Solving Division Problems 74,887 ÷ 108 And 82,001 ÷ 813
In this comprehensive guide, we will delve into the intricacies of division, providing a step-by-step approach to solving two complex division problems: 74,887 ÷ 108 and 82,001 ÷ 813. Understanding division is crucial for various mathematical applications and real-world scenarios. Whether you're a student seeking to improve your math skills or an adult looking to refresh your knowledge, this article will equip you with the necessary tools and techniques to master division.
Understanding the Basics of Division
Before we tackle the specific problems, it's essential to revisit the fundamental concepts of division. Division is one of the four basic arithmetic operations, the others being addition, subtraction, and multiplication. At its core, division is the process of splitting a whole into equal parts or groups. It helps us determine how many times one number (the divisor) is contained within another number (the dividend). The result of a division operation is called the quotient, and any remaining amount is known as the remainder.
The key terms in division are as follows:
- Dividend: The number being divided (the total amount).
- Divisor: The number by which the dividend is being divided (the number of groups or parts).
- Quotient: The result of the division (how many items are in each group).
- Remainder: The amount left over after dividing the dividend as evenly as possible by the divisor.
To illustrate, consider the simple division problem 10 ÷ 2. Here, 10 is the dividend, 2 is the divisor, 5 is the quotient (since 10 divided by 2 equals 5), and there is no remainder.
Long Division A Step-by-Step Approach
Long division is a standard algorithm for dividing larger numbers, especially when mental calculation becomes challenging. It involves a series of steps that systematically break down the division process into manageable parts. Let's outline the general steps involved in long division:
- Set up the problem: Write the dividend inside the division symbol (also known as the long division bracket) and the divisor outside the bracket to the left.
- Divide: Determine how many times the divisor goes into the first digit(s) of the dividend. Write the quotient above the dividend, aligning it with the digit(s) being divided.
- Multiply: Multiply the quotient by the divisor and write the product below the corresponding digits of the dividend.
- Subtract: Subtract the product from the corresponding digits of the dividend. The result is the remainder for that step.
- Bring down: Bring down the next digit of the dividend next to the remainder from the previous step. This forms the new dividend for the next iteration.
- Repeat: Repeat steps 2-5 until all digits of the dividend have been used. The final quotient is the number above the division symbol, and the final remainder (if any) is the number left over after the last subtraction.
Understanding these basic concepts and the long division algorithm is crucial for tackling more complex division problems. Now, let's apply these principles to solve the given problems.
Problem 1 74,887 ÷ 108
Let's begin by tackling the division problem 74,887 ÷ 108. This problem involves dividing a five-digit number (74,887) by a three-digit number (108). To solve this, we'll employ the long division method, breaking down the process into manageable steps.
Step-by-Step Solution
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Set up the problem: Write 74,887 inside the division symbol and 108 outside to the left:
________ 108 | 74887 -
Divide: Determine how many times 108 goes into the first few digits of 74,887. We start by looking at 748. Since 108 goes into 748 six times (108 x 6 = 648), we write 6 above the 8 in the quotient:
6______ 108 | 74887 -
Multiply: Multiply the quotient (6) by the divisor (108): 6 x 108 = 648. Write the product (648) below 748:
6______ 108 | 74887 648 -
Subtract: Subtract 648 from 748: 748 - 648 = 100. Write the difference (100) below 648:
6______ 108 | 74887 648 --- 100 -
Bring down: Bring down the next digit (8) from the dividend (74,887) next to the remainder (100). This forms the new dividend 1008:
6______ 108 | 74887 648 --- 1008 -
Repeat:
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Divide: Determine how many times 108 goes into 1008. It goes in 9 times (108 x 9 = 972). Write 9 next to 6 in the quotient:
69_____ 108 | 74887 648 --- 1008 -
Multiply: Multiply the new quotient digit (9) by the divisor (108): 9 x 108 = 972. Write 972 below 1008:
69_____ 108 | 74887 648 --- 1008 972 -
Subtract: Subtract 972 from 1008: 1008 - 972 = 36. Write the difference (36) below 972:
69_____ 108 | 74887 648 --- 1008 972 ---- 36 -
Bring down: Bring down the last digit (7) from the dividend (74,887) next to the remainder (36). This forms the new dividend 367:
69_____ 108 | 74887 648 --- 1008 972 ---- 367
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Repeat again:
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Divide: Determine how many times 108 goes into 367. It goes in 3 times (108 x 3 = 324). Write 3 next to 69 in the quotient:
693___ 108 | 74887 648 --- 1008 972 ---- 367 -
Multiply: Multiply the new quotient digit (3) by the divisor (108): 3 x 108 = 324. Write 324 below 367:
693___ 108 | 74887 648 --- 1008 972 ---- 367 324 -
Subtract: Subtract 324 from 367: 367 - 324 = 43. Write the difference (43) below 324:
693___ 108 | 74887 648 --- 1008 972 ---- 367 324 --- 43
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Since there are no more digits to bring down, the division is complete. The quotient is 693, and the remainder is 43.
Result
Therefore, 74,887 ÷ 108 = 693 with a remainder of 43. This can also be expressed as 693 43/108.
Problem 2 82,001 ÷ 813
Now, let's move on to the second division problem: 82,001 ÷ 813. This involves dividing a five-digit number (82,001) by a three-digit number (813). Again, we'll use the long division method to systematically find the quotient and remainder.
Step-by-Step Solution
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Set up the problem: Write 82,001 inside the division symbol and 813 outside to the left:
________ 813 | 82001 -
Divide: Determine how many times 813 goes into the first few digits of 82,001. We start by looking at 820. Since 813 goes into 820 one time (813 x 1 = 813), we write 1 above the 0 in the quotient:
1______ 813 | 82001 -
Multiply: Multiply the quotient (1) by the divisor (813): 1 x 813 = 813. Write the product (813) below 820:
1______ 813 | 82001 813 -
Subtract: Subtract 813 from 820: 820 - 813 = 7. Write the difference (7) below 813:
1______ 813 | 82001 813 --- 7 -
Bring down: Bring down the next digit (0) from the dividend (82,001) next to the remainder (7). This forms the new dividend 70:
1______ 813 | 82001 813 --- 70 -
Repeat:
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Divide: Determine how many times 813 goes into 70. Since 813 is larger than 70, it goes in 0 times. Write 0 next to 1 in the quotient:
10_____ 813 | 82001 813 --- 70 -
Bring down: Bring down the next digit (0) from the dividend (82,001) next to 70. This forms the new dividend 700.
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Divide: Determine how many times 813 goes into 700. Since 813 is larger than 700, it goes in 0 times. We will bring down the next number.
100____ 813 | 82001 813 --- 700 -
Bring down: Bring down the next digit (1) from the dividend (82,001) next to 700. This forms the new dividend 7001:
100____ 813 | 82001 813 --- 7001
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Repeat again:
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Divide: Determine how many times 813 goes into 7001. It goes in 8 times (813 x 8 = 6504). Write 8 next to 100 in the quotient:
1008__ 813 | 82001 813 --- 7001 -
Multiply: Multiply the new quotient digit (8) by the divisor (813): 8 x 813 = 6504. Write 6504 below 7001:
1008__ 813 | 82001 813 --- 7001 6504 -
Subtract: Subtract 6504 from 7001: 7001 - 6504 = 497. Write the difference (497) below 6504:
1008__ 813 | 82001 813 --- 7001 6504 ---- 497
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Since there are no more digits to bring down, the division is complete. The quotient is 100, and the remainder is 497.
Result
Therefore, 82,001 ÷ 813 = 100 with a remainder of 497. This can also be expressed as 100 497/813.
Conclusion
In this article, we have explored the concept of division and its application in solving two complex division problems. We have demonstrated the long division method, a step-by-step algorithm for dividing larger numbers. By breaking down the problems into manageable steps, we were able to find the quotients and remainders for both 74,887 ÷ 108 and 82,001 ÷ 813.
Mastering division is a fundamental skill in mathematics, and the long division method is a valuable tool for tackling challenging problems. By practicing these techniques and understanding the underlying concepts, you can confidently solve a wide range of division problems. Remember, practice is key to improving your mathematical skills, so continue to challenge yourself with different problems and explore various methods for solving them. This comprehensive guide serves as a strong foundation for your journey toward mastering division and beyond.
