Problem 29 - Prealgebra

Question

Find each sum or product. Explain your reasoning. $$2 \cdot 7 \cdot 30$$

Step-by-Step Solution

Answer

The product is 420.

Step 1: Calculate the product of the first two numbers

First, we calculate the product of the first two numbers: \(2\) and \(7\). Multiply \(2\) by \(7\) to get the result: \[2 \times 7 = 14\].

Step 2: Multiply the result by the next number

Now take the result from Step 1, which is \(14\), and multiply it by the next number, \(30\):\[14 \times 30 = 420\].

Step 3: Check the result

Finally, confirm that the calculations are correct. Recalculate if necessary to ensure accuracy. In this case, the calculations confirm that the result is \(420\).

Key Concepts

Multiplication

Multiplication is a fundamental arithmetic operation used to calculate the total of one number added repeatedly. For example, if you multiply 2 by 7, you are adding 2 together seven times: 2 + 2 + 2 + 2 + 2 + 2 + 2 = 14. In the case of the exercise, we multiply sequentially, starting with the first two numbers. Every multiplication process can be visualized as scaling. Here, 2 and 7 combine to create 14, forming a building block for later steps. When you subsequently multiply by another number, such as 30, you're scaling 14 by that factor. It's essential to understand multiplication as not just a set of numbers but as scaling one amount by another, leading to a product that grows as more numbers are multiplied.

Step-by-Step Solutions

Using step-by-step solutions is a powerful method to tackle complex problems by breaking them down into manageable parts. For the exercise provided, the steps simplify the computation:

• Start by multiplying the first two numbers: 2 times 7. Complete this step by calculating 2 × 7 = 14.
• Next, take the result of 14 and multiply it by the next number, which is 30. Perform the multiplication: 14 × 30 = 420.
• Finally, verify the calculations by reviewing each step to check for possible errors. This helps ensure you’ve arrived at the correct answer.

This strategy not only improves accuracy but also enhances understanding by having you work through each layer of the problem systematically. As you practice these steps, they become habits that lead to skillful and confident problem solving.

Mathematical Reasoning

Mathematical reasoning allows you to understand why certain operations work the way they do. For multiplication, it involves understanding how numbers interact. Here, multiplying 2 and 7 gives a product that stands as an intermediary step—a foundational understanding. You then enhance this product meaningfully by scaling it with another number. When multiplying 14 by 30, you aren't just adding large numbers; you are effectively combining units of 14, thirty times over. This deeper insight into operations not only helps solve the problem but also builds a stronger math foundation. It’s about seeing beyond numbers to understand the underlying patterns and principles.

Problem Solving

Problem solving in mathematics involves identifying the problem, employing strategies, and working through solutions to reach the final answer. In the context of this multiplication problem, the task is recognizing the need to find the product of three numbers. Here’s how you can approach it:

• Identify: Recognize that the exercise requires multiplication.
• Strategize: Plan to tackle the problem by calculating products step by step (2 × 7, then the result × 30).
• Execute: Perform the calculations sequentially, confirming each step as you proceed.
• Review: Double-check your work to ensure that no mistakes were made during any calculation step.

This process not only leads you to the correct answer but also develops critical-thinking skills that apply to various other mathematical challenges. In essence, problem solving is about gathering pieces, arranging them logically, and ensuring each piece fits perfectly.