Question 6

We will form a quadratic equation from the given information and solve it.

Step 1 — Set up the equation

Let's define the number of articles.

Let the number of articles be $x$.

Now, let's define the cost of each article.

The cost of each article is $2x + 3$ rupees.

The total cost is the number of articles multiplied by the cost of each.

The total cost is given as ₹90.

So, we can write the equation.

$x(2x + 3) = 90$

Let's expand this equation.

$2x^2 + 3x = 90$

We need to bring all terms to one side.

$2x^2 + 3x - 90 = 0$

$\boxed{2x^2 + 3x - 90 = 0}$

Step 2 — Solve for the number of articles

We have a quadratic equation.

Let's solve it by splitting the middle term.

We need two numbers that multiply to $2 \times (-90) = -180$.

These numbers must also add to 3.

The numbers are 15 and -12.

So, we can rewrite the middle term.

$2x^2 + 15x - 12x - 90 = 0$

Now, let's factor by grouping.

$x(2x + 15) - 6(2x + 15) = 0$

We can factor out the common term $(2x + 15)$.

$(2x + 15)(x - 6) = 0$

For the product to be zero, one of the factors must be zero.

So, either $2x + 15 = 0$ or $x - 6 = 0$.

$2x + 15 = 0 \implies 2x = -15 \implies x = -\frac{15}{2}$

$x - 6 = 0 \implies x = 6$

The number of articles cannot be negative.

So, we reject $x = -\frac{15}{2}$.

Therefore, the number of articles is 6.

$\boxed{\text{Number of articles} = 6}$

Step 3 — Calculate the cost of each article

We found the number of articles, $x = 6$.

The cost of each article is $2x + 3$.

Let's substitute the value of $x$.

$\text{Cost} = 2(6) + 3$

$= 12 + 3$

$= 15$

So, the cost of each article is ₹15.

$\boxed{\text{Cost of each article} = ₹15}$

Answer

(i) The number of articles produced is 6. (ii) The cost of each article is ₹15.