Quotients And Remainders A Step By Step Guide To Division And Verification

Division is a fundamental arithmetic operation that involves splitting a whole into equal parts. In mathematical terms, it's the process of finding out how many times one number (the divisor) is contained within another number (the dividend). The result of this operation is called the quotient, and any amount left over is known as the remainder.

In this comprehensive guide, we will delve into the intricacies of division by exploring various examples. We'll focus on calculating the quotient and remainder for different division problems, as well as verifying the accuracy of our answers. This skill is crucial for building a strong foundation in mathematics and tackling more complex problems in the future.

Problem 1: Finding Quotients and Remainders

Let's begin by addressing the first set of problems, where our primary goal is to determine the quotient and remainder for each division.

a. 58728 ÷ 28

To find the quotient and remainder when dividing 58728 by 28, we'll use the process of long division. Long division is a systematic method that breaks down the division process into smaller, manageable steps. Let's walk through it:

1. Set up the problem: Write the dividend (58728) inside the division symbol and the divisor (28) outside.
2. Divide the first digits: Determine how many times 28 goes into the first two digits of the dividend (58). In this case, 28 goes into 58 twice (2 x 28 = 56).
3. Write the quotient: Place the '2' above the 8 in the dividend.
4. Multiply: Multiply the quotient (2) by the divisor (28) and write the result (56) below the first two digits of the dividend.
5. Subtract: Subtract 56 from 58, which leaves 2.
6. Bring down the next digit: Bring down the next digit from the dividend (7) next to the 2, forming 27.
7. Repeat: Now, we have 27. How many times does 28 go into 27? It doesn't, so we write '0' as the next digit in the quotient.
8. Bring down the next digit: Bring down the next digit (2) to form 272.
9. Divide: How many times does 28 go into 272? It goes in 9 times (9 x 28 = 252).
10. Write the quotient: Place the '9' next to the '0' in the quotient.
11. Multiply: Multiply the quotient (9) by the divisor (28) and write the result (252) below 272.
12. Subtract: Subtract 252 from 272, which leaves 20.
13. Bring down the next digit: Bring down the last digit (8) to form 208.
14. Divide: How many times does 28 go into 208? It goes in 7 times (7 x 28 = 196).
15. Write the quotient: Place the '7' next to the '9' in the quotient.
16. Multiply: Multiply the quotient (7) by the divisor (28) and write the result (196) below 208.
17. Subtract: Subtract 196 from 208, which leaves 12. This is our remainder.

Therefore, when we divide 58728 by 28, the quotient is 2097 and the remainder is 12. This meticulous process ensures that we accurately determine both the whole number result (quotient) and any leftover amount (remainder).

b. 679856 ÷ 24

Following the same long division method, let's divide 679856 by 24:

1. Set up: 679856 ÷ 24
2. Divide: 24 goes into 67 twice (2 x 24 = 48).
3. Write quotient: Place '2' above the 7.
4. Multiply: 2 x 24 = 48.
5. Subtract: 67 - 48 = 19.
6. Bring down: Bring down the 9 to make 199.
7. Divide: 24 goes into 199 eight times (8 x 24 = 192).
8. Write quotient: Place '8' next to the '2'.
9. Multiply: 8 x 24 = 192.
10. Subtract: 199 - 192 = 7.
11. Bring down: Bring down the 8 to make 78.
12. Divide: 24 goes into 78 three times (3 x 24 = 72).
13. Write quotient: Place '3' next to the '8'.
14. Multiply: 3 x 24 = 72.
15. Subtract: 78 - 72 = 6.
16. Bring down: Bring down the 5 to make 65.
17. Divide: 24 goes into 65 twice (2 x 24 = 48).
18. Write quotient: Place '2' next to the '3'.
19. Multiply: 2 x 24 = 48.
20. Subtract: 65 - 48 = 17.
21. Bring down: Bring down the 6 to make 176.
22. Divide: 24 goes into 176 seven times (7 x 24 = 168).
23. Write quotient: Place '7' next to the '2'.
24. Multiply: 7 x 24 = 168.
25. Subtract: 176 - 168 = 8.

Thus, 679856 divided by 24 gives a quotient of 28327 and a remainder of 8.

c. 578692 ÷ 35

Let's continue practicing long division with 578692 ÷ 35:

1. Set up: 578692 ÷ 35
2. Divide: 35 goes into 57 once (1 x 35 = 35).
3. Write quotient: Place '1' above the 7.
4. Multiply: 1 x 35 = 35.
5. Subtract: 57 - 35 = 22.
6. Bring down: Bring down the 8 to make 228.
7. Divide: 35 goes into 228 six times (6 x 35 = 210).
8. Write quotient: Place '6' next to the '1'.
9. Multiply: 6 x 35 = 210.
10. Subtract: 228 - 210 = 18.
11. Bring down: Bring down the 6 to make 186.
12. Divide: 35 goes into 186 five times (5 x 35 = 175).
13. Write quotient: Place '5' next to the '6'.
14. Multiply: 5 x 35 = 175.
15. Subtract: 186 - 175 = 11.
16. Bring down: Bring down the 9 to make 119.
17. Divide: 35 goes into 119 three times (3 x 35 = 105).
18. Write quotient: Place '3' next to the '5'.
19. Multiply: 3 x 35 = 105.
20. Subtract: 119 - 105 = 14.
21. Bring down: Bring down the 2 to make 142.
22. Divide: 35 goes into 142 four times (4 x 35 = 140).
23. Write quotient: Place '4' next to the '3'.
24. Multiply: 4 x 35 = 140.
25. Subtract: 142 - 140 = 2.

Therefore, 578692 divided by 35 results in a quotient of 16534 and a remainder of 2.

d. 9035674 ÷ 5001

This problem involves a larger divisor, but the long division process remains the same:

1. Set up: 9035674 ÷ 5001
2. Divide: 5001 goes into 9035 once (1 x 5001 = 5001).
3. Write quotient: Place '1' above the 5.
4. Multiply: 1 x 5001 = 5001.
5. Subtract: 9035 - 5001 = 4034.
6. Bring down: Bring down the 6 to make 40346.
7. Divide: 5001 goes into 40346 eight times (8 x 5001 = 40008).
8. Write quotient: Place '8' next to the '1'.
9. Multiply: 8 x 5001 = 40008.
10. Subtract: 40346 - 40008 = 338.
11. Bring down: Bring down the 7 to make 3387.
12. Divide: 5001 does not go into 3387, so we write '0' in the quotient.
13. Bring down: Bring down the 4 to make 33874.
14. Divide: 5001 goes into 33874 six times (6 x 5001 = 30006).
15. Write quotient: Place '6' next to the '0'.
16. Multiply: 6 x 5001 = 30006.
17. Subtract: 33874 - 30006 = 3868.

Therefore, 9035674 divided by 5001 yields a quotient of 1806 and a remainder of 3868.

Problem 2: Division and Verification

Now, let's tackle the second set of problems, where we not only need to find the quotient and remainder but also verify our answers. Verification is a crucial step in mathematics as it ensures the accuracy of our calculations. The formula we'll use for verification is:

Dividend = (Divisor x Quotient) + Remainder

If this equation holds true after we plug in our calculated values, then our division is correct.

a. 56124 ÷ 19

1. Divide: Perform long division of 56124 by 19.

   • Quotient = 2953
   • Remainder = 17

2. Verify: Use the formula: Dividend = (Divisor x Quotient) + Remainder

   • 56124 = (19 x 2953) + 17
   • 56124 = 56107 + 17
   • 56124 = 56124

The equation holds true, so our division is correct. The quotient is 2953, the remainder is 17, and the answer is verified.

b. 39547 ÷ 39

1. Divide: Perform long division of 39547 by 39.

   • Quotient = 1014
   • Remainder = 1

2. Verify: Use the formula: Dividend = (Divisor x Quotient) + Remainder

   • 39547 = (39 x 1014) + 1
   • 39547 = 39546 + 1
   • 39547 = 39547

The equation holds true, confirming the accuracy of our division. The quotient is 1014, the remainder is 1, and the answer is verified.

In this comprehensive guide, we've explored the process of division, focusing on finding the quotient and remainder for various problems. We've also emphasized the importance of verification to ensure the accuracy of our calculations. By mastering these skills, you'll build a solid foundation in mathematics and be well-equipped to tackle more complex problems in the future. Remember, practice is key to success, so keep working on division problems to further hone your skills. The ability to confidently perform division is not just a mathematical skill, it's a valuable tool for problem-solving in everyday life.