Problem 14 - Fundamentals of Mathematics

Question

Use the distributive property to compute each product. \(25 \cdot 11\)

Step-by-Step Solution

Answer

The product of 25 and 11 is 275.

Step 1: Break Down One Factor

To use the distributive property, we first break down one of the factors into a sum or difference. Let's break down 11 as \( 10 + 1 \).

Step 2: Apply the Distributive Property

Now apply the distributive property: \((25 \cdot 10) + (25 \cdot 1)\). This means we distribute the 25 over each term inside the parenthesis.

Step 3: Calculate Each Product

Calculate each of the products separately: \(25 \cdot 10 = 250\) and \(25 \cdot 1 = 25\).

Step 4: Add the Results

Now, add the results of the two products together: \(250 + 25 = 275\).

Key Concepts

Multiplication

Multiplication is one of the fundamental arithmetic operations and can be easily applied using the distributive property. It's the process of calculating the total of one number taken a certain number of times. In our example, multiplying 25 by 11 means taking 25 eleven times. A convenient strategy is to break down complicated multiplication into simpler parts. Here, we used the distributive property to express 11 as the sum of 10 and 1. This makes it easier by turning the single multiplication problem into two simpler ones, i.e., multiplying 25 by 10 and 25 by 1. Understanding multiplication as repeated addition can also help solidify why this process works. You are essentially using the sum of smaller, more manageable pieces to reach the total product.

Arithmetic Operations

Arithmetic operations include addition, subtraction, multiplication, and division. These operations allow us to manipulate numbers in various ways to derive new quantities. The distributive property, an essential component of arithmetic, particularly comes into play when simplifying complex calculations. It states that multiplication over addition works as follows: a(b + c) = ab + ac. In our case, we broke down the number 11 into 10 + 1, allowing us to perform two smaller multiplication operations: 25 times 10 and 25 times 1. This method simplifies what might initially seem like a more complex calculation. Understanding each arithmetic operation's role allows us to choose the best strategy for solving problems quickly and efficiently. Additionally, knowing how they interrelate, like in the distributive property, can significantly deepen your mathematical insight.

Problem-Solving Skills

Using the distributive property showcases a strategic approach to problem-solving in mathematics. This property is especially beneficial when dealing with larger numbers, as it breaks the problem into smaller, more manageable steps. Improving problem-solving skills involves recognizing patterns and selecting efficient methods to achieve a solution. It encourages logical thinking and flexibility in approaching different problems. In the exercise provided, we used a step-by-step approach: first by breaking down the problem, then applying arithmetic operations, and finally combining results to find the solution. Enhancing problem-solving skills are not restricted to finding an answer but also understanding the process and being able to apply similar techniques to other problems. Developing these skills improves educational outcomes and builds confidence in tackling mathematical challenges.