Question

In an electric appliances shop there were some fans and coolers containing 3 blades and 4 blades each respectively along with a motor each. The cost of a fan is Rs. 1200 and that of cooler is Rs. 3000 . One day due to short circuit the shop caught the fire. After extinguishing the fire, shopkeeper has estimated that 90 sets of motor bodies and 320 blades were burned and then nothing was left useful in his shop. What is the loss incurred from the coolers alone?(a) Rs. \(1,50,000\) (b) Rs. 12,000 (c) Rs, 4800 (d) can't he determined

Short answer

Answer: (a) Rs. 150,000.

Step-by-step solution

Define the variables and equations

Let x be the number of fans and y be the number of coolers. The total number of motors will be x + y = 90, and the total number of blades will be 3x + 4y = 320. We can now write the equations as follows: 1) x + y = 90 2) 3x + 4y = 320

Solve the system of linear equations

We can solve the system of linear equations using various methods like substitution or elimination. Here, we'll use the substitution method. We can rewrite the first equation as x = 90 - y, and substitute this expression into the second equation: 3(90 - y) + 4y = 320 Now, solve for y: 270 - 3y + 4y = 320 y = 320 - 270 = 50 Now, plug the value of y back into the expression for x: x = 90 - 50 = 40 So, there were 40 fans and 50 coolers in the shop before the fire.

Calculate the loss incurred from the coolers

The cost of a cooler is Rs. 3000. Since all the coolers were burned, the total loss incurred from the coolers is: Loss = 50 (number of coolers) * Rs. 3000 (cost of a cooler) Loss = Rs. 150,000 Hence, the loss incurred from the coolers alone is (a) Rs. 150,000.

Key concepts

Substitution Method

The substitution method is a fundamental technique in solving a system of linear equations. This method involves isolating one variable in one equation and then substituting the resulting expression into the other equation. This process helps eliminate one variable, allowing us to solve for the other.

For instance, in our electric appliances shop problem, we start by expressing one variable in terms of the other using the first equation, which is simpler. By assuming that the total number of motors represents the sum of fans and coolers, we can say that if there are 'x' fans, there must be '90 - x' coolers, leading us to substitute '90 - x' for 'y' in the second equation. This single-variable equation can then be solved to find the value of 'x'. Once we know 'x', we can substitute it back into one of the original equations to find 'y'. This method is particularly useful for its simplicity and is a go-to approach in linear algebra, especially for systems that easily lend themselves to such manipulations.

Variable Definition

Defining variables in a clear and consistent manner is vital in tackling a system of linear equations. A variable is a symbol, commonly a letter, that represents a number or value that can vary. In the context of our problem regarding the electric appliances shop, we defined 'x' as the number of fans and 'y' as the number of coolers.

Clear variable identification allows us to translate the textual information presented in word problems into algebraic expressions and equations. By choosing 'x' and 'y' to represent the quantities in question, we created a way to capture the relationship between the two types of products—fans and coolers—and their associated costs. This step is a crucial foundation and aids in setting up equations that represent the conditions of the problem, such as the total number of motors and blades. Proper variable definition is an essential skill in linear algebra and quantitative aptitude as it sets the stage for the problem-solving process.

Linear Algebra

Linear algebra is an area of mathematics that deals with linear equations, linear functions, and their representations through matrices and vector spaces. It's not just about finding unknowns but understanding the relationship between them. Our example of the electric appliances shop requires solving a system of two linear equations to determine the quantity of two types of goods based on the number of blades and motors.

In the field of linear algebra, we often deal with several variables and constraints, and we use different methods like substitution, elimination, and matrix operations to solve these systems. Knowing how to handle linear equations is essential for students since these concepts apply not only in academic settings but also in various real-world scenarios, including economics, engineering, and the natural sciences. The skills developed here are directly applicable to many quantitative problems that require logical reasoning and mathematical foundations.

Quantitative Aptitude

Quantitative aptitude refers to the ability to handle numerical and mathematical problems efficiently. This skill is highly valued in many fields and is crucial for academic success, especially in disciplines that involve significant amounts of problem-solving. The shop problem is a direct application of quantitative aptitude, requiring one to interpret a real-life situation, model it with equations and then solve them to find a numerical answer.

Developing a strong quantitative aptitude can greatly enhance a student’s problem-solving abilities. It involves an understanding of various mathematical concepts and the fluency to apply them accordingly. For example, our problem demanded the ability to translate a word problem into mathematical equations and then solve the equations using the substitution method. Quantitative aptitude is fundamentally about making sense of numbers and mathematical concepts in practical situations, a skill that benefits individuals both in academic pursuits and in making informed decisions in daily life.