Rice Mixture Problem Solving For Optimal Quantity Combinations

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In the realm of quantitative aptitude, mixture problems often present a fascinating challenge, requiring a blend of logical reasoning and mathematical acumen. One such intriguing problem involves a trader dealing with three distinct varieties of rice, each priced differently, and the subsequent mixing and selling of these varieties. Let's delve into the intricacies of this problem and explore a systematic approach to solving it.

Problem Statement

A trader possesses three types of rice, priced at Rs. 37 per kg, Rs. 40 per kg, and Rs. 42 per kg respectively. The quantity of each rice type is a positive integral multiple of 10 kg. The trader mixes all three varieties and sells the resultant mixture at Rs. 45 per kg. The crux of the problem lies in determining the quantities of each rice type that the trader mixed, given the selling price of the mixture.

Deconstructing the Problem

To effectively tackle this problem, let's break it down into its core components:

  1. Identifying Variables:

    • Let x represent the quantity (in kg) of rice priced at Rs. 37 per kg.
    • Let y represent the quantity (in kg) of rice priced at Rs. 40 per kg.
    • Let z represent the quantity (in kg) of rice priced at Rs. 42 per kg.

  2. Formulating Equations:

    Based on the problem statement, we can formulate the following equations:

    • The total cost of the mixture is the sum of the costs of each individual rice type: 37x + 40y + 42z.
    • The total quantity of the mixture is the sum of the quantities of each rice type: x + y + z.
    • The selling price of the mixture is Rs. 45 per kg, so the total revenue from selling the mixture is 45(x + y + z).
    • Since the trader sold the mixture at a profit, the total revenue must be greater than the total cost: 45(x + y + z) > 37x + 40y + 42z.

  3. Constraints:

    • The quantities of each rice type are positive integral multiples of 10 kg. This implies that x, y, and z must be of the form 10n, where n is a positive integer.

Solving the Equations

Now that we have formulated the equations and identified the constraints, let's proceed with solving them.

From the inequality 45(x + y + z) > 37x + 40y + 42z, we can simplify it as follows:

45x + 45y + 45z > 37x + 40y + 42z

8x + 5y + 3z > 0

This inequality provides us with a crucial insight: the weighted sum of the quantities of each rice type, with weights 8, 5, and 3 respectively, must be greater than zero. This condition must hold true for any feasible solution.

To find the specific values of x, y, and z, we need to employ a combination of algebraic manipulation and logical deduction. Let's rewrite the inequality as:

8x > -5y - 3z

x > (-5y - 3z) / 8

This inequality tells us that the quantity of rice priced at Rs. 37 per kg (x) must be greater than a certain value determined by the quantities of the other two rice types (y and z). Since x, y, and z are positive integral multiples of 10, we can start by substituting different values for y and z and then checking if the resulting value of x satisfies the inequality and the constraint of being a multiple of 10.

Let's illustrate this with an example. Suppose we assume y = 10 kg and z = 10 kg. Then, the inequality becomes:

x > (-5(10) - 3(10)) / 8

x > -80 / 8

x > -10

Since x must be a positive multiple of 10, the smallest possible value for x is 10 kg. Let's check if this solution satisfies the original inequality:

8(10) + 5(10) + 3(10) > 0

80 + 50 + 30 > 0

160 > 0

This condition holds true, so one possible solution is x = 10 kg, y = 10 kg, and z = 10 kg. This means the trader could have mixed 10 kg of each rice type.

However, this is not the only possible solution. We can explore other values for y and z to find additional solutions. For instance, let's try y = 20 kg and z = 10 kg. Then, the inequality becomes:

x > (-5(20) - 3(10)) / 8

x > -130 / 8

x > -16.25

The smallest possible value for x in this case is 20 kg. Checking the original inequality:

8(20) + 5(20) + 3(10) > 0

160 + 100 + 30 > 0

290 > 0

This condition also holds true, so another possible solution is x = 20 kg, y = 20 kg, and z = 10 kg.

By systematically exploring different values for y and z, we can identify a range of possible solutions for the quantities of each rice type that the trader mixed. Each solution represents a different combination of rice quantities that would result in the trader selling the mixture at Rs. 45 per kg.

Key Takeaways

This problem highlights several important concepts in mixture problems:

  • Variable Representation: Assigning variables to the unknown quantities is crucial for formulating equations.
  • Equation Formulation: Translating the problem's information into mathematical equations is the cornerstone of solving mixture problems.
  • Constraint Identification: Recognizing the constraints on the variables, such as being positive integers or multiples of a specific number, helps narrow down the possible solutions.
  • Systematic Approach: Employing a systematic approach, such as substituting values and checking conditions, is essential for finding all feasible solutions.

The Power of Problem-Solving

Mixture problems like this not only test mathematical skills but also enhance logical reasoning and analytical thinking. By mastering the techniques involved in solving these problems, individuals can develop a more profound understanding of quantitative concepts and improve their problem-solving abilities in various real-world scenarios.

Conclusion

The trader's rice mixture problem serves as an excellent example of how mathematical principles can be applied to solve practical scenarios. By carefully analyzing the problem, formulating equations, and considering constraints, we can arrive at a range of possible solutions. This exercise underscores the importance of quantitative aptitude and its relevance in everyday decision-making.

The Art of Mixing Proportions and Prices

In the domain of quantitative reasoning, mixture problems stand out as intriguing challenges that demand a blend of analytical thinking and mathematical precision. These problems often involve combining different quantities of substances with varying characteristics, such as prices or concentrations, and then determining the properties of the resulting mixture. The rice mixture problem we're about to dissect falls squarely into this category, presenting a scenario where a trader skillfully blends three distinct varieties of rice, each with its unique price tag, to create a mixture that commands a profitable selling price. Understanding mixture problems is key to unlocking a wide range of real-world applications, from financial calculations to scientific experiments.

Problem Statement A Deep Dive

The problem presents us with a trader who has access to three distinct types of rice. The first type is priced at Rs. 37 per kg, the second at Rs. 40 per kg, and the third at Rs. 42 per kg. The problem statement introduces a crucial constraint: the quantity of each rice type must be a positive integral multiple of 10 kg. This means that the trader can only use quantities like 10 kg, 20 kg, 30 kg, and so on, for each variety. The trader's objective is to mix these three varieties in such a way that when the mixture is sold at Rs. 45 per kg, a profit is made. The challenge lies in determining the possible quantities of each rice type that the trader could have mixed to achieve this profitable outcome. This requires us to not only consider the cost prices of individual varieties but also the selling price of the final mixture, making it a multifaceted problem that necessitates a systematic approach.

Identifying Key Variables The Building Blocks

To effectively tackle this problem, the first step involves identifying the key variables. These variables will serve as the building blocks for our equations and inequalities. In this case, the most crucial variables are the quantities of each rice type. Let's denote: x as the quantity (in kg) of rice priced at Rs. 37 per kg. y as the quantity (in kg) of rice priced at Rs. 40 per kg. z as the quantity (in kg) of rice priced at Rs. 42 per kg. These variables, x, y, and z, will form the foundation of our mathematical representation of the problem. By defining these variables clearly, we set the stage for translating the word problem into a set of equations and inequalities that we can then solve.

Formulating Equations The Mathematical Framework

With the variables defined, we can now translate the problem's information into a set of mathematical equations and inequalities. This is a critical step in solving mixture problems, as it provides the framework for our analysis. There are several key relationships we need to capture: The total cost of the mixture: This is the sum of the costs of each individual rice type. We can express this as: 37x + 40y + 42z. The total quantity of the mixture: This is simply the sum of the quantities of each rice type: x + y + z. The total revenue from selling the mixture: Since the mixture is sold at Rs. 45 per kg, the total revenue is: 45(x + y + z). The profit condition: For the trader to make a profit, the total revenue must be greater than the total cost. This gives us the inequality: 45(x + y + z) > 37x + 40y + 42z. These equations and inequalities provide a comprehensive mathematical representation of the problem, allowing us to analyze the relationships between the variables and identify potential solutions. The formulation of equations is often the most challenging part of solving mixture problems, as it requires careful attention to detail and a deep understanding of the problem's structure.

The Profit Inequality Unveiling the Key Constraint

Among the equations and inequalities we've formulated, the profit inequality stands out as a key constraint. It dictates the conditions under which the trader can make a profit by selling the rice mixture. Let's delve deeper into this inequality: 45(x + y + z) > 37x + 40y + 42z. We can simplify this inequality to gain further insights. Expanding the left side, we get: 45x + 45y + 45z > 37x + 40y + 42z. Now, let's rearrange the terms by subtracting the terms on the right side from both sides: 45x - 37x + 45y - 40y + 45z - 42z > 0. This simplifies to: 8x + 5y + 3z > 0. This simplified inequality reveals a crucial relationship between the quantities of the three rice types. It states that the weighted sum of the quantities, with weights 8, 5, and 3 respectively, must be greater than zero for the trader to make a profit. This condition provides a valuable constraint that we can use to narrow down the possible solutions. Understanding the implications of this profit inequality is essential for efficiently solving the problem.

Unveiling Hidden Constraints The Multiples of 10 Rule

In addition to the profit inequality, the problem statement introduces another crucial constraint: the quantities of each rice type must be positive integral multiples of 10 kg. This means that x, y, and z must be of the form 10n, where n is a positive integer. This constraint significantly reduces the number of possible solutions. Instead of considering any arbitrary values for x, y, and z, we only need to consider values that are multiples of 10. This simplifies the problem considerably, as we can now focus on a discrete set of possible values. For instance, x can be 10, 20, 30, and so on, but it cannot be 15 or 25. This constraint on multiples of 10 is a common feature in mixture problems, and it often plays a crucial role in finding the solutions.

A Systematic Approach to Solving The Iterative Process

With the equations, inequalities, and constraints in place, we can now embark on the process of finding the solutions. A systematic approach is key to efficiently solving this problem. One effective method is to use an iterative process, where we start by substituting different values for some of the variables and then check if the resulting values for the other variables satisfy all the conditions. For instance, we can start by assuming values for y and z, and then use the profit inequality to determine the possible values for x. We can then check if these values satisfy the constraint of being multiples of 10. Let's illustrate this with an example. Suppose we assume y = 10 kg and z = 10 kg. Substituting these values into the profit inequality, we get: 8x + 5(10) + 3(10) > 0. This simplifies to: 8x + 50 + 30 > 0. Further simplification gives us: 8x > -80. Dividing both sides by 8, we get: x > -10. Since x must be a positive multiple of 10, the smallest possible value for x is 10 kg. Therefore, one possible solution is x = 10 kg, y = 10 kg, and z = 10 kg. This means the trader could have mixed 10 kg of each rice type. We can continue this process by trying different values for y and z and checking the resulting values for x. This iterative process allows us to systematically explore the possible solutions and identify all the combinations of rice quantities that satisfy the problem's conditions.

Exploring Multiple Solutions The Range of Possibilities

It's important to note that mixture problems often have multiple solutions. In the rice mixture problem, there are likely to be several combinations of rice quantities that would allow the trader to make a profit. Our goal is to identify as many of these solutions as possible. By systematically varying the values of y and z and using the profit inequality to determine the possible values of x, we can uncover a range of solutions. Each solution represents a different way the trader could have mixed the rice varieties to achieve a profitable outcome. Understanding that multiple solutions are possible is crucial for fully grasping the nuances of mixture problems. It highlights the flexibility the trader has in choosing the quantities of each rice type.

The Value of Logical Deduction Beyond the Equations

While mathematical equations and inequalities form the backbone of solving mixture problems, logical deduction plays a crucial role in refining the solutions and gaining deeper insights. By carefully analyzing the relationships between the variables and the constraints, we can often eliminate certain possibilities and narrow down the search for solutions. For instance, if we find that a particular combination of y and z leads to a very large value for x, we might be able to deduce that this solution is not practical in a real-world scenario. Logical deduction allows us to go beyond the pure mathematics and incorporate real-world considerations into the problem-solving process. This skill is invaluable not only in solving mixture problems but also in tackling a wide range of other quantitative challenges.

Real-World Relevance Mixture Problems in Action

Mixture problems are not just abstract mathematical exercises; they have significant real-world relevance. They arise in various practical scenarios, from business and finance to chemistry and engineering. For instance, in finance, portfolio management often involves mixing different types of assets, such as stocks and bonds, to achieve a desired risk-return profile. In chemistry, mixture problems are essential for calculating the concentrations of solutions. In engineering, they are used in blending materials to achieve specific properties. By mastering the techniques for solving mixture problems, individuals can enhance their problem-solving skills in a wide range of disciplines. The real-world relevance of these problems makes them a valuable topic of study for anyone seeking to develop their quantitative abilities.

Beyond the Numbers The Importance of Analytical Thinking

At its core, solving mixture problems is about more than just manipulating numbers. It's about developing analytical thinking skills. These problems require us to carefully dissect the given information, identify the key relationships, formulate mathematical representations, and then use logical deduction to arrive at the solutions. The process of solving a mixture problem hones our ability to think critically, break down complex problems into smaller, manageable parts, and apply systematic approaches to find solutions. These skills are transferable to a wide range of other contexts, making the study of mixture problems a valuable investment in one's overall problem-solving abilities. The emphasis on analytical thinking is what makes mixture problems so rewarding to solve. They challenge us to go beyond rote memorization and engage in a deeper level of understanding.

The Rice Mixture Problem A Gateway to Quantitative Mastery

The rice mixture problem we've explored is a fascinating example of a quantitative challenge that requires a blend of mathematical skills and logical reasoning. By dissecting the problem, identifying the variables, formulating equations and inequalities, and applying a systematic approach, we can uncover a range of possible solutions. This problem serves as a gateway to a deeper understanding of mixture problems and their real-world applications. Mastering the techniques involved in solving these problems not only enhances our quantitative aptitude but also cultivates valuable analytical thinking skills. The rice mixture problem is a testament to the power of quantitative reasoning and its ability to illuminate practical scenarios.

Embracing the Realm of Mixture Challenges

Within the expansive universe of quantitative problems, mixture challenges stand out as captivating puzzles that necessitate a harmonious blend of analytical prowess and mathematical acumen. These challenges typically revolve around scenarios where distinct entities, be it substances, ingredients, or even financial instruments, are combined in varying proportions, each possessing unique attributes like price points, concentrations, or risk profiles. The crux of the matter often lies in unraveling the characteristics of the resulting amalgamation or discerning the optimal proportions required to attain a specific outcome. Mixture challenges are not mere academic exercises; they mirror real-world complexities across diverse domains, including finance, manufacturing, and even culinary arts.

A Trader's Predicament Unveiling the Rice Conundrum

The problem at hand presents us with a compelling scenario: a trader navigating the intricacies of the rice market. This trader possesses three distinct varieties of rice, each commanding a different price tag. The first variety is priced at Rs. 37 per kilogram, the second at Rs. 40 per kilogram, and the third at Rs. 42 per kilogram. Adding a layer of complexity, the problem stipulates that the quantity of each rice variety must be a positive integral multiple of 10 kilograms. This constraint narrows down the trader's options, limiting the quantities to increments of 10. The trader's objective is to concoct a mixture of these three varieties and sell it at a premium price of Rs. 45 per kilogram, thereby securing a profitable transaction. The puzzle, therefore, lies in deciphering the possible quantities of each rice variety that the trader could have blended to achieve this lucrative outcome. To solve this, we need a good understanding of the rice conundrum.

The Dance of Variables Laying the Foundation

To effectively dissect this problem, our initial stride involves the identification of pivotal variables. These variables will serve as the cornerstones upon which we construct our mathematical edifice. In this context, the most salient variables are the quantities of each rice variety. Let's embark on a symbolic representation: Let x denote the quantity (expressed in kilograms) of rice priced at Rs. 37 per kilogram. Let y denote the quantity (expressed in kilograms) of rice priced at Rs. 40 per kilogram. Let z denote the quantity (expressed in kilograms) of rice priced at Rs. 42 per kilogram. These variables, x, y, and z, will serve as the linguistic currency of our mathematical discourse. By precisely defining these variables, we pave the path toward translating the narrative problem into a realm of equations and inequalities, setting the stage for subsequent analysis and resolution. Clearly setting out the dance of variables is key to solving the challenge.

Orchestrating Equations Weaving the Mathematical Tapestry

With our variables firmly in place, we now embark on the task of transmuting the problem's textual narrative into a symphony of mathematical equations and inequalities. This process constitutes a pivotal step in tackling mixture challenges, providing the scaffolding for our analytical endeavor. We must capture several fundamental relationships: The total cost of the mixture: This is the cumulative sum of the costs incurred for each individual rice variety. We can mathematically articulate this as: 37x + 40y + 42z. The total quantity of the mixture: This is simply the arithmetic sum of the quantities of each rice variety: x + y + z. The total revenue garnered from selling the mixture: Given that the mixture commands a selling price of Rs. 45 per kilogram, the total revenue can be expressed as: 45(x + y + z). The Profit Imperative: For the trader to realize a profit, the total revenue must surpass the total cost. This engenders the following inequality: 45(x + y + z) > 37x + 40y + 42z. These equations and inequalities collectively furnish a comprehensive mathematical portrait of the problem, enabling us to scrutinize the intricate relationships between the variables and identify potential solutions. The craft of orchestrating equations often represents the most formidable aspect of unraveling mixture challenges, demanding meticulous attention to detail and a profound grasp of the problem's underlying architecture.

The Profit Inequality Unmasking the Core Constraint

Amidst the constellation of equations and inequalities we've formulated, the profit inequality emerges as a beacon of paramount importance. This inequality dictates the conditions under which the trader can successfully reap a profit from the rice mixture venture. Let us delve deeper into its anatomy: 45(x + y + z) > 37x + 40y + 42z. We can embark on a simplification journey to glean further insights. Expanding the left-hand side, we arrive at: 45x + 45y + 45z > 37x + 40y + 42z. Now, let's engage in a ballet of terms, transposing those on the right-hand side to the left: 45x - 37x + 45y - 40y + 45z - 42z > 0. This melodic simplification yields: 8x + 5y + 3z > 0. This streamlined inequality unveils a pivotal relationship between the quantities of the three rice varieties. It asserts that the weighted sum of these quantities, with weights of 8, 5, and 3 respectively, must exceed zero for the trader to secure a profit. This condition serves as a valuable constraint, guiding us in narrowing the spectrum of potential solutions. A good understanding of the profit inequality is key to solving the problem.

Unveiling the Hidden Multiples of 10 The Discreet Quantity Rule

In addition to the profit inequality, the problem statement introduces another constraint of significance: the quantities of each rice variety must be positive integral multiples of 10 kilograms. This stipulation implies that x, y, and z must conform to the form 10n, where n represents a positive integer. This constraint acts as a filter, significantly curtailing the realm of permissible solutions. Rather than entertaining any arbitrary values for x, y, and z, we are now confined to considering values that are divisible by 10. This simplification proves instrumental, enabling us to concentrate on a discrete set of possibilities. For instance, x can assume values such as 10, 20, 30, and so forth, but it cannot take on intermediate values like 15 or 25. This constraint pertaining to multiples of 10 constitutes a recurring motif in mixture problems, often playing a pivotal role in the quest for solutions. Having an understanding of the discreet quantity rule is key to narrowing down the solutions.

The Systematic Solution A Dance of Iteration

With our equations, inequalities, and constraints firmly entrenched, we can now embark on the quest for solutions. A systematic approach is of paramount importance in efficiently tackling this challenge. One efficacious method involves an iterative process, wherein we initiate our exploration by substituting varying values for certain variables and subsequently verifying whether the resultant values for the remaining variables align with all stipulated conditions. For instance, we can commence by postulating values for y and z, and then leverage the profit inequality to ascertain the potential values for x. Subsequently, we can validate whether these values adhere to the constraint of being multiples of 10. Let's illustrate this methodology with a concrete example. Suppose we hypothesize that y = 10 kilograms and z = 10 kilograms. Substituting these values into the profit inequality, we obtain: 8x + 5(10) + 3(10) > 0. This expression simplifies to: 8x + 50 + 30 > 0. Further simplification yields: 8x > -80. Dividing both sides by 8, we arrive at: x > -10. Given that x must be a positive multiple of 10, the minimum feasible value for x is 10 kilograms. Consequently, one potential solution emerges: x = 10 kilograms, y = 10 kilograms, and z = 10 kilograms. This implies that the trader could have judiciously blended 10 kilograms of each rice variety. We can perpetuate this process by experimenting with alternative values for y and z, and scrutinizing the resultant values for x. This iterative process empowers us to systematically traverse the solution landscape and identify all combinations of rice quantities that satisfy the problem's overarching stipulations. Approaching this in a systematic way with a dance of iteration is key to solving this problem.

Exploring Multiple Solution Landscapes The Range of Possibilities

It's crucial to acknowledge that mixture problems frequently harbor a multitude of solutions. In the rice mixture scenario, it's plausible that several combinations of rice quantities would enable the trader to realize a profitable outcome. Our objective is to identify as many of these solutions as possible. By judiciously varying the values of y and z and employing the profit inequality to pinpoint the potential values of x, we can unveil a spectrum of solutions. Each solution represents a distinct strategy the trader could have adopted to blend the rice varieties and secure a profitable transaction. Recognizing the potential for multiple solution landscapes is essential for comprehensively grasping the subtleties inherent in mixture problems. It underscores the flexibility the trader possesses in orchestrating the quantities of each rice variety.

Beyond Equations The Art of Logical Deduction

While mathematical equations and inequalities serve as the bedrock for solving mixture problems, logical deduction assumes a pivotal role in refining solutions and cultivating profound insights. By meticulously analyzing the intricate relationships between variables and constraints, we can often eliminate certain possibilities and narrow our focus in the quest for solutions. For instance, if we discern that a specific combination of y and z engenders a conspicuously large value for x, we might judiciously deduce that this solution lacks practicality in a real-world context. Logical deduction empowers us to transcend the realm of pure mathematics and seamlessly integrate real-world considerations into the problem-solving odyssey. This invaluable skill transcends the confines of mixture problems, proving indispensable in tackling a diverse array of quantitative challenges. The subtle art of logical deduction is essential to refining the solutions.

Real-World Echoes Mixture Problems in Action

Mixture problems are not mere figments of academic imagination; they resonate with tangible real-world applications. These problems permeate diverse domains, spanning business and finance, chemistry and engineering. For instance, in the realm of finance, portfolio management often entails the strategic blending of various asset classes, such as stocks and bonds, to realize a desired risk-return profile. In chemistry, mixture problems are indispensable for meticulously calculating the concentrations of solutions. In engineering, these problems find utility in the artful blending of materials to attain specific properties. By mastering the techniques for unraveling mixture problems, individuals can amplify their problem-solving acumen across a spectrum of disciplines. The real-world echoes of mixture problems make them an invaluable area of study for those seeking to cultivate their quantitative prowess.

Beyond Numbers The Analytical Mindset

At its core, the endeavor of solving mixture problems transcends the mere manipulation of numerals. It's about fostering an analytical mindset. These problems necessitate a meticulous dissection of given information, a keen identification of salient relationships, the articulation of mathematical representations, and the judicious application of logical deduction to converge upon solutions. The process of dissecting a mixture problem hones our aptitude for critical thinking, enabling us to fragment complex challenges into more manageable constituents and employ systematic methodologies to unearth solutions. These cultivated skills extend far beyond the realm of mathematics, proving invaluable in diverse contexts. This helps create the analytical mindset needed in every aspect of business and planning.

The Rice Trader's Saga A Portal to Quantitative Mastery

The saga of the rice trader, with its intricate blend of mathematical nuances and logical intricacies, exemplifies a quantitative challenge that demands a harmonious confluence of skills. By dissecting the problem, identifying the variables, formulating equations and inequalities, and embracing a systematic approach, we can unveil a spectrum of potential solutions. This problem serves as a portal to a deeper appreciation of mixture problems and their pervasive real-world manifestations. Mastering the techniques intrinsic to these problems not only fortifies our quantitative aptitude but also nurtures invaluable analytical faculties. The entire rice trader's saga is a great experience to master the quantitative problem.

To solve this problem, let's denote the quantities of rice at Rs. 37, Rs. 40, and Rs. 42 per kg as 10x, 10y, and 10z, respectively, where x, y, and z are positive integers. The total cost of the mixture is 37(10x) + 40(10y) + 42(10z) = 370x + 400y + 420z. The total quantity of the mixture is 10x + 10y + 10z = 10(x + y + z). The selling price of the mixture is Rs. 45 per kg, so the total revenue from selling the mixture is 45 * 10(x + y + z) = 450(x + y + z).

For the trader to make a profit, the revenue must be greater than the cost:

450(x + y + z) > 370x + 400y + 420z

Divide both sides by 10:

45(x + y + z) > 37x + 40y + 42z

Expand:

45x + 45y + 45z > 37x + 40y + 42z

Rearrange:

8x + 5y > -3z

Rearrange:

8x + 5y > 3z

This inequality gives us a relationship between x, y, and z that must be satisfied for the trader to make a profit. Since we are looking for integer solutions, we can start testing values for x, y, and z to find some combinations that satisfy this condition.

One possible solution is x = 2, y = 1, and z = 1. Let's check:

8(2) + 5(1) > 3(1)

16 + 5 > 3

21 > 3 (True)

This solution works. The quantities are 20 kg of Rs. 37 rice, 10 kg of Rs. 40 rice, and 10 kg of Rs. 42 rice.

Another possible solution is x = 1, y = 2, and z = 1. Let's check:

8(1) + 5(2) > 3(1)

8 + 10 > 3

18 > 3 (True)

This solution also works. The quantities are 10 kg of Rs. 37 rice, 20 kg of Rs. 40 rice, and 10 kg of Rs. 42 rice.

Another possible solution is x = 3, y = 1, and z = 2. Let's check:

8(3) + 5(1) > 3(2)

24 + 5 > 6

29 > 6 (True)

This solution also works. The quantities are 30 kg of Rs. 37 rice, 10 kg of Rs. 40 rice, and 20 kg of Rs. 42 rice.

There are multiple possible solutions, and these are just a few examples. To find all possible solutions, one would need to further explore the inequality and consider other combinations of positive integers.

The problem presented a realistic scenario of a trader mixing rice varieties to maximize profit. By using algebraic equations and inequalities, we can derive the possible quantities of each variety that the trader mixed. The key inequality 8x + 5y > 3z is crucial for finding these solutions. As demonstrated, there are multiple possible solutions, and each one offers a different mix of rice quantities. This problem emphasizes the importance of quantitative skills and logical reasoning in practical business situations.