Problem 68 - Fundamentals of Mathematics

Question

Find the product. \(4,271 \times 630\).

Step-by-Step Solution

Answer

2680730

Step 1: Set Up the Problem

Write the numbers to be multiplied in a vertical column, with the larger (or complex) number on top and the smaller number below it, aligned to the right. Add the multiplication symbol: \[\begin{array}{r} 4271 \\times 630 \\hline\end{array}\]

Step 2: Multiply by the Units Digit

Multiply each digit of the number 4271 by the unit's digit of the number 630 (which is 0). This means multiplying 4271 by 0, resulting in a row of zeroes:\[0\]

Step 3: Multiply by the Tens Digit

Now multiply 4271 by the tens digit of 630 (which is 3). Remember to add a zero at the end of this row because we are actually multiplying by 30:\[\begin{array}{r} 12813 \ ext{(append a zero)}\end{array}\] \[128130\]

Step 4: Multiply by the Hundreds Digit

Multiply 4271 by the hundreds digit of 630 (which is 6). Remember to add two zeros at the end of this row because we are actually multiplying by 600:\[\begin{array}{r} 25626 \ ext{(append two zeros)}\end{array}\] \[2562600\]

Step 5: Add the Rows

Now add all the rows (partial products) together to get the final answer:\[\begin{array}{r} ext{ 0000} \ ext{128130} \\underline{2562600} \\end{array}\]Adding these results together, you get:\[2680730\]

Step 6: Verify the Calculation

To ensure the calculation is correct, recheck each step or use a calculator to verify the final product is indeed 2,680,730.

Key Concepts

Partial Products

In long multiplication, the term "partial products" refers to the intermediate results obtained when each digit of one number is multiplied by each digit of another. This process breaks a complex multiplication problem into smaller, more manageable pieces. For instance, in the multiplication of 4,271 by 630, we derive three partial products:

• First, 4,271 is multiplied by 0, which is the unit digit of 630, giving the partial product: 0.
• Next, 4,271 is multiplied by 30 (representing the tens digit, 3, with a zero appended), resulting in: 128,130.
• Finally, 4,271 is multiplied by 600 (representing the hundreds digit, 6, with two zeros appended), giving: 2,562,600.

Each partial product corresponds to one stage of the multiplication process, which is then summed to find the total product. This method simplifies handling big numbers.

Multiplication Steps

Understanding the steps of multiplication is essential for avoiding mistakes and for grasping the logic behind long multiplication. It is a systematic approach: Start by arranging the numbers vertically. Place the number with more digits on top. Then, multiply each digit of the lower number by the entire upper number:

• Multiply the entire top number by the unit digit of the bottom number.
• Next, multiply the whole upper number by the tens digit of the bottom number, placing a zero at the end of this row since you're actually multiplying by ten.
• Repeat the process for the hundreds digit, two zeros are added at the end since it denotes hundreds.

Finally, sum the intermediate results (partial products) to get the overall multiplication result. Each step builds upon the last, creating the overall answer.

Vertical Multiplication Setup

Setting up the multiplication problem vertically is crucial for clarity and success in solving it. This method involves: Writing the two numbers being multiplied as columns. Align these numbers to the right to maintain proper place value. For example, when setting up 4,271 multiplied by 630:

• Write 4,271 above the line since it has more digits, making calculations clearer.
• Place 630 directly underneath it, with the multiplication symbol to the left.

This setup allows for easy alignment when writing the partial products. Each new row of results, corresponding to a specific digit of the second multiplier, remains under the appropriate column, maintaining proper spacing and aiding in summation.

Verify Multiplication Result

Verifying your result is an essential part of any multiplication process, ensuring accuracy. Double-checking your work can prevent errors. To verify the calculated product of 2,680,730 from multiplying 4,271 by 630:

• Check each individual multiplication step for errors. Make sure all arithmetic is correct and each partial product includes the correct number of zeros.
• Recalculate using a calculator to match the obtained product with your manual calculation.

If everything aligns, you can confidently state that the product is accurate. Verification not only prevents mistakes but also strengthens your understanding of multiplication processes.