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Q2:
Solve the problems given below.
Represent the following situations mathematically:(i) John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, andthe product of the number of marbles they now have is 124. We would like to findout how many marbles they had to start with.(ii) A cottage industry produces a certain number of toys in a day. The cost ofproduction of each toy (in rupees) was found to be 55 minus the number of toysproduced in a day. On a particular day, the total cost of production wasRs.750. We would like to find out the number of toys produced on that day.
Solve the problems given below.
Represent the following situations mathematically:(i) John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, andthe product of the number of marbles they now have is 124. We would like to findout how many marbles they had to start with.(ii) A cottage industry produces a certain number of toys in a day. The cost ofproduction of each toy (in rupees) was found to be 55 minus the number of toysproduced in a day. On a particular day, the total cost of production wasRs.750. We would like to find out the number of toys produced on that day.
Solution :
Given:
(i) Total marbles with John and Jivanti = $45$. After losing $5$ marbles each, the product of the remaining marbles is $124$.
(ii) Let $x$ be the number of toys produced. The cost of production per toy = $55 - x$. Total cost of production = $Rs. 750$.
To Find:
(i) The initial number of marbles John and Jivanti had.
(ii) The number of toys produced on that day.
Part (i): Solving for Marbles
Step 1: Define variables.
Let the number of marbles John had be $x$.
Since the total number of marbles is $45$, the number of marbles Jivanti had is $(45 - x)$.
Step 2: Formulate the equation based on the condition.
After losing $5$ marbles each:
John's remaining marbles = $x - 5$
Jivanti's remaining marbles = $(45 - x) - 5 = 40 - x$
The product of these is $124$:
$(x - 5)(40 - x) = 124$
Step 3: Simplify the quadratic equation.
$40x - x^2 - 200 + 5x = 124$
$-x^2 + 45x - 200 = 124$
$x^2 - 45x + 324 = 0$ [Rearranging terms to standard form $ax^2 + bx + c = 0$]
Step 4: Solve by factorization.
We need two numbers that multiply to $324$ and add to $-45$. These are $-36$ and $-9$.
$x^2 - 36x - 9x + 324 = 0$
$x(x - 36) - 9(x - 36) = 0$
$(x - 36)(x - 9) = 0$
Therefore, $x = 36$ or $x = 9$.
If John had $36$ marbles, Jivanti had $9$. If John had $9$ marbles, Jivanti had $36$.
Final Answer (i): John and Jivanti started with 36 and 9 marbles respectively.
Part (ii): Solving for Toys
Step 1: Define variables.
Let the number of toys produced be $x$.
Cost of production per toy = $(55 - x)$
Step 2: Formulate the equation.
Total cost = (Number of toys) $\times$ (Cost per toy)
$x(55 - x) = 750$
Step 3: Simplify the quadratic equation.
$55x - x^2 = 750$
$x^2 - 55x + 750 = 0$
Step 4: Solve by factorization.
We need two numbers that multiply to $750$ and add to $-55$. These are $-25$ and $-30$.
$x^2 - 30x - 25x + 750 = 0$
$x(x - 30) - 25(x - 30) = 0$
$(x - 30)(x - 25) = 0$
Therefore, $x = 30$ or $x = 25$.
Final Answer (ii): The number of toys produced on that day was either 25 or 30.
More Questions from Class 10 Mathematics Quadratic Equations EXERCISE 4.2
- Q1(i): Find the roots of the following quadratic equations by factorisation: (i) $x^2 β 3x β 10 = 0$
- Q1(ii): Find the roots of the following quadratic equations by factorisation: (ii) $2x^2 + x β 6 = 0$
- Q1(iii): Find the roots of the following quadratic equations by factorisation: (iii) $\sqrt{2}x^2 + 7x + 5\sqrt{2} = 0$
- Q1(iv): Find the roots of the following quadratic equations by factorisation: (iv) $2x^2 β x + \frac{1}{8} = 0$
- Q1(v): Find the roots of the following quadratic equations by factorisation: (v) $100x^2 β 20x + 1 = 0$
- Q3: Find two numbers whose sum is $27$ and product is $182$.
- Q4: Find two consecutive positive integers, sum of whose squares is $365$.
- Q5: The altitude of a right triangle is $7$ cm less than its base. If the hypotenuse is $13$ cm, find the other two sides.
- Q6: 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 βΉ $90$, find the number of articles produced and the cost of each article.
CBSE Solutions for Class 10 Mathematics Quadratic Equations
Chapters in CBSE - Class 10 Mathematics
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