Step-by-Step Guide Multiplying 5971 By 49

In this comprehensive guide, we will walk you through the process of multiplying 5971 by 49. This step-by-step approach is designed to help you understand the underlying principles of multiplication and improve your arithmetic skills. Whether you are a student learning multiplication for the first time or someone looking to refresh your knowledge, this guide will provide you with a clear and concise method to solve this problem. We will break down the multiplication into manageable steps, ensuring that each step is thoroughly explained. By the end of this guide, you will not only be able to multiply 5971 by 49 but also apply these skills to similar multiplication problems.

Understanding Multiplication Basics

Before diving into the specific problem, let's review the basics of multiplication. Multiplication is a fundamental arithmetic operation that represents repeated addition. For example, 3 multiplied by 4 (written as 3 × 4) means adding 3 to itself 4 times (3 + 3 + 3 + 3), which equals 12. Understanding this basic concept is crucial for grasping more complex multiplication problems. In the context of our problem, multiplying 5971 by 49 means adding 5971 to itself 49 times. However, instead of performing repeated addition, we use a more efficient method that involves multiplying each digit of one number by each digit of the other number and then adding the results. This method, known as long multiplication, is what we will use in the step-by-step guide below.

Multiplication also has several properties that make calculations easier. The commutative property states that the order of the numbers being multiplied does not affect the result (e.g., 2 × 3 = 3 × 2). The associative property states that the grouping of numbers in a multiplication problem does not affect the result (e.g., (2 × 3) × 4 = 2 × (3 × 4)). The distributive property is particularly useful in long multiplication, as it allows us to break down the numbers into smaller parts and multiply them separately (e.g., 5 × (10 + 2) = (5 × 10) + (5 × 2)). These properties are essential tools in simplifying multiplication problems and ensuring accuracy in our calculations. In the following sections, we will apply these principles to solve the multiplication of 5971 by 49.

Step 1: Setting Up the Multiplication Problem

To begin, let's set up the multiplication problem correctly. Proper setup is crucial for accurate calculations. Write the two numbers, 5971 and 49, one above the other, aligning the digits by their place value (ones, tens, hundreds, etc.). It is conventional to write the number with more digits on top, but in this case, it doesn't significantly affect the process. Draw a horizontal line beneath the two numbers, which will separate the factors from the partial products and the final product. This setup helps organize the multiplication process and prevents errors. Ensure that each digit is clearly written and aligned in its respective column. A neat and organized setup minimizes the chances of making mistakes during the multiplication process.

Correctly aligning the numbers ensures that you are multiplying the correct place values together. For instance, the ones digit of 49 (which is 9) will first be multiplied by each digit of 5971, starting from the ones place. Then, the tens digit of 49 (which is 40) will be multiplied by each digit of 5971. The partial products obtained from these multiplications will then be added together to get the final product. If the numbers are not aligned correctly, you may end up multiplying the wrong place values, leading to an incorrect result. Therefore, take the time to set up the problem neatly and accurately before proceeding with the multiplication.

Step 2: Multiplying by the Ones Digit (9)

The next step involves multiplying the top number (5971) by the ones digit of the bottom number (9). Start by multiplying 9 by the ones digit of 5971, which is 1. 9 multiplied by 1 is 9. Write the 9 directly below the line in the ones place. Next, multiply 9 by the tens digit of 5971, which is 7. 9 multiplied by 7 is 63. Write down the 3 in the tens place below the line and carry over the 6 to the next column (hundreds place). Now, multiply 9 by the hundreds digit of 5971, which is 9. 9 multiplied by 9 is 81. Add the carried-over 6 to 81, which gives 87. Write down the 7 in the hundreds place and carry over the 8 to the next column (thousands place). Finally, multiply 9 by the thousands digit of 5971, which is 5. 9 multiplied by 5 is 45. Add the carried-over 8 to 45, which gives 53. Write down 53 to the left of the previously written digits. The partial product obtained from multiplying 5971 by 9 is 53739. This completes the first row of our multiplication.

This step is crucial because it sets the foundation for the rest of the multiplication process. Ensuring accuracy in this step is vital, as any error here will propagate through the remaining calculations. Double-check each multiplication and carry-over to minimize mistakes. The partial product 53739 represents 9 times 5971, and it will be added to the partial product obtained from multiplying 5971 by the tens digit (40) in the next step. This method of breaking down the multiplication into smaller, manageable steps is the essence of long multiplication and makes it easier to handle large numbers.

Step 3: Multiplying by the Tens Digit (40)

Now, let's multiply 5971 by the tens digit of 49, which is 40. Remember that multiplying by 40 is the same as multiplying by 4 and then multiplying by 10. To account for this, we add a zero as a placeholder in the ones place of the second partial product. This shifts the result one place to the left, effectively multiplying by 10. Next, multiply 4 by the ones digit of 5971, which is 1. 4 multiplied by 1 is 4. Write the 4 in the tens place below the line (next to the placeholder zero). Then, multiply 4 by the tens digit of 5971, which is 7. 4 multiplied by 7 is 28. Write down the 8 in the hundreds place and carry over the 2 to the next column (thousands place). Next, multiply 4 by the hundreds digit of 5971, which is 9. 4 multiplied by 9 is 36. Add the carried-over 2 to 36, which gives 38. Write down the 8 in the thousands place and carry over the 3 to the next column (ten-thousands place). Finally, multiply 4 by the thousands digit of 5971, which is 5. 4 multiplied by 5 is 20. Add the carried-over 3 to 20, which gives 23. Write down 23 to the left of the previously written digits. The partial product obtained from multiplying 5971 by 40 is 238840. This completes the second row of our multiplication.

Multiplying by the tens digit requires careful attention to place values and carry-overs. The placeholder zero is crucial because it ensures that we are multiplying by 40 and not just 4. The partial product 238840 represents 40 times 5971. This partial product, along with the partial product from the previous step (53739), will be added together to find the final product. Accuracy in this step is essential for the overall correctness of the multiplication. Double-checking each multiplication and carry-over will help prevent errors and ensure that the final result is accurate. The process of multiplying by the tens digit follows the same principles as multiplying by the ones digit, but it involves an additional step of shifting the result one place to the left due to the tens place value.

Step 4: Adding the Partial Products

The final step is to add the partial products obtained in the previous steps. We have two partial products: 53739 (from multiplying 5971 by 9) and 238840 (from multiplying 5971 by 40). Write these two numbers one below the other, aligning the digits by their place value. Draw a horizontal line below the partial products, which will separate them from the final product. Now, add the digits in each column, starting from the ones place. In the ones place, we have 9 + 0 = 9. Write 9 in the ones place below the line. In the tens place, we have 3 + 4 = 7. Write 7 in the tens place. In the hundreds place, we have 7 + 8 = 15. Write 5 in the hundreds place and carry over the 1 to the next column (thousands place). In the thousands place, we have 3 + 8 + 1 (carried over) = 12. Write 2 in the thousands place and carry over the 1 to the next column (ten-thousands place). In the ten-thousands place, we have 5 + 3 + 1 (carried over) = 9. Write 9 in the ten-thousands place. Finally, in the hundred-thousands place, we have 2. Write 2 in the hundred-thousands place. The final product is 292579.

Adding the partial products correctly is crucial for obtaining the correct final result. Ensure that the digits are aligned properly to avoid errors in addition. The process involves adding the digits in each column and carrying over any excess to the next column. This step combines the results of the individual multiplications into a single product, representing the result of multiplying 5971 by 49. Double-checking the addition process will help ensure the accuracy of the final product. The sum of the partial products, 292579, represents the total number obtained when 5971 is multiplied by 49. This final step brings together all the previous calculations to provide the solution to the multiplication problem.

Conclusion

In conclusion, multiplying 5971 by 49 involves a step-by-step process that includes setting up the problem, multiplying by the ones digit, multiplying by the tens digit, and adding the partial products. By following these steps carefully, we have determined that 5971 multiplied by 49 equals 292579. This method, known as long multiplication, is a fundamental arithmetic skill that is essential for solving more complex mathematical problems. Understanding each step and practicing regularly will improve your multiplication skills and overall mathematical proficiency. Multiplication is not just a mathematical operation; it is a tool that is used in various real-life situations, from calculating expenses to determining measurements. Therefore, mastering multiplication is a valuable skill that will benefit you in many aspects of life.

This step-by-step guide has provided a clear and concise method for multiplying 5971 by 49. By breaking down the problem into smaller, manageable steps, we have made the process easier to understand and execute. Remember to practice these steps with different numbers to reinforce your understanding and improve your speed and accuracy. Multiplication is a skill that improves with practice, so the more you work on it, the better you will become. This guide serves as a valuable resource for anyone looking to enhance their multiplication skills and tackle more challenging mathematical problems with confidence.