Problem 97 - Fundamentals of Mathematics

Question

Compute each product using the distributive property. \(30 \cdot 14\)

Step-by-Step Solution

Answer

The product of 30 and 14 is 420.

Step 1: Express Each Number in the Product

Break down each number in the calculation into smaller, more manageable numbers. We can express 30 as 30 = 3 × 10 and keep 14 as it is.

Step 2: Apply the Distributive Property

The distributive property states that \(a(b+c) = ab + ac\). We can use this property by rewriting 30 as \(10 + 20\). So we can consider: \[30 imes 14 = (10 + 20) imes 14 = 10 imes 14 + 20 imes 14\]

Step 3: Calculate Each Multiplication Separately

Calculate \(10 imes 14\): \(10 imes 14 = 140\).Next, calculate \(20 imes 14\): \(20 imes 14 = 280\).

Step 4: Add the Products Together

Add the results from the separate calculations:\(140 + 280 = 420\)

Key Concepts

Multiplication

Multiplication is one of the fundamental operations in mathematics, allowing us to find the total of equal groups of items. It is a form of repeated addition. For example, to multiply 3 by 4, you add 3 four times: 3 + 3 + 3 + 3 = 12.
This operation is crucial because it comes up in everyday situations, like calculating total cost, area, or any situation where the same amount is repeated.

• Key Principle: Multiplication is commutative, meaning the order of numbers does not matter; thus, 3 × 4 equals 4 × 3.
• Application: Helps in efficiently solving problems compared to using repeated addition.

When approaching larger numbers, techniques like the distributive property make it manageable, as we'll explore further.

Break Down Numbers

Breaking down numbers simplifies complex multiplication problems. This process involves splitting one or both factors into smaller, easier-to-handle components, which can then be individually multiplied and combined.
This approach transforms difficult problems into ones where basic multiplication facts can be applied. For instance, in the problem of multiplying 30 with 14,

• Breakdown: 30 is expressed as 10 + 20.
• Concept: Simplifies calculations by reducing them into two smaller, more manageable parts.

By breaking it down, you can leverage basic math skills rather than working with more challenging numbers directly. This method is especially helpful in mental math.

Calculate Separately

Calculating separately involves solving each part of the equation as broken down earlier. After breaking down the number 30 into 10 and 20, you multiply each separate piece by 14.

• Example: Calculate \(10 \times 14\) to get 140.
• Next, calculate \(20 \times 14\) to get 280.

Each multiplication is simpler and easier to handle on its own.
This step-by-step approach ensures accuracy, as you're dealing with smaller, more manageable numbers first. This way, you reduce the risk of errors that might occur when multiplying large numbers directly.

Add Products

After calculating each multiplication separately, the final step is to add those products together. This is where you combine the separate results to get the final answer.

• For Example: After calculating \(10 \times 14 = 140\) and \(20 \times 14 = 280\), you sum them: \(140 + 280\).

Adding these products ensures all components of the original multiplication are accounted for.
This final addition step solidifies the total value, verifying the accuracy of your multiplication and problem-solving process. When everything is added correctly, you achieve confidence in the solution. In our example, the sum equals 420, confirming the original product.