Calculating Division Problems Mastering Long Division Techniques

Division problems can seem daunting, but with a clear understanding of the process, they become manageable. This comprehensive guide breaks down several division problems, explaining each step in detail. We'll explore long division techniques and provide insights into how to approach different types of division questions. Whether you are new to division or seeking to refine your skills, this guide offers valuable tips and step-by-step solutions.

1 a. Calculate 117 divided by 39

When approaching division calculation, the initial step is to set up the long division problem correctly. The number being divided (the dividend) is 117, and the number we are dividing by (the divisor) is 39. The goal is to determine how many times 39 fits into 117. Begin by estimating how many times 39 goes into 117. Since 39 is close to 40, and 117 is close to 120, we can estimate that 39 goes into 117 about three times.

Next, multiply 39 by 3. 39 multiplied by 3 equals 117. Write 117 under the dividend (117) and subtract. 117 minus 117 equals 0. Since the remainder is 0, the division is exact, and there is no need to continue further. The quotient is 3, which means 117 divided by 39 is exactly 3. Therefore, we can confidently state that the result of the division is 3. This method illustrates a straightforward application of long division where the estimation and multiplication steps lead directly to the solution without additional iterations.

In summary, dividing 117 by 39 is a straightforward application of long division. The initial estimation allows us to quickly find the correct quotient, making the division calculation efficient and accurate. This type of problem is fundamental in understanding basic long division techniques and builds a solid foundation for tackling more complex problems.

1 b. Calculate 765 divided by 45

To calculate division of 765 by 45, we will use long division. Begin by setting up the problem: place 765 inside the division bracket and 45 outside as the divisor. We need to determine how many times 45 fits into 765.

First, consider how many times 45 goes into the first two digits of the dividend, which is 76. 45 goes into 76 one time. Write 1 above the 6 in 765. Multiply 1 by 45, which equals 45. Write 45 under 76 and subtract. 76 minus 45 equals 31. Bring down the next digit from the dividend, which is 5, to join 31, making it 315.

Now, we need to determine how many times 45 goes into 315. To estimate, we can round 45 to 50 and think of how many times 50 goes into 300 (since 315 is close to 300). 50 goes into 300 six times. So, let's try multiplying 45 by 7. 45 multiplied by 7 equals 315. Write 7 next to 1 on top of the division bracket, making the quotient 17. Write 315 under 315 and subtract. 315 minus 315 equals 0. Since the remainder is 0, the division is exact.

Therefore, 765 divided by 45 equals 17. This long division technique showcases a methodical approach to solving division problems. By breaking down the dividend into smaller parts and estimating appropriately, we can efficiently find the quotient. Understanding these steps is crucial for mastering division calculation and tackling more complex mathematical problems.

1 c. Calculate 210 divided by 15

To calculate division, we need to divide 210 by 15. We'll use long division for this calculation. Set up the problem with 210 as the dividend and 15 as the divisor. The first step is to see how many times 15 goes into the first part of the dividend, which is 21.

15 goes into 21 one time. Write 1 above the 1 in 210. Multiply 1 by 15, which equals 15. Write 15 under 21 and subtract. 21 minus 15 equals 6. Bring down the next digit from the dividend, which is 0, to join 6, making it 60.

Now, we need to determine how many times 15 goes into 60. We know that 15 multiplied by 4 equals 60. So, write 4 next to 1 on top of the division bracket, making the quotient 14. Write 60 under 60 and subtract. 60 minus 60 equals 0. Since the remainder is 0, the division is exact.

Thus, 210 divided by 15 equals 14. This long division technique demonstrates how to systematically solve division problems by breaking them down into smaller, manageable steps. The process of estimating, multiplying, subtracting, and bringing down digits is fundamental to division calculation. This method helps ensure accuracy and efficiency in solving such problems.

2 a. Calculate 828 divided by 92

To accurately calculate division, we must divide 828 by 92 using long division. This process involves several steps that help break down the problem into manageable parts. First, set up the long division with 828 as the dividend and 92 as the divisor. We need to determine how many times 92 fits into 828.

Begin by estimating how many times 92 goes into 828. Since 92 is close to 90 and 828 is close to 810, we can estimate by dividing 810 by 90, which gives us 9. Let’s try multiplying 92 by 9. 92 multiplied by 9 equals 828. Write 9 above the last digit of 828 in the quotient area. Write 828 under the dividend and subtract.

828 minus 828 equals 0. The remainder is 0, indicating that the division is exact. Therefore, 828 divided by 92 equals 9. This long division technique exemplifies a straightforward scenario where estimation plays a critical role in quickly arriving at the correct answer. The ability to estimate accurately helps in simplifying the division calculation process and ensures efficiency.

In summary, dividing 828 by 92 is a clear demonstration of how estimating can lead to an efficient solution in long division. The initial estimation of 9 allowed us to bypass multiple attempts and directly find the quotient. Understanding and applying these long division techniques is crucial for handling more complex division problems.

2 b. Calculate 624 divided by 12

To calculate division, we will divide 624 by 12 using long division. This process involves setting up the division problem correctly and then performing a series of steps to find the quotient. Begin by writing 624 inside the division bracket and 12 outside as the divisor.

First, consider how many times 12 goes into the first two digits of the dividend, which is 62. 12 goes into 62 five times (since 12 * 5 = 60). Write 5 above the 2 in 624. Multiply 5 by 12, which equals 60. Write 60 under 62 and subtract. 62 minus 60 equals 2. Bring down the next digit from the dividend, which is 4, to join 2, making it 24.

Now, determine how many times 12 goes into 24. 12 goes into 24 exactly two times. Write 2 next to 5 on top of the division bracket, making the quotient 52. Multiply 2 by 12, which equals 24. Write 24 under 24 and subtract. 24 minus 24 equals 0. Since the remainder is 0, the division is exact.

Therefore, 624 divided by 12 equals 52. This long division technique demonstrates a step-by-step approach to dividing larger numbers. By breaking down the dividend into smaller parts and systematically finding how many times the divisor fits into each part, we can accurately calculate division problems. Understanding these methods is essential for developing strong mathematical skills.

2 c. Calculate 966 divided by 42

To accurately calculate division, we need to divide 966 by 42 using long division. Long division is a methodical approach that breaks down the division process into smaller, manageable steps. Begin by setting up the long division with 966 as the dividend and 42 as the divisor. The goal is to determine how many times 42 fits into 966.

First, consider how many times 42 goes into the first two digits of the dividend, which is 96. 42 goes into 96 two times. Write 2 above the 6 in 966. Multiply 2 by 42, which equals 84. Write 84 under 96 and subtract. 96 minus 84 equals 12. Bring down the next digit from the dividend, which is 6, to join 12, making it 126.

Now, determine how many times 42 goes into 126. To estimate, we can think of 42 as close to 40 and 126 as close to 120. 40 goes into 120 three times. So, let’s try multiplying 42 by 3. 42 multiplied by 3 equals 126. Write 3 next to 2 on top of the division bracket, making the quotient 23. Write 126 under 126 and subtract. 126 minus 126 equals 0. Since the remainder is 0, the division is exact.

Thus, 966 divided by 42 equals 23. This long division technique illustrates the importance of estimation in making the division process more efficient. By estimating how many times the divisor goes into parts of the dividend, we can simplify the division calculation and arrive at the correct quotient quickly. This step-by-step approach is crucial for solving more complex division problems.

In conclusion, mastering long division techniques involves understanding the systematic process of breaking down larger numbers into smaller, manageable parts. Accurate estimation, careful multiplication, and precise subtraction are key components of this process. Whether you are dividing three-digit numbers or tackling more complex problems, a solid understanding of long division will greatly enhance your mathematical skills.