Problem 19 - Fundamentals of Mathematics

Question

Use the distributive property to compute each product. \(45 \cdot 38\)

Step-by-Step Solution

Answer

The product of 45 and 38 is 1710.

Step 1: Break Down One Factor

First, let's break down one of the factors into simpler, more manageable numbers using addition. This makes it easier to use the distributive property. Let's use 38 as 30 + 8.

Step 2: Apply the Distributive Property

Using the distributive property, we can write: \[45 \cdot 38 = 45 \cdot (30 + 8)\]Now distribute 45 to each term inside the parentheses: \[= 45 \cdot 30 + 45 \cdot 8\]

Step 3: Compute Each Product Separately

Now, compute the two products separately. First calculate \(45 \cdot 30\): \[45 \cdot 30 = 1350\] Then calculate \(45 \cdot 8\): \[45 \cdot 8 = 360\]

Step 4: Add the Results

Now sum the two products to find the total. Add 1350 and 360: \[1350 + 360 = 1710\] Therefore, the result of \(45 \cdot 38\) is 1710.

Key Concepts

Multiplication

Multiplication is one of the four fundamental arithmetic operations. It involves adding a number to itself a certain number of times. Think of it as a faster way to add groups of the same size together. When you multiply two numbers, you refer to these numbers as factors and the result is called the product. For example, when we multiply 45 by 38 in this exercise, we are essentially adding 45 to itself 38 times. It can seem daunting, but with strategies like the distributive property, we can simplify these calculations. Multiplication is at the core of many math concepts and is widely used in different real-world contexts like calculating area, understanding scales, and budgeting.

Decomposition

Decomposition is a powerful tool in mathematics, particularly when using the distributive property. It involves breaking down a number into simpler components. In this exercise, the number 38 is decomposed into 30 and 8. This makes it manageable to apply the distributive property.

• Decomposition helps us handle large numbers.
• It simplifies the multiplication process.
• It allows us to see complex operations in a more understandable way.

By decomposing numbers, we can easily distribute other numbers across, making multiplication less intimidating and more strategic.

Addition

Addition is the process of combining numbers to form a new total. In mathematics, addition serves as a basic building block for more complex operations like multiplication. In this exercise, addition comes into play when we combine the two simpler groups of products. First, we calculate the separate products: 45 multiplied by 30, and 45 multiplied by 8. Next, we add these individual products together to get the final outcome. By seeing how multiplication and addition interconnect, we can solve multi-step problems more efficiently. Understanding addition's role in the distributive property reinforces how each step in solving a problem contributes to finding the solution.

Product

A product in mathematics is the result of multiplying two numbers together. In our example, we want to find the product of 45 and 38. By using decomposition and the distributive property, we calculate a series of simpler products which are then combined into the final product of 1710.

• The term product refers to the end result of multiplication.
• Different strategies can be used for computation, such as the distributive property.
• Finding the product is essential in solving equations and real-world problems.

Ultimately, the product not only gives us the answer but also deepens our understanding of multiplication and its applications.