Problem 31 - Fundamentals of Mathematics

Question

Use the distributive property to compute each product. \(65 \cdot 40\)

Step-by-Step Solution

Answer

The product of 65 and 40 is 2600.

Step 1: Understand the Distributive Property

The distributive property states that for any numbers \(a\), \(b\), and \(c\), \(a(b + c) = ab + ac\). This property can be utilized to break down more complicated multiplication problems into simpler steps.

Step 2: Decompose One Factor

Decompose the number 40 into a sum of two easier numbers, for example, \(40 = 30 + 10\). This allows the use of the distributive property: \(65 \cdot 40 = 65 \cdot (30 + 10)\).

Step 3: Apply the Distributive Property

Apply the distributive property to the expression: \(65 \cdot (30 + 10) = (65 \cdot 30) + (65 \cdot 10)\).

Step 4: Calculate Each Product Separately

First, calculate \(65 \cdot 30\). This is equal to \(65 \cdot 3 \cdot 10 = 65 \cdot 3 \cdot 10 = 1950\).Then calculate \(65 \cdot 10\), which is simply \(650\).

Step 5: Add the Results

Add the results from the previous step: \((65 \cdot 30) + (65 \cdot 10) = 1950 + 650 = 2600\).

Key Concepts

Decomposition of Numbers

Have you ever tried splitting up large numbers to make calculations easier? This is exactly what decomposition of numbers means. It involves breaking down a number into two or more simpler numbers that are easier to work with.
For instance, when given a number like 40, you can express it as the sum of 30 and 10. This is simple but very useful when solving multiplication problems.

• This is particularly handy because it allows us to use the distributive property effectively.
• You can choose any combination of numbers that add up to the original number.

Try practicing this by decomposing different numbers. This builds your confidence in handling math problems efficiently.

Multiplication Strategies

Multiplication can sometimes feel challenging, but with good strategies, it becomes much easier. One popular strategy is using the distributive property.
This property helps break down complex multiplication into simpler chunks. Say you have to multiply 65 by 40. Start by decomposing 40 into smaller parts, such as 30 and 10.

• Then, multiply 65 by each decomposed part separately.
• So, calculate both 65 times 30 and 65 times 10.

Once you have both products, add them together to get the final result. This strategy simplifies the calculation process and reduces errors.

Basic Arithmetic

At the heart of it all, math revolves around basic arithmetic principles like addition, subtraction, multiplication, and division. These are foundational skills that help us solve more complex problems.
In our exercise, we used basic multiplication and addition to solve what initially seemed like a difficult problem.

• We relied on breaking down large numbers to make smaller, manageable problems.
• We performed separate multiplications and then used addition to get the final result.

Strengthening your understanding of basic arithmetic paves the way for mastering advanced math concepts and operations.