Along with addition, subtraction, and multiplication, division is one of the four basic mathematical operations. You can also divide by decimals, fractions, or exponents in addition to whole numbers. If one of the numbers is a single digit, then you can use short division. However, the long division should be learned first because it is the foundation for the rest of the process.

## Long Division

## 1. To represent the division, write out a long division bar.

The division symbol () is shaped like a set of closing parentheses joined to a horizontal line and placed above a series of numbers.

To **perform a division**, set the divisor (the number to be divided) to the right of the long division bar and the dividend (the number to be divided into) to the left of the long division bar.

- First example problem (beginners): 65 5. Substitute “65” for the division symbol and position “5” outside the bar. To read it, imagine a 565 grid with the 65 positioned below the horizontal line.
- 136 3 is an intermediate-level sample problem. Move the 3 away from the dividing line and the 136 within. It should read as 3136, but with the 136 placed below the horizontal line.

## 2. Find the divisor’s quotient and the dividend’s first digit.

To put it another way, count the number of times the divisor (the number outside the division bar) enters the first digit of the dividend.

Above the division line, above the first digit of the divisor, write the whole number result.

- Example 1 (565): 5 is the divisor, and 6 is the initial digit of the dividend (65). Five times into six is one. Hence the top of the divisor bar should be aligned with the 6.
- In problem 2 of the samples (3136), divisor 3 cannot divide evenly into the dividend’s initial digit 1. To do this, place a zero above the division bar, aligned with the 1.

## 3. The number above the division symbol should be multiplied by the denominator.

Simply multiply the result of the division by the divisor and the number you entered above the bar (the number to the left of the division bar).

In a new row below the dividend, align the result with the dividend’s first digit.

- Example Problem 1 (565): Multiply the number above the bar by the divisor to get the answer, and write it below the 6 in the original expression, which is 5.
- Example 2 (3136): Since there is a 0 in the blank space above the division bar, multiplying the resulting number by 3 (the divisor) gives you a 0 as a solution. Under the 1 in 136, jot down a zero on a new line.

## 4. Take the first digit of the payout and deduct the product of the multiplication.

To calculate the dividend, remove the digit from the dividend above it by the number you just entered in the new row below it. Using a new row, write the answer so that it corresponds with the digits in the subtraction.

- In Example 1 (565), the first step is to find the answer by subtracting the digit 5 (the product in the new row) from the digit 6 (the first digit of the dividend): 6 – 5 = 1. In a new row below the 5, enter the answer.
- Sample Problem 2: (3 136) Subtract the zero from the one directly above it to get the answer (the first digit in the dividend). Immediately below the zero, create a new row and enter the result.

## 5. The dividend’s second digit should be carried down.

The second digit of the dividend should be moved to the new bottom row, to the right of the result of the subtraction.

To solve the first example problem (565), move the 5 from the final product down to the same line as the 1 obtained by subtracting 5 from 6.

Puts you at 15 in a row.

The solution to Example 2: Multiply 3 by 136 to get 13.

## 6. Iterate the laborious process of dividing.

This time, use the new number in the bottom row as the dividend (the number to the left of the division bar) (the result of your first round of calculations and the digit you carried down).

In the same way, as previously, you can get your answer by doing the operations of division, multiplication, and subtraction.

To carry on with 565, divide 5 (the dividend) into the new number, and then write the quotient (3 since 15 5 = 3) to the right of the 1 above the division bar.

Then, write the product (3 x 5 = 15) under the division bar instead of the bar itself, as 3 is above the bar, and 5 is the dividend. Finally, in a new bottom row, remove 15 from 15, and enter 0.

No further digits may be carried down from the divisor. Hence Sample Problem #1 is finished. It looks like your solution is more than the division bar.

## 7. Try the long division again.

To recap, you divide, multiply, and subtract in the same order as before.

- For 3136, count the number of times 3 divides into 13 and record the result to the right of the 0 in the space above the division bar.
- After that, multiply 4 by 3, and record the number under 13.
- The last step is subtracting 12 from 13 and recording the number below 12.

## 8. In order to obtain the remainder, perform another round of long division.

Remember that there will be a remnant when you solve this problem (a number left over at the end of your calculation).

Copy this leftover down and put it next to your final answer.

- For 3136, go through the motions once more. Take off 6 from 136 and add it to the bottom row to make 16. Take 3 and divide it by 16; the answer can be found above the dividing line.
- Add a new row at the bottom of your spreadsheet and write the product of 5 by 3 there. To do this, we need to take 15 away from 16, then record the resulting number in a brand-new bottom row.
- With no more digits to carry down in the dividend, the problem is complete, and the final number is the remainder (the amount left over). In this case, you would write “45 r.1” above the division bar with an “r.” in front of it.