This article explores a seemingly simple mathematical problem: "What is the sum of the product of 3 and 4 and the product of 4 and 5?". While straightforward, it highlights the importance of understanding order of operations and provides a great opportunity to delve into fundamental mathematical concepts. We'll break down the problem step-by-step, illustrating the process and connecting it to broader mathematical principles.
Understanding the Problem
The question asks us to perform two multiplications and then add their results. Let's break it down into its constituent parts:
- Product of 3 and 4: This means we multiply 3 by 4 (3 x 4 = 12).
- Product of 4 and 5: This means we multiply 4 by 5 (4 x 5 = 20).
- Sum of the products: This means we add the results of the two multiplications (12 + 20).
Step-by-Step Solution
Following the order of operations (often remembered by the acronym PEMDAS/BODMAS – Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction), we get:
- (3 x 4) + (4 x 5) This explicitly shows the two products before the addition.
- 12 + 20 This is the result of performing the multiplications.
- 32 This is the final answer – the sum of the products.
Connecting to Broader Concepts
This simple problem touches on several key mathematical concepts:
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Order of Operations: PEMDAS/BODMAS ensures consistent interpretation of mathematical expressions. Without it, we could arrive at different answers depending on the order we perform the operations. For instance, incorrectly calculating 3 x (4 + 4) x 5 would yield a completely different result.
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Distributive Property (a more advanced perspective): While not directly applied here, we can consider a related concept. The expression could be interpreted as a simplified form of a distributive property scenario. Imagine if the question was “What is the sum of the product of 4 and (3+5)?” In that case, the distributive property (a(b+c) = ab + ac) would be directly applicable.
Practical Applications
Understanding order of operations and the ability to solve such problems are crucial in numerous areas, including:
- Programming: Many programming languages follow PEMDAS/BODMAS, so accurately representing mathematical formulas requires understanding this order.
- Engineering: Calculations in various engineering disciplines rely on precise order of operations to achieve correct results.
- Finance: Financial calculations, from simple interest to complex investment strategies, require accurate application of mathematical principles.
Conclusion
The solution to "What is the sum of the product of 3 and 4 and the product of 4 and 5?" is 32. This seemingly basic problem serves as a reminder of the importance of order of operations and provides a stepping stone to understanding more complex mathematical concepts. The ability to interpret and solve these types of problems is fundamental across various fields. Remember, breaking down complex problems into smaller, manageable steps is key to success.