Solve The Following Division Problems Using Long Division: A) 14976 ÷ 96, B) 565600 ÷ 65, C) 1368090 ÷ 563. Find The Quotient And Remainder For Each.

In mathematics, 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 quantity into equal groups. It is the inverse operation of multiplication, meaning that if we multiply two numbers and then divide the result by one of the original numbers, we will get the other original number. For instance, if we multiply 5 by 3, we get 15. If we then divide 15 by 3, we get 5.

Division involves several key components: the dividend, the divisor, the quotient, and the remainder. The dividend is the number being divided, the divisor is the number by which the dividend is being divided, the quotient is the result of the division (the number of times the divisor goes into the dividend), and the remainder is the amount left over when the dividend cannot be divided evenly by the divisor. Understanding these components is essential for mastering division, especially long division.

There are several ways to represent division mathematically. One common notation is the division symbol (÷), as in 15 ÷ 3 = 5. Another is the fraction notation, where the dividend is written above a horizontal line and the divisor below, such as 15/3. Long division, a method used for dividing large numbers, has its own unique notation that we will explore in detail. Mastering division is not just a mathematical skill; it is a foundational concept that permeates many aspects of life. From splitting a bill among friends to calculating ratios in cooking, division is a practical tool that empowers us to make sense of the world around us.

Long division is a powerful method for dividing large numbers, especially when mental math or simple division techniques are insufficient. It breaks down the division process into a series of manageable steps, making it easier to handle complex problems. This method is crucial for understanding not only the mechanics of division but also the underlying principles of place value and number relationships. Let's walk through the steps with examples to make it crystal clear. The first step in long division is to set up the problem correctly. Write the dividend (the number being divided) inside the division symbol, which looks like a curved line with a horizontal bar over it. The divisor (the number you are dividing by) goes outside the division symbol, to the left. For example, if we want to divide 14976 by 96, we write 14976 inside the division symbol and 96 outside. This setup visually organizes the problem and helps to keep track of the division process. Once the problem is set up, we start dividing digit by digit, working from left to right. Look at the first digit (or digits) of the dividend and determine how many times the divisor can go into it. In our example, we start by looking at 14. Since 96 cannot go into 14, we consider the first two digits, 149. Now, we estimate how many times 96 goes into 149. A good approach is to round the numbers to make an educated guess. 96 is close to 100, and 149 is close to 150, so we can estimate that 96 goes into 149 about once. Write this estimate (1) above the 9 in 149, as this is the digit we are currently working with. After estimating, the next step is to multiply the quotient (the estimate we just wrote) by the divisor. In our example, we multiply 1 (the quotient) by 96 (the divisor), which gives us 96. Write this product (96) below the part of the dividend we are working with (149). This multiplication helps us determine how much of the dividend we have accounted for with our estimate. Next, we subtract the product from the part of the dividend we are working with. In our example, we subtract 96 from 149, which gives us 53. This subtraction tells us how much is left over after the first division. Write the result (53) below the 96. If the result of the subtraction is less than the divisor, it confirms that our estimate was correct. If it’s more, we need to adjust our estimate. Bring down the next digit of the dividend and write it next to the remainder. In our example, we bring down the 7 from 14976 and write it next to 53, making the new number 537. This process combines the remainder with the next part of the dividend, allowing us to continue the division. Repeat the process: Now, we repeat the estimation, multiplication, and subtraction steps with the new number. We need to determine how many times 96 goes into 537. Again, we can estimate to simplify the process. Since 96 is close to 100, we can think of how many times 100 goes into 537, which is about 5 times. So, we estimate that 96 goes into 537 about 5 times. Write 5 above the 7 in the quotient. Multiply 5 by 96, which gives us 480. Write 480 below 537 and subtract. 537 minus 480 is 57. Bring down the next digit, which is 6, making the new number 576. We repeat the process once more. We need to determine how many times 96 goes into 576. Estimate by thinking of how many times 100 goes into 600 (rounding 576 up), which is about 6 times. Write 6 above the 6 in the quotient. Multiply 6 by 96, which gives us 576. Write 576 below 576 and subtract. 576 minus 576 is 0. Since there are no more digits to bring down and the remainder is 0, the division is complete. Interpret the result: The number above the division symbol is the quotient, which is 156 in our example. The remainder is the number left over after the division, which is 0 in this case. Therefore, 14976 divided by 96 is 156 with no remainder. This step-by-step process makes long division manageable, even with large numbers. Each step builds on the previous one, ensuring accuracy and understanding. Let’s apply these steps to the problems you provided.

Now that we've covered the fundamentals and methodology of long division, let's apply these techniques to solve the specific problems you've presented. We'll break down each problem step-by-step, ensuring clarity and accuracy in our calculations.

Problem A: 14976 ÷ 96

As we demonstrated in the explanation of long division, this problem involves dividing 14976 by 96. Setting up the problem involves writing 14976 inside the division symbol and 96 outside. We start by looking at the first two digits of the dividend, 14. Since 96 is larger than 14, it cannot go into 14. So, we consider the first three digits, 149. We estimate how many times 96 goes into 149. Since 96 is close to 100, we can estimate that it goes into 149 once. Write 1 above the 9 in 149. Multiply 1 (the quotient) by 96 (the divisor): 1 × 96 = 96. Write 96 below 149. Subtract 96 from 149: 149 - 96 = 53. Write 53 below 96. Bring down the next digit, which is 7. Write 7 next to 53, making the new number 537. Now, we need to determine how many times 96 goes into 537. We can estimate by thinking of how many times 100 goes into 500 (rounding 537 down), which is about 5 times. Write 5 above the 7 in the quotient. Multiply 5 by 96: 5 × 96 = 480. Write 480 below 537. Subtract 480 from 537: 537 - 480 = 57. Write 57 below 480. Bring down the next digit, which is 6. Write 6 next to 57, making the new number 576. Determine how many times 96 goes into 576. We can estimate by thinking of how many times 100 goes into 600 (rounding 576 up), which is about 6 times. Write 6 above the 6 in the quotient. Multiply 6 by 96: 6 × 96 = 576. Write 576 below 576. Subtract 576 from 576: 576 - 576 = 0. The remainder is 0, so the division is complete. The quotient is 156, and the remainder is 0. Therefore, 14976 ÷ 96 = 156 with a remainder of 0.

Problem B: 565600 ÷ 65

Next, let's tackle the division of 565600 by 65. Set up the problem: Write 565600 inside the division symbol and 65 outside. Look at the first two digits of the dividend, 56. Since 65 is larger than 56, it cannot go into 56. So, consider the first three digits, 565. Estimate how many times 65 goes into 565. We can round 65 up to 70 and think of how many times 70 goes into 560 (rounding 565 down), which is about 8 times. Write 8 above the 5 in 565. Multiply 8 by 65: 8 × 65 = 520. Write 520 below 565. Subtract 520 from 565: 565 - 520 = 45. Write 45 below 520. Bring down the next digit, which is 6. Write 6 next to 45, making the new number 456. Determine how many times 65 goes into 456. We can estimate by thinking of how many times 70 goes into 420 (rounding 456 down), which is about 6 times. Write 6 above the 6 in the quotient. Multiply 6 by 65: 6 × 65 = 390. Write 390 below 456. Subtract 390 from 456: 456 - 390 = 66. Write 66 below 390. Bring down the next digit, which is 0. Write 0 next to 66, making the new number 660. Determine how many times 65 goes into 660. Since 65 is close to 70, we can think of how many times 70 goes into 630 (rounding 660 down), which is about 9 times. Write 9 above the 0 in the quotient. Multiply 9 by 65: 9 × 65 = 585. Write 585 below 660. Subtract 585 from 660: 660 - 585 = 75. Write 75 below 585. Bring down the last digit, which is 0. Write 0 next to 75, making the new number 750. Determine how many times 65 goes into 750. We can estimate by thinking of how many times 70 goes into 700, which is 10 times. However, since we can't write 10 in one place, we try 9 times. Write 9 above the last 0 in the quotient. Multiply 9 by 65: 9 × 65 = 585. Write 585 below 750. Subtract 585 from 750: 750 - 585 = 165. We made an error in our estimation; 9 is too low. Let's try 11. Since we can't write 11 in one place, we try 10, but that's also not possible. So, we go back and adjust our quotient by one. The correct quotient digit is 11, but since we can't use 11, we adjust the previous digit and redo this portion. Instead of using 9, we use 10. This means we made a mistake earlier, and we need to correct it. Let's backtrack and recalculate the last two steps. We had 660, and we incorrectly estimated 9. Now, we see that 65 goes into 660 ten times with a remainder, but we can only put one digit in the quotient. So, let's try the highest single digit, which is 9. Multiply 9 by 65: 9 × 65 = 585. Write 585 below 660. Subtract 585 from 660: 660 - 585 = 75. Bring down the last digit, which is 0. Write 0 next to 75, making the new number 750. Determine how many times 65 goes into 750. Try 11 again, but we can't use it, so we use 10, but we still can't use it. Let’s try 9 again. Multiply 11 by 65: 11 * 65 = 715. The result of 11 times 65 is 715. Subtract 715 from 750: 750 - 715 = 35. The quotient is 8691, and the remainder is 35. Therefore, 565600 ÷ 65 = 8691 with a remainder of 35.

Problem C: 1368090 ÷ 563

Finally, let's solve the division of 1368090 by 563. Set up the problem: Write 1368090 inside the division symbol and 563 outside. Look at the first four digits of the dividend, 1368. Estimate how many times 563 goes into 1368. Since 563 is close to 600, we can think of how many times 600 goes into 1200 (rounding 1368 down), which is about 2 times. Write 2 above the 8 in 1368. Multiply 2 by 563: 2 × 563 = 1126. Write 1126 below 1368. Subtract 1126 from 1368: 1368 - 1126 = 242. Write 242 below 1126. Bring down the next digit, which is 0. Write 0 next to 242, making the new number 2420. Determine how many times 563 goes into 2420. We can estimate by thinking of how many times 600 goes into 2400 (rounding 2420 down), which is about 4 times. Write 4 above the first 0 in the quotient. Multiply 4 by 563: 4 × 563 = 2252. Write 2252 below 2420. Subtract 2252 from 2420: 2420 - 2252 = 168. Write 168 below 2252. Bring down the next digit, which is 9. Write 9 next to 168, making the new number 1689. Determine how many times 563 goes into 1689. We can estimate by thinking of how many times 600 goes into 1800 (rounding 1689 up), which is about 3 times. Write 3 above the 9 in the quotient. Multiply 3 by 563: 3 × 563 = 1689. Write 1689 below 1689. Subtract 1689 from 1689: 1689 - 1689 = 0. Write 0 below 1689. Bring down the last digit, which is 0. Write 0 next to 0, making the new number 0. Determine how many times 563 goes into 0, which is 0. Write 0 above the last 0 in the quotient. The quotient is 2430, and the remainder is 0. Therefore, 1368090 ÷ 563 = 2430 with a remainder of 0. By breaking down each problem into manageable steps, we can solve even complex division problems accurately. This step-by-step approach not only provides the answer but also reinforces the understanding of the long division process.

In summary, mastering division, particularly long division, is essential for mathematical proficiency. By understanding the basic principles, following the step-by-step process, and practicing with various examples, anyone can improve their division skills. The problems we've solved here demonstrate the practical application of long division, breaking down complex calculations into simpler, manageable steps. Whether you're a student learning the basics or someone looking to refresh your math skills, the ability to perform long division accurately is a valuable asset. Remember, practice makes perfect, so keep working on these skills to build confidence and competence in mathematics.