number of potteries produced in a day. If the cost of production on that day Rs. 168. Find the number of potteries produced.

Answer 1

Hint: First, it has to be identified that it is related to some quadratic equation because we want to find the number of potteries produced. So, we will assume the number of potteries to be variable x. Now, the cost of production of each article will be $2x+5$ . So, total production cost will be equal to the number of potteries multiplied with cost of production of each article i.e. given as Rs. 168. So, here the quadratic equation will be formed and we need to find the value of x.

Complete step-by-step answer:

Now, cost of production is given to be Rs. 168. Assuming the the number of potteries to be x. Also, cost production of an article in a day is 5 more than twice the number of potteries, so, in mathematical form the equation formed will be $2x+5$ .
So, total cost production $=x\left( 2x+5 \right)=168$ .
On simplification the above equation we get quadratic equation as,
$\Rightarrow 2{{x}^{2}}+10x=168$ ………………………(1)
 $\Rightarrow 2{{x}^{2}}+10x-168=0$
Here taking 2 common from the equation, we get:
 $\Rightarrow 2\left( {{x}^{2}}+5x-84 \right)=0$
Dividing the equation by 2, we get:
$\Rightarrow {{x}^{2}}+5x-84=0$
So, the above equation needs to be factorized by splitting the middle term such that, factors of 84 should be split in such a way that on adding them it results in a middle term.
So, factors of 84 $=2\times 2\times 3\times 7$
So, when we separate them in two parts i.e. $2\times 2\times 3$ and 7 we get 12 and 7
So, in the equation we get as:
$\Rightarrow {{x}^{2}}+12x-7x-84=0$
Taking x common in first 2 terms and taking $-7$ common from remaining terms, we get
$\Rightarrow x\left( x+12 \right)-7\left( x+12 \right)=0$
$\Rightarrow \left( x-7 \right)\left( x+12 \right)=0$
From here, we have two equations i.e., $\left( x-7 \right)=0$ and $\left( x+12 \right)=0$
So, on equation both equation with 0, we get value of x as: 7, $-12$
But the quantity produced can not be negative, so the number of potteries produced is 7.
Thus, 7 potteries were produced.

Note: Don’t get confused by the words article and pottery. Don’t take is two different variables suppose x and y then, linear equations will be formed, and further simplification will not be done. So, be careful with it. Also, solving quadratic equations using factorisation method should be known properly otherwise factorization will become tedious and time consuming.