Problem 34 - Fundamentals of Mathematics

Question

Use the distributive property to compute each product. $$ 50 \cdot 63 $$

Step-by-Step Solution

Answer

50 multiplied by 63 equals 3150.

Step 1: Express Numbers in Convenient Form

First, express each number in a form that will make it easier to apply the distributive property. For example, we can express 63 as \(60 + 3\). Then we have: \(50 \times 63 = 50 \times (60 + 3)\).

Step 2: Apply the Distributive Property

Apply the distributive property: \(a \cdot (b + c) = a \cdot b + a \cdot c\). Here, this becomes: \(50 \times (60 + 3) = 50 \times 60 + 50 \times 3\).

Step 3: Compute Each Product Separately

Calculate each part of the distributed expression separately:- \(50 \times 60 = 3000\)- \(50 \times 3 = 150\)

Step 4: Add the Results

Now, add the two products we calculated: \(3000 + 150\). This sums to 3150.

Key Concepts

Mathematics Fundamentals

Understanding the foundation of mathematics is crucial for solving all kinds of math problems efficiently. One of these foundation stones is the distributive property, a core concept in arithmetic operations. The distributive property is a mathematical rule that allows us to break down complex multiplication problems into simpler parts.

For instance, when faced with an operation like multiplying 50 by 63, instead of trying to do it all at once, we can make the numbers more manageable using this property. By expressing 63 as 60 + 3, we create a simple addition problem inside a larger multiplication problem.

This not only makes calculations easier but also helps in understanding the relations and structures within numbers, boosting mathematical intuition and problem-solving skills.

Multiplication Techniques

Multiplication is a fundamental math skill, and there are different techniques to make it easier, especially for large numbers. The distributive property is particularly helpful because it turns a single complex multiplication into a series of simpler ones.

When you multiply 50 by 63, using the distributive property means you multiply 50 by both parts of the additive split of 63, which are 60 and 3.

• First, do 50 multiplied by 60.
• Then separately compute 50 multiplied by 3.

These two simpler multiplications are easier to calculate than 50 times 63 in one go. Once you have the results, adding them gives you the final product. This technique simplifies multiplication of larger numbers and can be applied in various situations.

Arithmetic Operations

Arithmetic operations like addition, subtraction, multiplication, and division form the basis of most mathematical calculations. Applying the distributive property in a multiplication problem involves a clear understanding of these operations and how they work together.

The computation of 50 times 63 involves breaking down the multiplication through addition. After splitting 63 into 60 and 3, you multiply 50 with each.

Here's how it looks:

• Multiply 50 by 60 to get 3000.
• Then, multiply 50 by 3 to get 150.

Finally, add these two products: 3000 + 150, resulting in 3150.
This example shows how interconnected arithmetic operations are, and how understanding these connections makes solving math problems a straightforward task.