Distributing 294 In 3 Parts Proportionally And Inversely
In the realm of mathematics, the concept of distributing a quantity into parts that are both directly proportional (DP) and inversely proportional (IP) to given numbers is a fundamental one. This principle finds applications in various fields, ranging from resource allocation and profit sharing to financial analysis and scientific research. In this comprehensive guide, we will delve into the intricacies of this mathematical concept, using the specific example of dividing 294 into three parts that are directly proportional to 4, 5, and 3, and inversely proportional to 2, 3, and 5. We will meticulously dissect the steps involved in solving this problem, providing a clear and concise explanation for each stage. Additionally, we will identify the smallest part obtained from this distribution, thereby enhancing our understanding of the underlying principles.
Understanding Direct and Inverse Proportionality
To effectively tackle the problem at hand, it is crucial to have a firm grasp of the concepts of direct and inverse proportionality. Direct proportionality signifies a relationship between two quantities where an increase in one quantity leads to a proportional increase in the other quantity. Conversely, inverse proportionality denotes a relationship where an increase in one quantity results in a proportional decrease in the other quantity. These fundamental concepts form the bedrock for solving distribution problems involving both direct and inverse proportions.
In the context of our problem, the three parts we seek to determine are directly proportional to the numbers 4, 5, and 3. This implies that the ratio between each part and its corresponding number will remain constant. For instance, the ratio of the first part to 4 will be equal to the ratio of the second part to 5, and so on. Furthermore, these parts are inversely proportional to the numbers 2, 3, and 5. This means that the product of each part and its corresponding number will remain constant. For example, the product of the first part and 2 will be equal to the product of the second part and 3, and so forth.
Setting Up the Problem
Let's denote the three parts we are trying to find as A, B, and C. According to the problem statement, these parts are directly proportional to 4, 5, and 3, and inversely proportional to 2, 3, and 5. We can express these relationships mathematically as follows:
- A ∝ 4/2
- B ∝ 5/3
- C ∝ 3/5
Here, the symbol ∝ signifies proportionality. To convert these proportionalities into equations, we introduce a constant of proportionality, which we will denote as 'k'. This constant will allow us to establish a precise relationship between the parts and the given numbers.
Introducing the constant of proportionality, we can rewrite the above relationships as:
- A = k * (4/2) = 2k
- B = k * (5/3)
- C = k * (3/5)
These equations express the parts A, B, and C in terms of the constant of proportionality 'k'. The next step involves determining the value of 'k', which will enable us to calculate the exact values of the parts.
Determining the Constant of Proportionality
We know that the sum of the three parts, A, B, and C, must equal the total quantity being divided, which is 294. This crucial piece of information allows us to establish an equation that will help us solve for the constant of proportionality 'k'.
Therefore, we can write:
A + B + C = 294
Substituting the expressions for A, B, and C in terms of 'k', we get:
2k + (5/3)k + (3/5)k = 294
To solve this equation for 'k', we first need to find a common denominator for the fractions. The least common multiple of 3 and 5 is 15, so we multiply each term by the appropriate factor to obtain a common denominator:
(30/15)k + (25/15)k + (9/15)k = 294
Combining the terms with 'k', we have:
(64/15)k = 294
Now, to isolate 'k', we multiply both sides of the equation by the reciprocal of 64/15, which is 15/64:
k = 294 * (15/64)
Simplifying this expression, we get:
k = (147 * 15) / 32
k = 2205 / 32
Therefore, the value of the constant of proportionality 'k' is 2205/32. This value will be instrumental in calculating the exact values of the three parts, A, B, and C.
Calculating the Parts
Now that we have determined the value of the constant of proportionality 'k', we can proceed to calculate the individual parts A, B, and C using the equations we established earlier:
- A = 2k
- B = (5/3)k
- C = (3/5)k
Substituting the value of k = 2205/32 into these equations, we get:
- A = 2 * (2205/32) = 2205/16 ≈ 137.81
- B = (5/3) * (2205/32) = (5 * 2205) / (3 * 32) = 11025/96 ≈ 114.84
- C = (3/5) * (2205/32) = (3 * 2205) / (5 * 32) = 6615/160 ≈ 41.34
Therefore, the three parts are approximately A ≈ 137.81, B ≈ 114.84, and C ≈ 41.34. These values represent the distribution of 294 in accordance with the given direct and inverse proportionalities.
Identifying the Smallest Part
To identify the smallest part, we simply compare the values we calculated for A, B, and C:
- A ≈ 137.81
- B ≈ 114.84
- C ≈ 41.34
By comparing these values, it is evident that the smallest part is C, which is approximately 41.34.
Conclusion
In this comprehensive guide, we have meticulously dissected the process of distributing a quantity into parts that are both directly proportional and inversely proportional to given numbers. We utilized the specific example of dividing 294 into three parts directly proportional to 4, 5, and 3, and inversely proportional to 2, 3, and 5. Through a step-by-step approach, we established the mathematical relationships, determined the constant of proportionality, calculated the individual parts, and ultimately identified the smallest part as approximately 41.34. This detailed explanation serves to enhance our understanding of the fundamental principles governing direct and inverse proportionality, and their application in solving real-world distribution problems. The ability to solve such problems is crucial in various fields, including resource management, financial analysis, and scientific research, making this mathematical concept a valuable tool in diverse domains.