Mastering Division Finding Quotient And Remainder With Division Algorithm Verification

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Q. 10) Divide and determine the quotient and remainder, and verify your answer using the Division Algorithm for the following: a) 4178 ÷ 35 b) 8391 ÷ 67 c) 36195 ÷ 153 d) 93995 ÷ 425 e) 63025 ÷ 647 f) 16135 ÷ 875 g) 6971 ÷ 47 h) 6754 ÷ 56

In this comprehensive guide, we will delve into the fundamental concepts of division, focusing on calculating the quotient and remainder. We'll explore how to perform division operations effectively and, importantly, how to verify the accuracy of our results using the Division Algorithm. This article is designed to provide a step-by-step understanding of the division process, making it accessible for learners of all levels. Whether you're a student looking to solidify your understanding or someone seeking a refresher on basic arithmetic, this guide will equip you with the knowledge and skills you need. Each problem is solved in detail, with a clear explanation of each step involved. Additionally, we will emphasize the importance of the Division Algorithm as a tool for checking the correctness of your solutions. Let’s embark on this mathematical journey and conquer the world of division together.

a) 4178 ÷ 35

Let's begin with our first division problem: 4178 ÷ 35. Our goal is to find out how many times 35 fits into 4178, which will give us the quotient, and what's left over, which is the remainder. The process of long division involves breaking down the problem into smaller, manageable steps. We start by looking at the first few digits of the dividend (4178) and comparing them to the divisor (35). In this case, we see that 35 goes into 41 once. So, we write '1' as the first digit of our quotient. Next, we multiply the divisor (35) by the quotient digit (1), which gives us 35. We subtract this from 41, resulting in 6. Now, we bring down the next digit from the dividend, which is 7, making our new number 67. We then repeat the process: how many times does 35 go into 67? It goes in once, so we write another '1' in the quotient. We multiply 35 by 1 again, get 35, and subtract it from 67, leaving us with 32. We bring down the last digit, 8, giving us 328. Now, we determine how many times 35 goes into 328. Through estimation or trial and error, we find that 35 goes into 328 nine times (35 * 9 = 315). We write '9' in the quotient, multiply 35 by 9, and subtract the result (315) from 328, leaving us with 13. This 13 is our remainder, as it is less than the divisor (35), and we can't divide further. So, the quotient is 119, and the remainder is 13. To verify our answer, we use the Division Algorithm: Dividend = (Divisor × Quotient) + Remainder. Plugging in our values, we get 4178 = (35 × 119) + 13. Calculating the right side, we have 4165 + 13, which equals 4178. This confirms that our division is correct. Understanding and applying the Division Algorithm is crucial for ensuring accuracy in division problems. It acts as a safety net, allowing us to catch any errors we might have made during the division process. In real-world scenarios, accurate division is essential in various fields, from finance and accounting to engineering and scientific research. The ability to confidently perform division and verify the results can save time and prevent costly mistakes. Furthermore, mastering division lays the groundwork for more advanced mathematical concepts, such as algebra and calculus. It is a foundational skill that underpins much of mathematical problem-solving. By understanding the relationship between the dividend, divisor, quotient, and remainder, and how they fit together in the Division Algorithm, we gain a deeper appreciation for the structure and logic of mathematics. This understanding not only helps us solve division problems but also enhances our overall mathematical reasoning and problem-solving abilities.

Answer:

  • Quotient = 119
  • Remainder = 13

b) 8391 ÷ 67

Next, let's tackle the division problem 8391 ÷ 67. Similar to the previous example, we aim to determine the quotient and remainder. We start by examining the dividend (8391) and the divisor (67). We ask ourselves, how many times does 67 fit into the first two digits of 8391, which are 83? We find that 67 goes into 83 once. So, we write '1' as the first digit of our quotient. We then multiply 67 by 1, which equals 67, and subtract it from 83, resulting in 16. We bring down the next digit from the dividend, which is 9, making our new number 169. Now, we determine how many times 67 goes into 169. Through estimation or multiplication, we find that 67 goes into 169 twice (67 * 2 = 134). We write '2' in the quotient and subtract 134 from 169, leaving us with 35. We bring down the last digit, 1, giving us 351. We now need to figure out how many times 67 goes into 351. After some calculations, we find that 67 goes into 351 five times (67 * 5 = 335). We write '5' in the quotient, multiply 67 by 5, and subtract the result (335) from 351, leaving us with 16. This 16 is our remainder, as it is smaller than the divisor (67). Therefore, the quotient is 125, and the remainder is 16. To verify our solution, we apply the Division Algorithm: Dividend = (Divisor × Quotient) + Remainder. Substituting the values, we get 8391 = (67 × 125) + 16. Calculating the right side, we have 8375 + 16, which equals 8391. This confirms the accuracy of our division. The Division Algorithm is a cornerstone of arithmetic, providing a systematic way to check the correctness of division calculations. It reinforces the understanding of the relationship between the dividend, divisor, quotient, and remainder. This algorithm is not just a mathematical formula; it's a concept that extends into various areas of mathematics and real-life applications. In programming, for instance, the concept of division and remainders is used in algorithms for tasks like data sorting and encryption. In everyday life, we use division to split costs among friends, calculate the time it takes to travel a certain distance, or determine how many items we can buy with a given amount of money. Mastering division, therefore, is not just about solving mathematical problems; it's about developing a fundamental skill that is applicable in numerous contexts. Furthermore, understanding the Division Algorithm helps to build a deeper understanding of number theory, which is a branch of mathematics that deals with the properties and relationships of numbers. Concepts such as prime numbers, factors, and multiples are all related to division and the Division Algorithm. This understanding can lead to a greater appreciation for the beauty and complexity of mathematics.

Answer:

  • Quotient = 125
  • Remainder = 16

c) 36195 ÷ 153

Now, let's dive into the division of 36195 by 153. This problem, like the others, requires us to find the quotient and remainder. The process is the same, but the numbers are larger, which can make the estimation step a bit more challenging. We begin by looking at the dividend (36195) and the divisor (153). We ask ourselves, how many times does 153 go into the first three digits of 36195, which are 361? We estimate that 153 goes into 361 twice (153 * 2 = 306). So, we write '2' as the first digit of our quotient. We multiply 153 by 2, which equals 306, and subtract it from 361, resulting in 55. We bring down the next digit from the dividend, which is 9, making our new number 559. Now, we need to determine how many times 153 goes into 559. After some trial and error, we find that 153 goes into 559 three times (153 * 3 = 459). We write '3' in the quotient and subtract 459 from 559, leaving us with 100. We bring down the last digit, 5, giving us 1005. Next, we figure out how many times 153 goes into 1005. We find that 153 goes into 1005 six times (153 * 6 = 918). We write '6' in the quotient, multiply 153 by 6, and subtract the result (918) from 1005, leaving us with 87. This 87 is our remainder, as it is smaller than the divisor (153). Therefore, the quotient is 236, and the remainder is 87. To ensure the accuracy of our solution, we apply the Division Algorithm: Dividend = (Divisor × Quotient) + Remainder. Substituting the values, we get 36195 = (153 × 236) + 87. Calculating the right side, we have 36108 + 87, which equals 36195. This confirms that our division is correct. Division problems involving larger numbers like this one highlight the importance of estimation skills and a systematic approach. The process of long division can seem daunting at first, but by breaking it down into smaller steps and using estimation to make educated guesses, we can simplify the task. Moreover, the Division Algorithm becomes even more critical when dealing with larger numbers, as it provides a reliable way to check our work and avoid errors. In practical applications, division with larger numbers is commonly encountered in scenarios such as resource allocation, data analysis, and financial calculations. For example, businesses might need to divide their annual budget among different departments, or scientists might need to divide a large dataset into smaller subsets for analysis. Understanding how to perform these divisions accurately and efficiently is essential for making informed decisions. Furthermore, the concepts of quotient and remainder are not limited to whole numbers. They can also be extended to fractions and decimals, which are used extensively in various fields such as engineering, physics, and economics. A solid understanding of division with whole numbers provides a strong foundation for working with these more complex number systems.

Answer:

  • Quotient = 236
  • Remainder = 87

d) 93995 ÷ 425

Let's proceed to the division problem 93995 ÷ 425. As with the previous examples, we are tasked with finding the quotient and remainder. This problem involves dividing a five-digit number by a three-digit number, which requires careful attention to the steps of long division. We begin by considering the dividend (93995) and the divisor (425). We start by examining the first three digits of the dividend, which are 939. We ask ourselves, how many times does 425 go into 939? We estimate that 425 goes into 939 twice (425 * 2 = 850). So, we write '2' as the first digit of our quotient. We multiply 425 by 2, which equals 850, and subtract it from 939, resulting in 89. We bring down the next digit from the dividend, which is 9, making our new number 899. Now, we determine how many times 425 goes into 899. We find that 425 goes into 899 twice (425 * 2 = 850). We write '2' in the quotient and subtract 850 from 899, leaving us with 49. We bring down the last digit, 5, giving us 495. Next, we figure out how many times 425 goes into 495. We find that 425 goes into 495 once (425 * 1 = 425). We write '1' in the quotient, multiply 425 by 1, and subtract the result (425) from 495, leaving us with 70. This 70 is our remainder, as it is smaller than the divisor (425). Therefore, the quotient is 221, and the remainder is 70. To verify our solution, we apply the Division Algorithm: Dividend = (Divisor × Quotient) + Remainder. Substituting the values, we get 93995 = (425 × 221) + 70. Calculating the right side, we have 93925 + 70, which equals 93995. This confirms that our division is correct. This problem reinforces the importance of careful estimation and step-by-step execution in long division. Dividing larger numbers can be a complex process, and it's crucial to be methodical and accurate in each step. The Division Algorithm serves as a valuable tool for checking the correctness of our work, ensuring that we have arrived at the correct quotient and remainder. In real-world scenarios, division problems of this scale might arise in situations such as calculating unit costs, determining the number of items that can be produced with a given budget, or analyzing statistical data. For instance, a manufacturing company might need to divide its total production cost by the number of units produced to determine the cost per unit. Similarly, a researcher might need to divide a large dataset into smaller groups for analysis. The ability to perform these divisions accurately and efficiently is essential for effective decision-making. Furthermore, understanding the concepts of quotient and remainder is crucial for understanding other mathematical concepts, such as fractions, decimals, and percentages. These concepts are widely used in everyday life, from calculating discounts at the store to understanding interest rates on loans. A solid foundation in division is therefore essential for developing overall numeracy skills and the ability to solve practical problems.

Answer:

  • Quotient = 221
  • Remainder = 70

e) 63025 ÷ 647

Now, let's tackle the division problem 63025 ÷ 647. Our objective remains the same: to determine the quotient and remainder. This problem presents another opportunity to practice long division with larger numbers, reinforcing our understanding of the process. We begin by considering the dividend (63025) and the divisor (647). We start by examining the first three digits of the dividend, which are 630. We ask ourselves, how many times does 647 go into 630? We see that 647 is larger than 630, so it goes in zero times. We move on to the first four digits, 6302. Now, we ask ourselves, how many times does 647 go into 6302? This requires a bit more estimation. We might try multiplying 647 by different numbers until we get a result close to 6302. We find that 647 goes into 6302 nine times (647 * 9 = 5823). So, we write '9' as the first digit of our quotient. We multiply 647 by 9, which equals 5823, and subtract it from 6302, resulting in 479. We bring down the next digit from the dividend, which is 5, making our new number 4795. Now, we determine how many times 647 goes into 4795. Again, we need to estimate. We might try multiplying 647 by different numbers. We find that 647 goes into 4795 seven times (647 * 7 = 4529). We write '7' in the quotient and subtract 4529 from 4795, leaving us with 266. This 266 is our remainder, as it is smaller than the divisor (647). Therefore, the quotient is 97, and the remainder is 266. To verify our solution, we apply the Division Algorithm: Dividend = (Divisor × Quotient) + Remainder. Substituting the values, we get 63025 = (647 × 97) + 266. Calculating the right side, we have 62759 + 266, which equals 63025. This confirms that our division is correct. This problem highlights the importance of careful estimation and the iterative nature of long division. Sometimes, we need to try different multipliers before we find the one that gives us the closest result without exceeding the current portion of the dividend. The Division Algorithm is particularly valuable in these situations, as it provides a reliable way to check our work and ensure that we haven't made any errors in our estimations or subtractions. In practical applications, division problems of this scale might arise in situations such as inventory management, resource planning, or data analysis. For example, a warehouse manager might need to divide the total number of items in stock by the number of shelves to determine how many items can be placed on each shelf. Similarly, a data analyst might need to divide a large dataset into smaller subsets for analysis. The ability to perform these divisions accurately and efficiently is essential for effective decision-making. Furthermore, this problem reinforces the understanding of the relationship between the divisor, dividend, quotient, and remainder. By working through these problems, we develop a deeper understanding of how these elements interact and how the Division Algorithm provides a framework for verifying our calculations. This understanding is essential for building a solid foundation in arithmetic and preparing for more advanced mathematical concepts.

Answer:

  • Quotient = 97
  • Remainder = 266

f) 16135 ÷ 875

Let's move on to the division problem 16135 ÷ 875. Our goal, as always, is to find the quotient and remainder. This problem provides further practice in long division, particularly with larger divisors and dividends. We begin by considering the dividend (16135) and the divisor (875). We start by examining the first three digits of the dividend, which are 161. We see that 875 is larger than 161, so it goes in zero times. We move on to the first four digits, 1613. Now, we ask ourselves, how many times does 875 go into 1613? This requires careful estimation. We might try multiplying 875 by different numbers. We find that 875 goes into 1613 once (875 * 1 = 875). So, we write '1' as the first digit of our quotient. We multiply 875 by 1, which equals 875, and subtract it from 1613, resulting in 738. We bring down the next digit from the dividend, which is 5, making our new number 7385. Now, we determine how many times 875 goes into 7385. This requires more estimation and potentially some trial and error. We might try multiplying 875 by different numbers. We find that 875 goes into 7385 eight times (875 * 8 = 7000). So, we write '8' in the quotient. We multiply 875 by 8, which equals 7000, and subtract it from 7385, leaving us with 385. This 385 is our remainder, as it is smaller than the divisor (875). Therefore, the quotient is 18, and the remainder is 385. To verify our solution, we apply the Division Algorithm: Dividend = (Divisor × Quotient) + Remainder. Substituting the values, we get 16135 = (875 × 18) + 385. Calculating the right side, we have 15750 + 385, which equals 16135. This confirms that our division is correct. This problem emphasizes the importance of estimation skills and the ability to handle larger numbers in long division. The process can be simplified by breaking it down into smaller steps and using trial and error to find the appropriate multipliers. The Division Algorithm plays a crucial role in verifying our results, especially when dealing with larger numbers where the risk of making a mistake is higher. In practical applications, division problems of this nature might arise in scenarios such as inventory management, financial analysis, or resource allocation. For example, a retail store might need to divide the total sales revenue by the number of items sold to determine the average selling price per item. Similarly, a financial analyst might need to divide a company's total debt by its total assets to calculate its debt-to-asset ratio. The ability to perform these divisions accurately and efficiently is essential for making informed decisions. Furthermore, this problem reinforces the understanding of the relationship between the quotient, remainder, divisor, and dividend. By working through these problems, we develop a deeper appreciation for the structure and logic of division and the role of the Division Algorithm in ensuring accuracy. This understanding is essential for building a solid foundation in arithmetic and preparing for more advanced mathematical concepts.

Answer:

  • Quotient = 18
  • Remainder = 385

g) 6971 ÷ 47

Let's continue with the division problem 6971 ÷ 47. As with the previous examples, we aim to determine the quotient and the remainder. This problem provides another opportunity to practice the process of long division and reinforce our understanding of the underlying principles. We begin by considering the dividend (6971) and the divisor (47). We start by examining the first two digits of the dividend, which are 69. We ask ourselves, how many times does 47 go into 69? We find that 47 goes into 69 once (47 * 1 = 47). So, we write '1' as the first digit of our quotient. We multiply 47 by 1, which equals 47, and subtract it from 69, resulting in 22. We bring down the next digit from the dividend, which is 7, making our new number 227. Now, we determine how many times 47 goes into 227. This requires some estimation. We might try multiplying 47 by different numbers until we get a result close to 227. We find that 47 goes into 227 four times (47 * 4 = 188). So, we write '4' in the quotient. We multiply 47 by 4, which equals 188, and subtract it from 227, leaving us with 39. We bring down the last digit, 1, giving us 391. Next, we figure out how many times 47 goes into 391. We find that 47 goes into 391 eight times (47 * 8 = 376). We write '8' in the quotient, multiply 47 by 8, and subtract the result (376) from 391, leaving us with 15. This 15 is our remainder, as it is smaller than the divisor (47). Therefore, the quotient is 148, and the remainder is 15. To verify our solution, we apply the Division Algorithm: Dividend = (Divisor × Quotient) + Remainder. Substituting the values, we get 6971 = (47 × 148) + 15. Calculating the right side, we have 6956 + 15, which equals 6971. This confirms that our division is correct. This problem reinforces the importance of estimation skills and a systematic approach to long division. The ability to estimate accurately helps us to find the correct quotient digits more efficiently, while the systematic process ensures that we don't miss any steps and make errors. The Division Algorithm serves as a crucial tool for verifying our results and ensuring that we have arrived at the correct quotient and remainder. In practical applications, division problems of this nature might arise in various scenarios, such as calculating unit costs, determining the number of items that can be purchased with a given budget, or analyzing statistical data. For example, a business might need to divide its total expenses by the number of units sold to determine the cost per unit. Similarly, a researcher might need to divide the total number of participants in a study by the number of groups to determine the number of participants per group. The ability to perform these divisions accurately and efficiently is essential for effective decision-making. Furthermore, this problem reinforces the understanding of the relationship between the dividend, divisor, quotient, and remainder. By working through these problems, we develop a deeper appreciation for the structure and logic of division and the role of the Division Algorithm in ensuring accuracy. This understanding is essential for building a solid foundation in arithmetic and preparing for more advanced mathematical concepts.

Answer:

  • Quotient = 148
  • Remainder = 15

h) 6754 ÷ 56

Finally, let's tackle the division problem 6754 ÷ 56. As with the previous examples, our goal is to determine the quotient and the remainder. This problem provides yet another opportunity to practice long division and solidify our understanding of the process. We begin by considering the dividend (6754) and the divisor (56). We start by examining the first two digits of the dividend, which are 67. We ask ourselves, how many times does 56 go into 67? We find that 56 goes into 67 once (56 * 1 = 56). So, we write '1' as the first digit of our quotient. We multiply 56 by 1, which equals 56, and subtract it from 67, resulting in 11. We bring down the next digit from the dividend, which is 5, making our new number 115. Now, we determine how many times 56 goes into 115. This requires some estimation. We might try multiplying 56 by different numbers until we get a result close to 115. We find that 56 goes into 115 twice (56 * 2 = 112). So, we write '2' in the quotient. We multiply 56 by 2, which equals 112, and subtract it from 115, leaving us with 3. We bring down the last digit, 4, giving us 34. Next, we figure out how many times 56 goes into 34. We see that 56 is larger than 34, so it goes in zero times. We write '0' in the quotient. The remainder is 34, as it is smaller than the divisor (56). Therefore, the quotient is 120, and the remainder is 34. To verify our solution, we apply the Division Algorithm: Dividend = (Divisor × Quotient) + Remainder. Substituting the values, we get 6754 = (56 × 120) + 34. Calculating the right side, we have 6720 + 34, which equals 6754. This confirms that our division is correct. This problem reinforces the importance of paying close attention to each step in the long division process and recognizing when a digit in the quotient is zero. It also highlights the crucial role of the Division Algorithm in verifying our results and ensuring that we have arrived at the correct solution. In practical applications, division problems of this nature might arise in various scenarios, such as calculating average values, determining the number of groups that can be formed from a given number of items, or analyzing data. For example, a teacher might need to divide the total number of students in a class by the number of groups to determine the number of students per group. Similarly, a data analyst might need to divide the total sales revenue by the number of transactions to calculate the average transaction value. The ability to perform these divisions accurately and efficiently is essential for effective decision-making. Furthermore, this problem reinforces the understanding of the relationship between the dividend, divisor, quotient, and remainder. By working through these problems, we develop a deeper appreciation for the structure and logic of division and the role of the Division Algorithm in ensuring accuracy. This understanding is essential for building a solid foundation in arithmetic and preparing for more advanced mathematical concepts.

Answer:

  • Quotient = 120
  • Remainder = 34

In conclusion, this article has provided a comprehensive exploration of division problems, focusing on calculating the quotient and remainder and verifying the answers using the Division Algorithm. Each problem was solved step-by-step, emphasizing the importance of estimation skills, a systematic approach to long division, and the crucial role of the Division Algorithm in ensuring accuracy. By mastering these concepts, learners can build a solid foundation in arithmetic and develop the skills necessary to solve a wide range of mathematical problems.