Problem 98 - Fundamentals of Mathematics

Question

Compute each product using the distributive property. \(60 \cdot 18\)

Step-by-Step Solution

Answer

The product of 60 and 18 is 1080.

Step 1: Break Down One Factor

We'll use the distributive property by breaking down one of the numbers into parts that are easy to multiply. Let's decompose 18 into 10 and 8. So, we have \(18 = 10 + 8\).

Step 2: Apply the Distributive Property

The distributive property states that \(a(b + c) = ab + ac\). Applying this to our problem, we have \(60 \times (10 + 8) = 60 \times 10 + 60 \times 8\).

Step 3: Multiply Each Part Separately

Now, calculate each of the products: \(60 \times 10 = 600\) and \(60 \times 8 = 480\).

Step 4: Add the Results

Finally, add the two results together: \(600 + 480 = 1080\).

Key Concepts

Multiplication

Multiplication is one of the basic arithmetic operations that involves calculating the total of one number added repeatedly. In simpler terms, multiplying means combining equal groups together to find out how many objects there are in all. This operation is a cornerstone of math and is frequently used in daily life.
For example, when multiplying 60 by 18, you are essentially adding 60, eighteen times. However, with larger numbers like these, direct multiplication can be complex, and that's where strategies like the distributive property come into play.
The distributive property can simplify multiplication by breaking numbers into smaller, easier-to-manage parts, making calculations more straightforward and less error-prone.

Arithmetic

Arithmetic is the branch of mathematics concerned with the study of numbers and the basic operations on them: addition, subtraction, multiplication, and division. It's the foundation of math that helps us perform simple calculations and solve various quantitative problems.
Multiplication and the distributive property are both key aspects of arithmetic. The distributive property is a useful rule when handling multiplication and addition together. It helps to simplify complex calculations by breaking one of the factors in a product into smaller numbers.
This makes arithmetic more intuitive by dividing a complex task into manageable parts, which is particularly useful when working with large numbers or solving problems mentally.

Step-by-Step Solution

A step-by-step solution is a methodical approach to solving a problem. It involves breaking down the problem into smaller, more manageable steps and solving each one sequentially. This approach helps in understanding the process and logic behind the solution.
In our example, to solve the multiplication of 60 and 18 using the distributive property, we followed a clear step-by-step solution:

• Step 1: Break down the number 18 into 10 and 8.
• Step 2: Apply the distributive property to rewrite the multiplication.
• Step 3: Multiply each part separately by 60.
• Step 4: Add the resulting products together to get the final answer, 1080.

By dissecting each part of the solution, learners can effectively grasp the application of the distributive property.