Problem 30 - Fundamentals of Mathematics

Question

Use the distributive property to compute each product. \(95 \cdot 12\)

Step-by-Step Solution

Answer

The product is 1140.

Step 1: Break Down the Numbers

To make the multiplication easier, express 95 as a product of sums: \(95 = 90 + 5\). This allows us to use the distributive property easily.

Step 2: Apply Distributive Property

Apply the distributive property to \(95 \cdot 12\) as follows: \((90 + 5) \cdot 12 = 90 \cdot 12 + 5 \cdot 12\). This breaks the calculation into simpler parts.

Step 3: Multiply Each Term Separately

Calculate each part separately: \(90 \cdot 12 = 1080\) and \(5 \cdot 12 = 60\).

Step 4: Add the Products

Sum the results of the separate multiplications: \(1080 + 60 = 1140\).

Key Concepts

Multiplication

Multiplication is a fundamental mathematical operation. It allows us to find the total number of objects in a set of equal groups. When you multiply, you are essentially adding the same number repeatedly. For example, multiplying 3 by 4 is the same as adding 3 four times:

• \( 3 + 3 + 3 + 3 = 12 \)
• \( 3 \times 4 = 12 \)

This makes multiplication a faster and more efficient method for combining numbers. In the given exercise, we need to multiply 95 by 12. Performing this directly could be complex, so we can use strategies like the distributive property to make multiplication easier.

Breaking Down Numbers

Breaking down numbers is a helpful strategy in simplifying multiplication problems. It involves expressing a number as a sum of more manageable parts. For our exercise, breaking down 95 as \( 90 + 5 \) allows us to handle more straightforward calculations.

When numbers are broken down, it's easier to apply the distributive property, as with our original problem where 95 becomes \( 90 + 5 \). This decomposition lets us multiply each component separately. This step is crucial for simplifying the calculation process while maintaining accuracy. It’s much easier to compute calculations with round numbers like 90 or 10.

Simplifying Calculations

The goal of simplifying calculations is to make complex problems more manageable by breaking them into smaller, easy-to-handle parts. By using the distributive property, we convert a tough multiplication problem into a series of smaller, straightforward multiplications.

With the exercise example, once you break down 95 into \( 90 + 5 \) and apply the distributive property, you end up with two simpler multiplications: \( 90 \times 12 \) and \( 5 \times 12 \).

The distributive property allows:

• \( 90 \times 12 = 1080 \)
• \( 5 \times 12 = 60 \)

You then add these results to get the final answer: \( 1080 + 60 = 1140 \). Simplifying calculations through these methods not only ensures accuracy but also saves time and reduces errors in solving mathematical problems.