Long Division

Short Course 

This is a short course (slide show) in long division. All of the examples in this show use two digit divisors.   

Setting up the Division Calculation

The process for setting up the division calculation remains the same. First, set up the dividend. Enter this number from on the right side of the abacus. If the dividend is a 3-digit number, then from left to right, set each digit starting in column 3 (for an integer remainder). If you want the remainder to a decimal, move the starting position to the left to accommodate the decimal digits. We will perform our division calculations to 3 decimal positions so the dividend will be entered 3 additional columns to the left.

Then, set up the divisor. Start from the left side of your abacus, in column 13, and set each digit of the divisor. If the divisor has 2 digits, it will be set in columns 13 and 12.  

The result of the division calculation is the quotient. We will enter this number in the in the columns to the left of the dividend using the “equal, shift 2 left” and “not equal, shift 1 left” principles.

The columns for the quotient are determined by adding the number of digits in the divisor and adding 1 for the process of division. If the division calculation has a decimal remainder, remember to also add the number of decimal digits. The quotient is the number to the left of this column. The remainder is the number to the right of this column.

Long Division Calculations

The process for setting up the division calculation remains the same as in the previous pages. First, set up the dividend. Enter this number from on the right side of the bottom abacus. If the dividend is a 3digit number, then from left to right, set each digit starting in column 3.

Next, set up the divisor. Start from the left side of the bottom abacus, in column 13, and set each digit of the divisor. A two-digit divisor will be set in columns 13 and 12.  

The result of the division calculation is the quotient. We will enter this number in the in the columns in the top abacus.

It is reasonable to assume that we would follow this same process that we used in the previous chapter to divide with two-digit divisors.  That is, we divide the first digit of the divisor into either the first digit or the first two digits of the dividend.  There is only one problem. Unfortunately, this process only works for a small group of divisors with 2 or more digits. We need to use a better method that works for ALL divisors of 2 or more digits.  

To address this issue, we will use a simple process that works for divisors of 2 or more digits.  Instead of using the first digit of the divisor we mentally add 1 to the first digit and divide it into the dividend. Then we divided the higher divisor into the first or first two digits of the dividend. This digit is entered in the left of the dividend. Then we multiply the result of this division by each digit of the original divisor and subtracted the product from the dividend.  Each round of division continues until the dividend is 0 or less than the divisor. This process is also discussed and used in the Simplified Abacus Division text.

Sometimes, when we are use a higher divisor, the result of the division will need to be added in the same column of the previous division.  We call this upward adjustment. 

When the first digit of the divisor is nine (9), we will not be able to use a higher divisor.  In this situation, we will need to divide the first digits of the dividend by 9 but reduce the quotient by 1. Then we follow the same process division process using the lower quotient,  

So, to perform our division calculations, we will follow these steps:  

• • Divide the higher divisor (or first digit of the divisor if it equals 9) into the first digit or the first two digits of the dividend.  If the first digit of the divisor is 9, reduce the result of the division by 1 (lower quotient). The location of the partial quotient is determined by the first digit of the divisor.

    • • If the number of digits we use in the divisor equals the number of digits we use in the dividend, we apply the “equal, shift 2 left” principle. This means we move 2 columns to the left of the dividend and then set the result of the division calculation in this column.

      • If the number of digits we use in the divisor doesn’t equal the number of digits we use in the dividend, we apply the “not equal, shift 1 left” principle.  In this case we move 1 column to the left and set the results of the division calculation in this column.

  • Then, multiply the number that you just entered by each digit of the divisor.  Remember, the  product of two single digits is always a 2-digit number (3 * 3 = 09). Then subtract the result from the correct columns of the dividend.  Each product is moved one column to the right as  we multiply each digit of the divisor. The final result gives you a partial dividend.

  • If the partial dividend is greater than the divisor, repeat the division process by dividing the      first digit of the divisor into either the first digit or first 2 digits of the new partial dividend,      setting the result in the correct column, multiplying the result by the divisor and subtracting   the product from the partial dividend.

• • • • The division process ends when the partial dividend is 0 or less than the value of the divisor. 

      • The columns for the quotient are determined by adding the number of digits in the divisor and adding 1 for the process of division. If the division calculation has a decimal remainder, remember to also add the number of decimal digits. The quotient is the number to the left of this column. The remainder is the number to the right of this column.

Examples

Now, let’s look at some more examples of this process. In these examples, the dividend will have many digits, the divisor will be two digits and the remainder will be a decimal. Click on the link to show the steps to calculate the answer. 

• Example:  868 divided by 41

• Example:  636 divided by 93

• Example:  6,655 divided by 75 

• Example:  67,180 divided by 48

• Example:  72,357 divided by 87

Build Your Skill

Now you are ready to try some calculations on your own. Click on the problems and their solutions to test your skill.