Long Division: Solving 315 ÷ 6 Step-by-Step

Dec 3, 2025 by Alex Johnson 44 views

Long division can seem daunting at first, but breaking it down step-by-step makes even complex problems manageable. If you're looking to understand how to tackle 315 ÷ 6 using the long division algorithm, you've come to the right place. This comprehensive guide will walk you through each step, ensuring you grasp the process thoroughly. So, let’s dive in and make long division feel less like a chore and more like a puzzle!

Understanding the Basics of Long Division

Before we jump into the specifics of 315 ÷ 6, let's briefly recap the core concepts of long division. Long division is essentially a method for dividing large numbers into smaller, more manageable parts. It involves dividing, multiplying, subtracting, and bringing down digits in a systematic way. Understanding this foundational method is crucial for successfully solving more complex division problems.

In long division, we have three main components:

Dividend: This is the number being divided (in our case, 315).
Divisor: This is the number by which we are dividing (in our case, 6).
Quotient: This is the result of the division (the number of times the divisor goes into the dividend).

The goal of long division is to find the quotient and any remainder. The beauty of long division is that it allows us to handle large numbers with relative ease, one digit at a time. This methodical approach not only helps in getting the correct answer but also enhances our understanding of numerical relationships. By mastering long division, you'll find that your confidence in tackling various mathematical problems significantly increases. Now, with these basics in mind, let's proceed to solve 315 ÷ 6 step by step.

Step 1: Setting Up the Long Division Problem

First things first, let's set up our long division problem correctly. This initial setup is critical because it ensures we organize our work and minimize errors. To set up the problem, we write the dividend (315) inside the division bracket and the divisor (6) outside to the left. This visual arrangement helps us clearly see the structure of the problem and the steps we need to follow.

      ______
  6 | 315

As you can see, the dividend (315) is nestled comfortably under the division symbol, while the divisor (6) stands guard outside. This setup mirrors the question we're trying to answer: "How many times does 6 fit into 315?" Correctly arranging the numbers like this provides a clear visual guide, making it much easier to proceed with the division process. It’s like laying out all the pieces of a puzzle before you start assembling it – you can see the whole picture and plan your moves. Now that our problem is set up neatly, we're ready to tackle the first actual step in the division process.

Step 2: Dividing the First Digit(s)

Now that we have set up the problem, the next step involves dividing the first digit (or digits) of the dividend by the divisor. We start by looking at the first digit of the dividend, which is 3. Can 6 go into 3? No, because 3 is smaller than 6. So, we move to the next digit and consider the first two digits together, which gives us 31. Now, the question becomes: How many times does 6 go into 31?

Think of your multiplication facts. We know that 6 multiplied by 5 is 30, which is less than 31, and 6 multiplied by 6 is 36, which is greater than 31. Therefore, 6 goes into 31 five times. We write the 5 above the 1 in 315, as this is the first digit of our quotient. This initial division is crucial as it sets the stage for the rest of the problem. If we miscalculate this first step, it can throw off the entire solution. By carefully considering the multiples of 6, we’ve determined that 6 fits into 31 five times, marking a significant milestone in our long division journey. With this first part of the quotient in place, we’re ready to move on to the next step, which involves multiplying and subtracting.

      5_____
  6 | 315

Step 3: Multiply and Subtract

After determining how many times the divisor goes into the initial digits of the dividend, we proceed with multiplication and subtraction. This is a critical step in long division, helping us to whittle down the dividend into manageable parts. In our case, we've established that 6 goes into 31 five times. So, we multiply the divisor (6) by the part of the quotient we just found (5). 6 multiplied by 5 equals 30. We write this 30 directly below the 31 in our dividend.

      5_____
  6 | 315
     30

Next, we subtract the 30 from 31. 31 minus 30 equals 1. We write this 1 below the 30. This subtraction tells us how much of the dividend remains after accounting for the initial division. The result, 1, is smaller than our divisor (6), which is a good sign because it confirms we chose the largest possible multiple of 6 that fits into 31. This multiplication and subtraction process is a fundamental loop in long division, allowing us to systematically break down the problem. Now that we have a remainder of 1, we're ready to bring down the next digit from the dividend and continue the process.

Step 4: Bring Down the Next Digit

With the subtraction complete, the next key step in long division is to bring down the next digit from the dividend. This step allows us to continue the division process with the remaining portion of the dividend. Looking at our problem, 315 ÷ 6, we have performed the subtraction and are left with a remainder of 1. The next digit in the dividend that we haven't used yet is 5. So, we bring the 5 down next to the 1, forming the number 15. This new number, 15, now becomes the focus of our division.

Bringing down the digit effectively combines the remainder from the previous step with the next digit in the dividend, creating a new number to work with. This step-by-step approach is what makes long division so manageable, as it breaks down a large division problem into smaller, more easily solvable parts. Now, with our new number 15 in place, we're ready to repeat the division process. We’ll ask ourselves, “How many times does 6 go into 15?” This cyclical process of dividing, multiplying, subtracting, and bringing down digits is the heart of the long division algorithm.

Step 5: Repeat the Division Process

Now that we have 15, we repeat the division process. We ask ourselves: How many times does 6 go into 15? Recalling our multiplication facts, we know that 6 multiplied by 2 is 12, which is less than 15, and 6 multiplied by 3 is 18, which is greater than 15. So, 6 goes into 15 two times. We write the 2 next to the 5 in the quotient above the division bracket.

Next, we multiply the divisor (6) by the new digit in the quotient (2). 6 multiplied by 2 equals 12. We write this 12 below the 15.

Then, we subtract 12 from 15. 15 minus 12 equals 3. We write this 3 below the 12. This completes one full cycle of the division process: divide, multiply, and subtract. The remainder we have now is 3.

This repetitive cycle is what makes long division so effective. By breaking the problem into smaller, manageable chunks, we can systematically work towards the solution. With our remainder of 3, we need to determine if we're done or if there's more to the division. In this case, since there are no more digits to bring down from the dividend, the 3 becomes our final remainder.

Step 6: Determine the Remainder

With no more digits to bring down from the dividend, we've reached the final step in our long division process: determining the remainder. In our problem, 315 ÷ 6, we have gone through all the steps and are left with a remainder of 3. This means that after dividing 315 by 6, we have 52 full groups of 6, and 3 left over.

So, we can express our answer as 52 with a remainder of 3, often written as 52 R 3. Alternatively, we can express the remainder as a fraction. To do this, we write the remainder (3) over the divisor (6), giving us the fraction 3/6. This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3. So, 3/6 simplifies to 1/2. Thus, we can also express our answer as 52 and 1/2.

Understanding the remainder is crucial because it completes the division process. It tells us the amount that is left over after dividing the dividend as evenly as possible. In practical terms, remainders can have significant implications, depending on the context of the problem. Whether we express the remainder as R 3 or 1/2, we have successfully completed the long division of 315 ÷ 6.

Conclusion: 315 ÷ 6 = 52 with a Remainder of 3

In summary, we've successfully completed the long division of 315 ÷ 6. By following each step methodically, we found that 315 divided by 6 equals 52 with a remainder of 3. This can be expressed as 52 R 3 or 52 1/2.

Long division might seem challenging initially, but breaking it down into manageable steps—setting up the problem, dividing, multiplying, subtracting, bringing down digits, and determining the remainder—makes it far less intimidating. Each step builds upon the previous one, leading us to the final solution in a clear and logical manner.

Mastering long division not only helps in solving mathematical problems but also enhances your understanding of numerical relationships and operations. It's a fundamental skill that builds confidence in tackling more complex arithmetic challenges.

So, the next time you encounter a division problem, remember the steps we've covered, and approach it with confidence. Practice makes perfect, and with consistent effort, you'll find long division becoming second nature. And for further exploration and practice, consider checking out resources like Khan Academy's Arithmetic Section for additional examples and exercises.