Points to Remember:
- This is a mathematical problem involving ratios and proportions.
- We need to find the unit cost of fans and ovens to solve the problem.
- The solution will involve setting up and solving a system of equations (or using ratios directly).
Introduction:
This question is a classic example of a problem involving proportional relationships. Given the total cost of a combination of fans and ovens, we need to determine the cost of a different combination of the same items, assuming the unit price of each item remains constant. This type of problem frequently appears in basic algebra and is crucial for understanding concepts like ratios and proportions in various real-world applications, such as budgeting, resource allocation, and scaling up or down production costs.
Body:
1. Defining the Variables and Setting up Equations:
Let’s denote the cost of one fan as ‘f’ (in rupees) and the cost of one oven as ‘o’ (in rupees). We can translate the given information into two equations:
- 8f + 14o = 36520 (Equation 1)
We need to find the cost of 12 fans and 21 ovens, which can be represented as:
- 12f + 21o = ? (Equation 2)
2. Solving for the Unit Costs (f and o):
We have one equation and two unknowns. However, we can observe that the second combination (12 fans and 21 ovens) is a multiple of the first combination (8 fans and 14 ovens). Specifically, it’s 1.5 times larger:
- 12 = 8 * 1.5
- 21 = 14 * 1.5
Therefore, the total cost of 12 fans and 21 ovens will also be 1.5 times the cost of 8 fans and 14 ovens.
3. Calculating the Cost of 12 Fans and 21 Ovens:
Cost of 12 fans and 21 ovens = 1.5 * 36520 = â¹54780
4. Alternative Method (Solving for f and o individually):
While the ratio method is simpler here, we can also solve for ‘f’ and ‘o’ individually. This requires additional information or an assumption. Since we only have one equation, we cannot uniquely determine the individual costs of a fan and an oven. The ratio method avoids this issue.
Conclusion:
The cost of 12 fans and 21 ovens is â¹54780. This was determined by recognizing the proportional relationship between the two combinations of fans and ovens. The problem highlights the importance of understanding ratios and proportions in solving practical problems. While solving for individual unit costs is possible with additional information, the proportional approach provides a more efficient solution in this specific scenario. This method emphasizes the practical application of mathematical concepts in everyday situations and promotes efficient problem-solving skills.
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