Problem 106 - Fundamentals of Mathematics

Question

Compute each product using the distributive property. \(81 \cdot 40\)

Step-by-Step Solution

Answer

81 times 40 equals 3240.

Step 1: Apply the Distributive Property

The distributive property states that the product of a number and a sum is equal to the sum of the products of that number and each addend. To apply it here, let's express 40 as a sum: 40 = 30 + 10.

Step 2: Distribute Multiplication

Now, distribute 81 to each part of the sum: \[81 \times 40 = 81 \times (30 + 10) = (81 \times 30) + (81 \times 10).\]

Step 3: Calculate Each Product Separately

First, calculate \(81 \times 30\): \[81 \times 30 = 2430.\]Then, calculate \(81 \times 10\): \[81 \times 10 = 810.\]

Step 4: Sum the Results

Add the two products obtained: \[(81 \times 30) + (81 \times 10) = 2430 + 810.\]The sum is 3240.

Key Concepts

Multiplication

When we talk about multiplication, we're referring to a mathematical operation that represents combining equal groups. For instance, if you have 81 groups of 40 items, multiplication gives you a shortcut to find the total number of items quickly.
To dig deeper, multiplication involves two primary components: the

• **Multiplier**, which is how many groups you have.
• **Multiplicand**, which is how many items each group contains.

In the problem given, 81 acts as our multiplier, and 40 is the multiplicand. The multiplication problem is then computed as 81 times 40.
Using the distributive property, we break down larger multiplication problems like this one into more manageable parts, making it easier to solve.

Arithmetic

Arithmetic is the branch of mathematics that deals with numbers and basic operations like addition, subtraction, multiplication, and division. Understanding arithmetic is fundamental to solving mathematical problems efficiently.
In our exercise, we utilized arithmetic to first split 40 into 30 and 10. This method allows us to manage smaller numbers, which simplifies our calculations. Here, we were able to calculate the products separately:

• 81 multiplied by 30
• 81 multiplied by 10

Each number pair was handled individually, leveraging basic arithmetic skills. Once calculated and added together, they give us the total product of 81 multiplied by 40. Utilizing arithmetic in this way affirms its necessity in breaking down complex calculations into easy steps.

Problem Solving

Problem solving in mathematics often involves the strategic use of properties and operations to simplify complex problems. With the distributive property, we can solve problems by breaking them into smaller, manageable parts. This approach requires critical thinking and applying learned concepts in a practical way.
The problem-solving process used in our exercise involves several steps:

• Understanding the problem and identifying the property that can simplify it (the distributive property in our case).
• Deconstructing the problem into smaller parts.
• Solving each part using basic arithmetic.
• Compiling the results to reach the final solution.

By following these steps, you develop not just your calculation skills but also your analytical thinking, crucial for more advanced mathematics and real-world problem solving.