Mastering Long Division with Remainders: A Step-by-Step Guide

Long division with remainders

Many students need help with long division, especially when remainders are involved. However, mastering long division with remainders is possible and essential for solving more complex math problems with the right approach and practice. In this article, I will provide a step-by-step guide to help you understand the basics of long division and how to do it with remainders. I will also cover long polynomial division, tips and tricks, common mistakes, and real-life scenarios where division with remainders is required.

Understanding the Basics of Long Division

Long division divides two numbers where the dividend (the number being divided) is written under the division symbol, and the divisor (the number dividing the dividend) is written outside the symbol. The quotient (the answer) is written above the division symbol, and the remainder (the amount left after division) is written next to the quotient.

For example, let’s divide 123 by 4:

long division with remainders

    30

__________

4 | 123

    |8 1

    —

    43

In this example, 123 is the dividend, 4 is the divisor, 30 is the quotient, and 3 is the remainder. To check our work, we can multiply the quotient by the divisor and add the remainder: 30 x 4 + 3 = 123.

How to Do Long Division with Remainders Step-by-Step

Now that we understand the basics of long division, let’s learn how to do it with remainders. Here’s a step-by-step process:

  • Write the dividend and divisor as shown above.
  • Divide the first digit of the dividend by the divisor. Write the quotient above the dividend and the remainder next to the dividend.
  • Write down the next digit of the dividend next to the remainder.
  • Repeat steps 2 and 3 until all dividend digits have been used.
  • The final number next to the quotient is the remainder.

Let’s use the same example from before and do it step-by-step:

    30

  __________

4 | 123

    8 1

    —

    43

  • Write 123 ÷ 4.
  • 4 goes into 1 zero times, so write 0 above the 1.
  • Multiply 0 by 4 and subtract it from 1: 1 – 0 = 1. Write 1 next to the 4 as the remainder.
  • Bring down the 2 and write it next to the 1: 12.
  • Divide 4 into 12: 12 ÷ 4 = 3. Write 3 above the 2.
  • Multiply 3 by 4 and subtract it from 12: 12 – 12 = 0. There is no remainder.
  • Bring down the 3 and write it next to the 0: 3.
  • Divide 4 into 3: 3 ÷ 4 = 0 with a remainder of 3.
  • Write 3 next to the 3 as the final remainder.

Therefore, 123 ÷ 4 = 30 R3.

Solving Long Division with Remainders with Decimals

Multiplying and dividing decimals. Long division with remainder

Long division can also be used to divide decimals. The process is similar to dividing whole numbers, but we need to pay attention to the placement of the decimal point. Here’s how to do it step-by-step:

  • Move the decimal point in the dividend to the right until it becomes a whole number. Count the number of times you moved the decimal point.
  • Move the decimal point in the divisor the same number of times to the right.
  • Divide the whole numbers as usual.
  • Place the decimal point in the quotient directly above the decimal point in the dividend.
  • Bring down the next digit of the dividend and continue the process until there are no more digits.

Let’s use an example: 3.6 ÷ 0.2.

Long Division with remainders

  • Move the decimal point in 3.6 to the right one place, making it 36. Count one move.
  • Move the decimal point in 0.2 to the right one place, making it 2. Count one move.
  • 36 ÷ 2 = 18.
  • Place the decimal point in the quotient above the decimal point in 3.6, making it 18.0.
  • Bring down the next digit of the dividend, which is 0. Continue the process.

Therefore, 3.6 ÷ 0.2 = 18.

Polynomial Long Division with Remainders

Polynomial long division divides two polynomials where the divisor is a polynomial of degree one or higher. The process is similar to long division with numbers, but we need to pay attention to the degree of the polynomials. Here’s how to do it step-by-step:

  • Write the dividend and divisor as shown above.
  • Divide the dividend’s first term by the divisor’s first term. Write the quotient above the dividend and multiply it by the entire divisor.
  • Subtract the result from the dividend and write the remainder.
  • Write down the next term of the dividend next to the remainder.
  • Repeat steps 2-4 until all terms of the dividend have been used.
  • The final polynomial next to the quotient is the remainder.

Let’s use an example: $$(x^3 + 3x^2 + 2x – 1) ÷ (x + 1).$$

long division with remainders

  • Write $$(x^3 + 3x^2 + 2x – 1) ÷ (x + 1).$$
  • Divide $$x^3$$ by x, which gives $$x^2.$$ Write $$x^2$$ above the dividend.
  • Multiply $$(x + 1)$$ by $$x^2$$, which gives $$x^3 + x^2.$$
  • Subtract $$x^3 + x^2$$ from $$x^3 + 3x^2$$, which gives $$2x^2$$. Write $$2x^2$$ as the remainder.
  • Bring down the next term, which is 2x. Write 2x next to $$2x^2$$ as $$2x^2 + 2x.$$
  • Divide $$x^2$$ by x, which gives x. Write x above the 2x.
  • Multiply $$(x + 1)$$ by x, which gives x^2 + x.
  • Subtract $$x^2 + x$$ from $$2x^2 + 2x$$, which gives $$x^2 + x.$$ Write $$x^2 + x$$ as the remainder.
  • Bring down the final term, which is -1. Write -1 next to $$x^2 + x$$ as $$x^2 + x – 1.$$
  • Divide $$x^2$$ by x, which gives x. Write x above -1.
  • Multiply (x + 1) by x, which gives x^2 + x.
  • Subtract $$x^2 + x$$ from $$x^2 + x – 1$$, which gives -1. Write -1 as the final remainder.

Therefore, $$(x^3 + 3x^2 + 2x – 1) ÷ (x + 1) = x^2 + 2x – 1$$ with a remainder of -1.

Tips and Tricks for Mastering Long Division with Remainders

Here are some tips and tricks to help you master long division with remainders:

  • Practice, practice, practice! The more you do long-division problems, the more comfortable and confident you will become.
  • Use graph paper or lined paper to keep your numbers in neat columns.
  • Use a ruler or straight edge to draw the division symbol and align your numbers.
  • Write down each step and double-check your work as you go.
  • If you get stuck, take a break and return to the problem later with fresh eyes.
  • Try to understand the concept behind the long division, not just memorize the steps.
  • Use real-life scenarios to practice division with remainders, such as dividing a cake among friends or calculating the cost per person at a restaurant.

Common Mistakes to Avoid in Long Division with Remainders

Long division with remainders division wheel

Here are some common mistakes to avoid when doing long division with remainders:

  • Remember to add a zero as a placeholder when the divisor does not enter the dividend’s first digit.
  • Carry a reminder to the next digit without subtracting the quotient’s and divisor’s product.
  • Placing the decimal point in the wrong place when dividing decimals.
  • You are mixing up the order of the terms when doing polynomial long division.
  • I need to write the remainder as the final answer.

By being aware of these mistakes and taking your time to double-check your work, you can avoid making them and improve your accuracy in long division

Long Division with Remainders Worksheets and Practice Problems

Long Divisions with remainders

Many worksheets and practice problems are available online to further practice long division with remainders. You can find them on websites such as Math-Drills.com, Math-Aids.com, and SuperTeacherWorksheets.com. These resources offer a variety of problems at different levels of difficulty, so you can choose the ones that best suit your needs.

Division with Remainders in Real-Life Scenarios

Division with remainders is a mathematical concept that is also applied in everyday life. For instance, while breaking a group into teams, certain individuals can be left out and require assistance in completing the team. Or, while sharing a pizza with pals, the leftovers might not be split equally. You can apply this idea to actual situations and use long division with remainders to make better decisions if you comprehend and grasp it.

Conclusion: The Importance of Mastering Long Division with Remainders

In conclusion, learning long division with remainders is crucial to figuring out more challenging math issues and making wise choices in everyday situations. You can feel confident performing long division with remainders by learning the fundamentals of long division, practicing with worksheets and problems, and avoiding common errors. This step-by-step manual has benefited you whether you are a student, a teacher, or someone who wishes to develop their mathematical abilities.

FAQs: Long Division with Remainders

A larger number is divided by a smaller one using long division with two digits and a remainder to determine how many times the smaller number can fit into the larger one and how much is left over. The steps are as follows:

  • Take the dividend’s first digit starting from the left.
  • Write the result as the quotient on top after dividing it by the divisor.
  • Write the difference that results from subtraction below.
  • Bring down the following dividend digit, if any.
  • Keep doing the same thing until there are no more digits to bring down.
  • If a remainder is present, it should be expressed as a fraction with the divisor as the numerator.

Now that we understand the basics of long division, let’s learn how to do it with remainders. Here’s a step-by-step process:

  • Write the dividend and divisor as shown above.
  • Divide the first digit of the dividend by the divisor. Write the quotient above the dividend and the remainder next to the dividend.
  • Write down the next digit of the dividend next to the remainder.
  • Repeat steps 2 and 3 until all dividend digits have been used.
  • The final number next to the quotient is the remainder.

Let’s use the same example from before and do it step-by-step:

    30

  __________

4 | 123

    8 1

    —

    43

  • Write 123 ÷ 4.
  • 4 goes into 1 zero times, so write 0 above the 1.
  • Multiply 0 by 4 and subtract it from 1: 1 – 0 = 1. Write 1 next to the 4 as the remainder.
  • Bring down the 2 and write it next to the 1: 12.
  • Divide 4 into 12: 12 ÷ 4 = 3. Write 3 above the 2.
  • Multiply 3 by 4 and subtract it from 12: 12 – 12 = 0. There is no remainder.
  • Bring down the 3 and write it next to the 0: 3.
  • Divide 4 into 3: 3 ÷ 4 = 0 with a remainder of 3.
  • Write 3 next to the 3 as the final remainder.

Therefore, 123 ÷ 4 = 30 R3.

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