Calculating 107.4508 Divided By 1.16 A Step-by-Step Guide

In this article, we will dive into the process of calculating the division problem $107.4508 \div 1.16$. This is a fundamental arithmetic operation, and we will break it down step by step to ensure clarity and understanding. Whether you are a student learning the basics or someone looking to refresh your math skills, this guide will provide a comprehensive explanation. We will cover the essential steps involved in performing long division with decimals, including setting up the problem, handling the decimal points, and arriving at the final answer. By the end of this article, you will have a solid grasp of how to divide decimals accurately and efficiently.

When tackling a division problem like $107.4508 \div 1.16$, it is crucial to understand what the problem is asking. In simple terms, we want to find out how many times 1.16 fits into 107.4508. This involves dividing the dividend (107.4508) by the divisor (1.16). The result of this division is known as the quotient, which will be our final answer. To effectively solve this, we need to employ the principles of long division, especially as it applies to decimal numbers. Understanding the place values of the digits is also essential. In the dividend, we have hundreds, tens, units, tenths, hundredths, thousandths, and ten-thousandths places. Similarly, in the divisor, we have units and hundredths places. The presence of decimals requires careful alignment and handling to ensure an accurate result. Before we begin the long division process, we will address how to deal with the decimal in the divisor, which is a critical step in simplifying the calculation. By thoroughly understanding the problem and the numbers involved, we lay a solid foundation for performing the division accurately.

Before we begin the long division process for $107.4508 \div 1.16$, we need to address the decimal in the divisor. Dividing by a decimal can be tricky, so the first step is to convert the divisor into a whole number. We achieve this by multiplying both the divisor and the dividend by the same power of 10. In this case, the divisor 1.16 has two decimal places, so we multiply both 1.16 and 107.4508 by 100. This gives us a new problem: $10745.08 \div 116$. Multiplying both numbers by 100 does not change the value of the quotient, as it is equivalent to multiplying the entire fraction by $\frac{100}{100}$, which equals 1. Now that our divisor is a whole number, we can proceed with the long division method more easily. This preparation step is crucial because it simplifies the division process and reduces the chances of making errors with decimal placements. After converting the divisor to a whole number, the problem is set up in the long division format, ready for the next steps of dividing, multiplying, subtracting, and bringing down the digits.

Now, let’s walk through the step-by-step long division process for $10745.08 \div 116$. First, set up the long division problem with 10745.08 as the dividend inside the division bracket and 116 as the divisor outside. We start by dividing the first few digits of the dividend (1074) by the divisor (116). Since 116 does not go into 107, we consider 1074. We estimate how many times 116 goes into 1074. A reasonable estimate is 9, because $116 \times 9 = 1044$. Write 9 above the 4 in 1074. Next, we multiply 9 by 116, which gives us 1044. We subtract 1044 from 1074, resulting in 30. Bring down the next digit from the dividend, which is 5, making our new number 305. Now, divide 305 by 116. It goes 2 times, as $116 \times 2 = 232$. Write 2 next to the 9 above the division bracket. Subtract 232 from 305, which gives us 73. Bring down the next digit, which is 0, making our new number 730. Divide 730 by 116. It goes 6 times, as $116 \times 6 = 696$. Write 6 next to the 2 above the division bracket. Subtract 696 from 730, resulting in 34. Bring down the last digit, which is 8, making our new number 348. Divide 348 by 116. It goes exactly 3 times, as $116 \times 3 = 348$. Write 3 next to the 6 above the division bracket. Subtract 348 from 348, which gives us 0. Since we have reached 0, the division is complete. The quotient we obtained is 92.63. This step-by-step approach ensures accuracy and helps in understanding the mechanics of long division with decimals.

In the long division process of $107.4508 \div 1.16$, handling the decimal point correctly is crucial for obtaining the accurate quotient. After multiplying both the dividend and the divisor by 100, we transformed the problem into $10745.08 \div 116$. When performing long division, we align the decimal point in the quotient directly above the decimal point in the dividend. As we go through the division steps, we bring down digits after the decimal point as needed. In our example, after dividing 10745 by 116, we reached a remainder and then brought down the 0 from the tenths place in the dividend. This meant that the decimal point in our quotient would be placed after the 2 (from 92). We continued the division process, bringing down the 8 from the hundredths place and completing the division. The correct placement of the decimal point ensures that the quotient reflects the true value of the division. By carefully tracking the decimal point throughout the long division process, we avoid errors and arrive at the accurate answer. This careful attention to detail is essential in decimal arithmetic.

To ensure the accuracy of our result for $107.4508 \div 1.16$, it's essential to verify the answer. We found the quotient to be 92.63. To verify this, we can multiply the quotient (92.63) by the original divisor (1.16) and see if it equals the original dividend (107.4508). Performing the multiplication: $92.63 \times 1.16 = 107.4508$. Since the result matches the original dividend, our division is correct. This verification step is a critical practice in mathematics to confirm the accuracy of calculations. It provides confidence in the solution and helps identify any potential errors. In summary, after performing the long division and verifying the result, we can confidently state that: $107.4508 \div 1.16 = 92.63$. This is our final answer, expressed as a decimal, which accurately represents the quotient of the given division problem. The process of verification reinforces the understanding of the relationship between division and multiplication, further solidifying our grasp on these arithmetic operations.

In conclusion, we have successfully calculated $107.4508 \div 1.16$ and found the answer to be 92.63. We began by understanding the problem, preparing for long division by removing the decimal from the divisor, and then performing the long division step by step. Key to the process was managing the decimal point accurately to ensure the correct placement in the quotient. Finally, we verified our answer by multiplying the quotient by the original divisor, confirming the result. This exercise demonstrates the importance of following a systematic approach in mathematics, especially when dealing with decimals. Long division is a fundamental skill, and mastering it provides a solid foundation for more complex calculations. By breaking down the problem into manageable steps, we can tackle division problems confidently and accurately. This guide has provided a clear and detailed explanation, making the process accessible for learners of all levels. Understanding and practicing these steps will undoubtedly improve your arithmetic skills and mathematical proficiency.