NCERT Class X Chapter 4: Quadratic Equation Exercise 4.2 Question 6

Question:

A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was Rs 90, find the number of articles produced and the cost of each article.

Given:

Cost of production of each article = 3 + 2 × (Number of articles produced).

Total cost of production = Rs 90.

To Find:

The number of articles produced and the cost of each article.

Formula:

WKT,
Total Cost of Production = Number of Articles Produced × Cost of Production of Each Article.

WKT, For a quadratic equation ax² + bx + c = 0, we can find two numbers m and n such that:

ax² + bx + c = ax² + mx + nx + c

where

m × n = a × c
m + n = b

Cases:

If (a × c > 0) and b < 0, both m and n are negative.
If (a × c > 0) and b > 0, both m and n are positive.
If (a × c < 0), the numbers m and n have opposite signs.

Solution:

Let the number of articles produced on that day be 'x'.

According to the given condition, the cost of production of each article is 3 more than twice the number of articles produced.

⇒ Cost of each article = (2x + 3) rupees.

The total cost of production on that day was Rs 90.

WKT, Total Cost = Number of articles × Cost per article

⇒ x(2x + 3) = 90

⇒ 2x² + 3x = 90

Rearrange the equation into the standard quadratic form ax² + bx + c = 0:

⇒ 2x² + 3x - 90 = 0

Now, we solve this quadratic equation by factorization.

Here, a = 2, b = 3, c = -90.

Product (a × c) = 2 × (-90) = -180.

Sum (b) = 3.

We need to find two numbers whose product is -180 and sum is 3.

Take the factors of the product without sign.

WKT, 180 = 18 x 10

WKT, 18 = 6 x 3

WKT, 10 = 5 x 2

⇒180 = 6 x 3 x 5 x 2

Rearranging terms in ascending order we have

180 = 2 x 3 x 5 x 6

The possible values of (m,n) are (1, 180), (2, 90), (3, 60), (4, 45), (5, 36), (6, 30), (9, 20), (10, 18), and (12, 15)

Since product is -180 and -180 < 0 both numbers have opposite signs.

Exhaustive case table

(m,n)	Case	m	n	Product (m×n)	Sum (m+n)	(product=-180 and sum=3)?
(1, 180)	Case 1	1	-180	-180	-179	No
(1, 180)	Case 2	-1	180	-180	179	No
(2, 90)	Case 1	2	-90	-180	-88	No
(2, 90)	Case 2	-2	90	-180	88	No
(3, 60)	Case 1	3	-60	-180	-57	No
(3, 60)	Case 2	-3	60	-180	57	No
(4, 45)	Case 1	4	-45	-180	-41	No
(4, 45)	Case 2	-4	45	-180	41	No
(5, 36)	Case 1	5	-36	-180	-31	No
(5, 36)	Case 2	-5	36	-180	31	No
(6, 30)	Case 1	6	-30	-180	-24	No
(6, 30)	Case 2	-6	30	-180	24	No
(9, 20)	Case 1	9	-20	-180	-11	No
(9, 20)	Case 2	-9	20	-180	11	No
(10, 18)	Case 1	10	-18	-180	-8	No
(10, 18)	Case 2	-10	18	-180	8	No
(12, 15)	Case 1	12	-15	-180	-3	No
(12, 15)	Case 2	-12	15	-180	3 → Yes	✅

Therefor the two numbers are -12 and 15

Rewrite the middle term (3x) using these numbers:

⇒ 2x² + 15x - 12x - 90 = 0

Factor by grouping:

⇒ x(2x + 15) - 6(2x + 15) = 0

Factor out the common binomial (2x + 15):

⇒ (x - 6)(2x + 15) = 0

Set each factor equal to zero to find the values of x:

Case 1: x - 6 = 0

⇒ x = 6

Case 2: 2x + 15 = 0

⇒ 2x = -15

⇒ x = -15 2

Since the number of articles produced cannot be negative (< 0) , we take x = 6.

Number of articles produced = 6.

Cost of each article = 2x + 3 = 2(6) + 3 = 12 + 3 = Rs 15.

Result:

The number of articles produced is 6 and the cost of each article is Rs 15.

Next Question Solution:

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.2 Example 7.

Explore more in Quadratic Equations chapter:

Click this link to explore more NCERT Class X Chapter 4 Quadratic Equations solutions

Explore more:

Click this link to explore more NCERT Class X chapter solutions

Search This Blog

Kaliyuga Ekalavya