Problem 104 - Fundamentals of Mathematics

Question

Compute each product using the distributive property. \(19 \cdot 85\)

Step-by-Step Solution

Answer

The product of 19 and 85 using the distributive property is 1615.

Step 1: Express the Numbers

First, express 19 and 85 in a form that is easy to work with using the distributive property. Notice that 19 can remain as is, but 85 can be expressed as the sum of two numbers, such as 80 and 5. So, we write 85 as \(85 = 80 + 5\).

Step 2: Apply the Distributive Property

Use the distributive property \(a(b+c) = ab + ac\). Here, \(a = 19\), \(b = 80\), and \(c = 5\). Distribute 19 across the sum: \(19 imes 85 = 19 imes (80 + 5) = 19 imes 80 + 19 imes 5\).

Step 3: Calculate the Products Separately

Complete the multiplication for each part separately. 1. Calculate \(19 imes 80\): Use the multiplication \(19 imes 80 = 1520\). 2. Calculate \(19 imes 5\): Use the multiplication \(19 imes 5 = 95\).

Step 4: Combine the Products

Add the two results to get the final answer. Combine \(19 imes 80 = 1520\) and \(19 imes 5 = 95\) to find the total: \(1520 + 95 = 1615\).

Key Concepts

Understanding Multiplication Using the Distributive Property

Multiplication is a fundamental arithmetic operation where we combine equal groups together. When learning multiplication, it's crucial to grasp how we can simplify calculations using various methods. One such efficient method is the distributive property.
The distributive property states that multiplying a sum by a number is the same as multiplying each addend individually by the number and then adding the products. Mathematically, it's expressed as:

• \(a(b + c) = ab + ac\)

This property allows us to break larger, more difficult numbers into smaller, easier-to-multiply parts. For instance, in our example of \(19 \cdot 85\), we split the large number 85 into the sum of 80 and 5. By doing this, it becomes much more manageable to multiply 19 by each part separately. Once calculated, we simply add the results for a streamlined solution.

Exploring Arithmetic Strategies for Efficient Calculations

Arithmetic involves performing basic operations such as addition, subtraction, multiplication, and division. Mastering these operations is essential for solving everyday math problems. A great strategy to enhance arithmetic skills is by understanding properties like the distributive property.
Using the distributive property is an arithmetic strategy that leverages the natural structure of numbers to make multiplication easier. Rather than dealing directly with complex numbers, this property allows us to transform them into simpler, more workable expressions.

• For example, breaking 85 down into 80 and 5 in our problem helps us handle the complexity by turning the problem into two smaller products.
• This technique doesn't just make calculations easier; it also reinforces the understanding of how numbers relate to each other arithmetically.

Incorporating these strategies can greatly improve arithmetic proficiency and confidence in handling numerical operations.

Problem Solving with Mathematical Properties

Problem-solving is a critical skill in mathematics that involves finding solutions to given situations through logical reasoning and systematic steps. Using mathematical properties, such as the distributive property, plays a significant role in this process.
The step-by-step approach to problem-solving involves:

• Firstly, recognizing the components of the problem. In the case of multiplying 19 by 85, we identify the numbers and their parts.
• Secondly, applying relevant mathematical properties or techniques to simplify the problem. By choosing to use the distributive property, we split the problem into more manageable parts.
• Finally, carrying out the calculations and combining the results to reach a solution, as shown in adding the products \(1520\) and \(95\).

This structured approach not only helps in effective problem-solving but also in understanding the underlying mathematical concepts better. Strategies like these not only aid in finding accurate answers but also build a solid foundation for tackling more complex mathematical problems in the future.