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Long Division

What will I learn from this article?

After reading this article, you will be able to:

  • define a long division;
  • identify the parts of a long division;
  • properly use the method of long division; and
  • solve problems involving long division. 

What is a long division?

Division is one of the four fundamental mathematical operations, with addition, subtraction, and multiplication being the other three. Long Division is a technique for dividing large numbers and algebraic expressions into multiple steps following a predetermined sequence. It is the most frequently used method for dividing problems. 

The illustration below shows how long division works.

What are the parts of long division?

If we are asked to divide two numbers, we usually see the notation 35 ÷6, and it is the same thing in long division. We have a dividend, a divisor, a quotient, and for some, a remainder. 

Let us look deep into it! 

  • The dividend is the number being divided to, which in this case is 100.
  • The divisor is the number that divides the dividend; in this case, 3 is the divisor.
  • The quotient is the result of dividing the dividend and the divisor. In this case, it is 33. 
  • The remainder is the leftover part of a number after doing the process of division. In this case, the remainder is 1. 

In English-speaking countries, the long division does not use the division slash or division sign symbols but instead creates a tableau. A right parenthesis “)” or vertical bar “|” separates the divisor from the dividend; a vinculum separates the dividend from the quotient (i.e., an overbar). These two symbols are sometimes referred to as a long division symbol or division bracket. It evolved in the 18th century from a single-line notation in which the dividend and quotient were separated by a left parenthesis. 

How to do long division?

The long division method can help us divide two whole numbers, two numbers with decimals, and even dividing polynomials. The process of long division begins by dividing the dividend’s left-most digit by the divisor. The result’s first digit is the quotient (rounded to an integer), and the remainder is calculated (this step is notated as a subtraction). This remainder is carried forward when the process is repeated on the next digit of the dividend (referred to as ‘bringing the next digit down’ to the remainder). The process is complete when all digits have been processed and no remainders remain.

In simple terms, long division involves only 6 steps.

  1. Divide.
  2. Multiply. 
  3. Subtract.
  4. Bring down.
  5. Repeat or Remainder. 
  6. Check.

Dividing two whole numbers

To divide two whole numbers:

Step 1: Compare the dividend and the first digit of the divisor. 

  • If the divisor is less than or equal to the first digit of the dividend, then divide the two numbers.
  • If the divisor is greater than the first digit of the dividend, then divide the divisor to the first and second digit of the dividend.

Step 2: Write the quotient on the top of the division bracket. 

Step 3: Multiply the quotient and the divisor and write it below the dividend.

Step 4: Subtract the product of the quotient and the divisor from the dividend. 

Step 5: Bring down the next digit that you weren’t able to use. 

Step 6: Repeat the same process until there is no digit to bring down.

Step 7: If a remainder exists, ensure that it is less than the divisor. Otherwise, there must be a process in the multiplication or subtraction process.

Step 8: Lastly, the remainder must be added to the quotient and must be written in the form of rm , where r is the quotient and m is the divisor. 

Example #1

What is the quotient if 424 is divided by 4?

Solution

Long DivisionStep-by-step explanation
Set up the division problem using the long division symbol. In the given problem, 424 is the dividend and 4 is the divisor. 
Put 424 inside the division bracket and put 4 outside the bracket, as shown in the figure.
Compare the first digit of the dividend to the divisor. Since the divisor, 4, is equal to the first digit of the dividend, which is also 4, that means we can divide the two numbers.
By dividing 4 by 4, we will get the quotient of 1. The quotient is then placed on top of the division bracket.
Multiply the quotient 1 to the divisor. Thus, multiplying 1 by 4 will result to 4. Then, put the product below the dividend. 
Subtract the product to the first digit of the dividend. 
Bring down the next digit of the dividend and insert it next to 0. 
Divide 2 by 4. Thus, 2  ÷ 4 = 0
Then, put 0 on top of the division bracket next to 1. 
Multiply 0 to the divisor 4, and put the product below the line 02.   
Subtract 0 from 02. Thus,02 – 0 = 2
Bring down the last digit, 4, to the line of 2. 
Divide 24 by 4. 24 ÷ 4 = 6
Then, put the quotient at the top of the division bracket. 
Multiply 6 to the divisor.
6 x 4 = 24
Then, put the product below the 24 inside the division bracket. 
Subtract 24 from 24. Thus, the difference is 0. This means that there is no remainder if we divide 424 by 4.  
Therefore, the quotient of 424 ÷ 4 is 106.

Example #2 

What is the quotient if 583 is divided by 7?

Solution

Long DivisionStep-by-step explanation
Set up the division problem using the long division symbol. In the given problem, 583 is the dividend and 7 is the divisor. 
Put 583 inside the division bracket and put 7 outside the bracket.
Compare the first digit of the dividend to the divisor. Since 5 is less than the divisor, then we already know that the answer is 0.
Thus, we will use the next digit after 5. 
Now, we will divide 58 by 7. 
Dividing 58 by 7 will result to 8. 
Put the quotient, 8, on top of the division bracket.
Multiply the quotient to the divisor. Thus,8 x 7 = 56
Put the difference below the dividend. 
Subtract 56 from 58. Thus,58 – 56 = 2.
Bring down the last digit of the dividend in the same line of 2. 
Divide 23 by 7. Thus,23 ÷ 7 =3
Put the quotient on top of the division bracket, next to 8. 
Multiply 3 to the divisor. Thus, 3 x 7 = 21
Then, put the product below the last line, which is 23.
Subtract 21 from 23. Thus, 23 – 21 = 2
Since we do not have any digit to bring down, this means that 2 is the quotient of the two numbers. Thus, we will write it as 27 and add it to the number on top of the division bracket. 
Thus, 83 + 27 =  83 27
Therefore, the quotient when 583 is divided by 7 is 83 with a remainder of 2, or if written mathematically, the quotient is 83 27.

Example #3 

What is the quotient when 2448 is divided by 19? 

Solution

Long DivisionStep-by-step explanation
Set up the division problem using the long division symbol. In the given problem, 2448 is the dividend and 19 is the divisor. 
Put 2448 inside the division bracket and put 19 outside the bracket, as shown in the figure.
Since the divisor is a 2-digit number, we will use the first two digits of the dividend.
Find how many 19s can there be in 24. Since there can only be one 19 in 24, put 1 on top of the division bracket. 
Multiply the quotient to the divisor.
Thus, 1 x 19 = 19.
Then, put the product below the dividend. 
Subtract 19 from 24. Thus,24 – 19 = 5.
Bring down the next digit to the last line inside the division bracket. 
Determine how many 19s can there be in 54, and put the answer on top of the division bracket.
Multiply 2 to the divisor. Thus,2 x 19 = 38.
Then, write the product, 38, below the line of 54. 
Find the difference of 54 and 38. Thus, 54 – 38 = 16.
Bring down the last digit to the line of 16. 
Determine how many 19s are there in 168. Then, put the answer on top of the division bracket. 
Multiply 8 to the divisor. Thus,8 x 19 = 152.
Then, write the product below the last line.
Subtract 152 from 168.
Thus, 168 – 152 = 16
Since we do not have any digit to bring down, this means that 16 is the quotient of the two numbers. Thus, we will write it as 1619 and add it to the number on top of the division bracket. 
Thus, 128 + 1619 =  128 1619
Therefore, the quotient when 2448 is divided by 19 is 128 with a remainder of 16, or if written mathematically, the quotient is 128 1619.

Dividing two numbers with decimal points

  1. Locate all decimal points within the dividend n and divisor m.
  2. If necessary, simplify the long division problem by shifting the decimals of the divisor and dividend to the right (or to the left) by the same number of decimal places, so that the decimal of the divisor is to the right of the final digit.
  3. When performing long division, maintain a straight line from top to bottom beneath the tableau.
  4. Ensure that the remainder for each step is less than the divisor. If it is not, one of three things could be wrong: the multiplication is incorrect, the subtraction is incorrect, or a larger quotient is required.
  5. Finally, the remainder, r, is added as a fraction, rm, to the quotient.

Example

Divide 506.25 by 5.

Long DivisionStep-by-step explanation
Set up the division problem using the long division symbol. In the given problem, 506.25 is the dividend and 5 is the divisor. 
Put 506.25 inside the division bracket and put 5 outside the bracket.
Put the decimal point on top of the division bracket in the same position as the dividend. 
Compare the first digit of the dividend to the divisor. Since 5 is equal to the divisor, then we already know that the answer is 1.
Then, the quotient, which is 1, is placed on top of the division bracket.
Multiply 1 to the divisor. Thus,1 x 5 = 5
Then, put the product below the first digit of the dividend.
Subtract 5 from 5. Thus,5 – 5 = 0.
Bring down the next digit of the dividend to the last line.
Divide 0 by 5. Thus,0 ÷ 5= 0
Put the quotient of 5 and 0 on top of the division bracket.
Multiply 0 to the divisor. Thus,0 x 5 = 0
Then put the product below the last line.
Subtract 0 from 0. Thus, the answer is 0. 
Bring down the next digit of the dividend, next to the last line.
Determine the number of 5s that can be in 6. Then, put the answer on top of the division bracket. 
Multiply 1 to the divisor. Thus,1 x 5 = 5
Then, put the product below the last line. 
Subtract 5 from 6. Thus,6 – 5 = 1
Bring down the next digit of the dividend to the last line, next to 1. 
Determine the number of 5s that can be in 12. By division, we will know that there are two 5s that can be in 12. 
Then, put 2 on top of the division bracket. 
Remember to not move the decimal point – as it should be in the same position as the dividend. 
Multiply 2 to the divisor. Thus,2 x 5 = 10
Then, put the product below the last line. 
Subtract 10 from 12. Thus,12 – 10 = 2.
Then, put the difference below the last line.
Bring down the last digit of the dividend to the last line, next to 2. 
Divide 25 by 5. Thus,25 ÷ 5 = 5
Then, put the quotient on top of the division bracket. 
Multiply 5 to the divisor. Thus,5 x 5 = 25.
Then, put the product of two numbers below the last line.
Subtract 25 from 25. Thus, 25 – 25 = 0.This means that there is no remainder if we divide 506.25.  
Therefore, the quotient if we divide 506.25 by 5 is 101. 25.

When the quotient is not an integer and the division operation is extended beyond the decimal point, one of two things may occur:

The process may terminate when a remainder of 0 is reached; or it may continue.

A remainder that is identical to a previous remainder that occurred after the decimal points were written could be obtained. In the latter case, continuing the process would be pointless, as the same sequence of digits would appear in the quotient repeatedly from that point on. As a result, a bar is drawn over the repeating sequence to indicate that it will continue to repeat indefinitely (i.e., every rational number is either a terminating or repeating decimal).

What is the significance of long division?

Long division enables the division problem to be broken down into a series of simpler steps. Long division of integers can easily be extended to include rational non-integer dividends. This is due to the fact that every rational number has a recursive decimal expansion. Additionally, the procedure can be extended to encompass divisors with a finite or terminating decimal expansion (i.e. decimal fractions). 

The technique used in long division also made way in other areas of division such as:

  • Division of polynomials – when dividing polynomials we usually use synthetic division or polynomial long division.
  • Binary division – calculations using the binary number system are simplified because each digit in the course can only be either 1 or 0 – no multiplication is required because multiplying by either results in the same number or zero.

Moreover, it enables the execution of computations involving arbitrarily large numbers via a series of simple steps. Short division is the abbreviation for long division, and it is almost always used instead of long division when the divisor has only one digit. Chunking (affectionately referred to as the partial quotients method or the hangman method) is a less mechanical method of long division popular in the United Kingdom that contributes to a more holistic understanding of the division process. 

Is long division still necessary to learn?

Calculators and computers have become the most frequently used tools for solving division problems, obviating the need for a traditional mathematical exercise and reducing educational opportunities to demonstrate how to do so using paper and pencil techniques. (Internally, those devices employ a variety of division algorithms, the quickest of which rely on approximations and multiplications to accomplish the task.) Long division has been singled out for de-emphasis or even omission from the school curriculum in the United States, despite its traditional introduction in the fourth or fifth grades.

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