Problem 63 - Prealgebra

Question

Perform the indicated operations. $$75 \times 43,560$$

Step-by-Step Solution

Answer

The result of multiplying 75 by 43,560 is 3,267,000.

Step 1: Understand the problem

We need to perform the multiplication of two numbers, 75 and 43,560. The result will be the product of these two numbers.

Step 2: Multiply the numbers

To find the product of 75 and 43,560, we multiply them directly:\[75 \times 43,560\]

Step 3: Break down the multiplication

We can break down the multiplication into smaller, more manageable steps:First, multiply 75 by each digit in 43,560 starting from the right:- Multiply 75 by 0: \(75 \times 0 = 0\)- Multiply 75 by 6: \(75 \times 6 = 450\)- Multiply 75 by 5: \(75 \times 5 = 375\). Shift to the left by one place.- Multiply 75 by 3: \(75 \times 3 = 225\). Shift to the left by two places.- Multiply 75 by 4: \(75 \times 4 = 300\). Shift to the left by three places.

Step 4: Add the partial results

Now sum all of our partial results to get the final product:- Write the partial results with their shifts accounted for: - 0 - 450 - 3,750 (3750 due to one left shift) - 22,500 (22500 due to two left shifts) - 300,000 (300000 due to three left shifts)Add these numbers together:\[0 + 450 + 3,750 + 22,500 + 300,000 = 3,267,000\]

Step 5: Verify the calculation

Re-check each multiplication and addition to ensure the calculation is correct. In this case, revisiting all the steps shows our final result is accurate.

Key Concepts

Partial Products

When multiplying larger numbers, breaking the computation into smaller, more manageable pieces leads us to the concept of partial products. Each number in the multiplication is considered one of these pieces, allowing us to work step by step. By doing this, the multiplication becomes less intimidating and easier to handle. In the provided exercise, we multiplied 75 by each digit in 43,560 individually. This approach separates our complex calculations into simpler ones by focusing on:

• Zeroing each smaller product calculation with a specific weight based on its place value.
• Summing these calculated pieces to arrive at the final answer.

Another significant benefit of this method is error reduction, as re-checking smaller calculations is more manageable than reevaluating one large operation.

Long Multiplication

Long multiplication is a cornerstone method in arithmetic for dealing with larger numbers. It involves multiplying values in a columnar fashion - a structured way to keep track of each calculation clearly. This method allows multiple small multiplication tasks to be broken down and executed methodically:

• Write the numbers to be multiplied one above the other, carefully aligning the digits according to place value.
• Process each digit in the multiplicand individually, multiplying by the entire multiplier.
• Shift the products based on the digit’s place value, ensuring each row beneath aligns correctly.

The sum of all these smaller, shifted products gives the final answer. Practicing this exercise boosts numerical fluency and lays the groundwork for more advanced math concepts.

Place Value

The concept of place value is fundamental to understanding multiplication methods like long multiplication. In essence, it is the value each digit in a number holds, determined by its position. For each step of our multiplication:

• Zero is multiplied with 75, resulting directly in zero, causing no change.
• Each subsequent digit (e.g., 5, 6, 3, 4) contributes a value based on its position in 43,560.
• A one-place left shift is similar to multiplying a number by 10, representing the tens, hundreds, thousands, etc., positions.

By leveraging place value, we can ensure our intermediate products align correctly in computing our final result. Understanding place values reinforces comprehension of the number magnitude and enhances accuracy in arithmetic operations.

Arithmetic Operations

Arithmetic operations are the basic building blocks of mathematics. Multiplication, in particular, asks for careful handling due to its complexity compared to addition, subtraction, or division. Performing the multiplication:

• First, multiply each digit from the bottom number with every digit of the top number. Each of these is a basic arithmetic operation.
• Then, these results are added together, accruing into the final total.

By breaking down the task into core operations, students can better see how these components interact to form the final result. It’s helpful to regularly practice these skills individually to build a stronger overall mathematical foundation.