Problem 17 - Fundamentals of Mathematics

Question

For problems 17-21, use the distributive property to obtain the exact result. \((\) Section 8.4\() 25 \cdot 14\)

Step-by-Step Solution

Answer

The exact result is 350.

Step 1: Understand the Distributive Property

The distributive property states that for any numbers \( a, b, \) and \( c, \) \( a(b + c) = ab + ac \). This means we can decompose one of the numbers into a sum or a difference to make multiplication easier.

Step 2: Decompose 14

Let's break down 14 into two numbers that are easier to work with. We will use 10 and 4 since they add up to 14, such that 14 = 10 + 4.

Step 3: Apply the Distributive Property

Using the decomposed form of 14, the original expression \( 25 \cdot 14 \) becomes \( 25 \cdot (10 + 4) \). According to the distributive property, this is \( 25 \cdot 10 + 25 \cdot 4 \).

Step 4: Calculate Each Part

Now, calculate each multiplication separately: \( 25 \cdot 10 = 250 \) and \( 25 \cdot 4 = 100 \).

Step 5: Combine the Results

Add the results from the previous step together: \( 250 + 100 = 350 \).

Step 6: Write the Final Answer

Thus, the exact result of \( 25 \cdot 14 \) is 350.

Key Concepts

Multiplication

Multiplication is one of the four basic arithmetic operations. It involves finding the total sum when a number is taken a specified number of times. For instance, when we multiply 25 by 14, it means that we are adding the number 25 fourteen times. Instead of adding repeatedly, multiplication helps to simplify this process into a single operation.
For example:

• 25 multiplied by 1 is just 25.
• With each increase, you go a step further to repeat addition.
• In practice, patterns within multiplication make it more efficient than repeated addition.

Learning and applying multiplication is fundamental to advancing in mathematics. This core skill forms the basis for many complex mathematical concepts.

Decomposition

Decomposition in mathematics refers to the process of breaking down complex expressions or numbers into simpler parts. This method is incredibly useful in multiplication, especially when dealing with larger numbers.
Imagine the number 14: it's much easier to work with when you think of it as 10 plus 4, rather than as a whole. This simplification allows you to use smaller, more manageable numbers in calculations. It is a strategy that can be readily applied:

• Break down numbers into smaller parts, like how 14 becomes 10 and 4.
• Use these parts to simplify operations using basic arithmetic operations.
• The process not only makes multiplication easier but also enhances understanding of number structures.

Decomposition is like having a shortcut where you can circumvent complex calculations, saving time and reducing mistakes.

Arithmetic Operations

Arithmetic operations form the foundation of mathematics. They include addition, subtraction, multiplication, and division, and they are integral to solving almost any math problem. When applying these operations effectively, understanding their properties is crucial.
The distributive property is one such property that plays an essential role, particularly when simplifying problems by decomposing numbers. It allows you to break down the calculations step by step:

• Add the parts separately and then recombine them.
• This approach can turn daunting calculations into straightforward arithmetic facts.
• It not only speeds up the process but also improves accuracy since smaller numbers are generally easier to handle.

Grasping these basic operations and their properties improves overall mathematical fluency and problem-solving skills.