In a division problem, the number to be divided is called the __dividend__, the number that
divides into the dividend is called the __divisor__ and the answer consists of a quotient and
sometimes a__ remainder__.

** NOTE:** When we divide two numbers, say, 17 divided by 3, we get a quotient of 5 and a
remainder of 2/3. For convenience of abacus operation, we shall denote the remainder
simply by 2 and neglect the divisor 3. Thus, the remainder indicates the difference between
the product of the quotient times the divisor from the digits of the dividend.

__Division involving Divisors of One Digit__

__General Procedure__

1. Move the beads for the dividend to the crosspiece so that its unit digit is placed
__one __column to the __right__ of the reference column,. (For dividends of 6 or less digits, the
8th column is the reference column. For dividends of more than 6 and up to 9 digits,
the 11th column is used as the reference column.)

2. Move the beads for the divisor to the crosspiece so that its unit digit will be three columns to the left of the leftmost digit of the dividend.

3. Divide the divisor into the leading (or leftmost) digit of the dividend and place the quotient and/or the remainder on the abacus. There are two cases to consider.

**Case 1**

If the divisor is less than or equal to the leading digit of the dividend, the product of the
quotient times the divisor is __subtracted__ from the leading digit of the dividend. This
difference becomes the new leading digit. The quotient is now entered one column to the
left of the leading digit of the dividend. If the difference is zero, continue the operation of
division with the next digit of the dividend (which is now considered to be the leading
digit).

**Case 2**

If the divisor is greater than the leading digit o the dividend, the leading digit of the
dividend is considered as being 10 times larger. This means that the divisor divides into 10
times the leading digit of the dividend. The quotient is first entered in the column where
the leading digit of the dividend is, (the quotient replaces the leading digit), and then the
remainder is __added__ to the column on the right.

When the remainder is added to the column on the right, the sum could become equal to
or greater than 10. Since 10 or more beads cannot be placed on the column on the right,
simply place the unit digit of the sum in this column (with the understanding that this
number represents one number which is equal to or greater than 10). When the divisor is
divided into this number, Case 1 applies. We add the quotient to the left column and this
column is then __replaced__ by the remainder. (See Example 17.)

Continue in the same fashion as above until there are no digits left in the dividend or the remaining dividend is one digit only an it is less than the divisor. That is the remainder.

In division, for convenience, we shall use the letter "R" to represent the word
"__remainder__." Thus R0 represents no remainder, R1 a remainder of one, and so on.

**Example 14.** Divide: 4 / 8

Enter dividend (8) in col 9.

**The dividend (8) is placed so that its unit digit is one column to the right of the unit
reference column.**

Enter divisor (4) in col 6.

**The divisor (4) is placed so that its unit digit is three columns to the left of the
leftmost digit of the dividend.**

4 into 8 = 2; 2 x 4 = 8 (quotient x divisor).

Subtract 8 from dividend in col 9,

**Here we subtract the product of the quotient times the divisor from the leading digit
of the dividend.**

and enter the quotient (2) in col 8.

Quotient is 2.

Remainder is 0.

**This leaves 0 in column 9 and the quotient (2) is entered in column 8.**

**Example 15.** Divide: 2 / 78

Enter dividend (78) in cols 8-9.

Enter divisor (2) in col 5.

2 into 7 = 3 R1; 3 x 2 = 6

Subtract 6 in col 8,

and enter the quotient (3)in col 7.

**(The remainder 1 is left in column 8.)**

2 into 10 = 5 R0.

Subtract 10 in col 8,

**Here the divisor 2 is larger than the leading
digit 1 in column 8. We now apply Case
2. We consider the 1 to be 10 times larger (or 10) and then divide.**

and enter quotient (5) in col 8.

2 into 8 = 4 R0; 4 x 2 = 8.

Subtract 8 in col 9,

and add the quotient (4) in col 8.

Quotient is 39.

Remainder is 0.

**The quotient (39) is in cols 7-8. There is no remainder. (If there was one, it would
have appeared in col 9.)**

**Example 16.**

Divide: 3 / 703

Enter dividend (703) in cols 7-9.

Enter divisor (3) in col 4.

3 into 7 = 2 R1; 2 x 3 = 6.

Subtract 6 in col 7,

and enter quotient (2) in col 6.

3 into 10 = 3 R1.

3 x 3 = 9.

Subtract 1 (10) from col 7,

place quotient (3) in col 7,

**Abacus Shortcut: The number of bead movements in the above example can be
reduced by simply adding 2 beads in column 7 instead of subtracting 1 bead and
then placing 3 beads in the same column.**

and add remainder (1) in col 8.

3 into 10 = 3 R1.

3 x 3 = 9;

Subtract 1 (10) in col 8,

place quotient (3) in col 8,

**Abacus Shortcut: The number of bead movements in the above example can be
reduced by simply adding 2 beads in column 8 instead of subtracting one bead and
then placing 3 beads in that column.**

and add remainder (1) in col 9.

3 into 4 = 1 R1; 1 x 3 = 3.

Subtract 3 in col 9,

and add quotient (1) in col 8.

Quotient; is 234.

Remainder is 1.

**Example 17.** Divide 7 / 7691

Enter dividend (7691) in cols 6-9.

Enter divisor (7) in col 3.

7 into 7 = 1;

1 x 7 = 7.

Subtract 7 in col 6.

and enter quotient (1) in col 5.

7 into 60 (6 x 10) = 8 R4; 8 x 7 = 56

Place quotient (8) in col 7.

and add remainder 4

in col 8.

**To add 4 in col 8, use the concept of complementary numbers and subtract 6 in col 8
leaving 3. Remember that this number 3 actually represents 13. (For an explanation,
refer to Case 2 in the General Procedure for Division.)**

7 into 13 (3 + 10) = 1 R6;

Add 1 in col 7,

and enter the remainder

(6) in col 8.

Divide 7 into 60 (6 x 10 = 8 R4

Place the quotient 8 in col 8,

and add the remainder 4 in col 9.

Quotient is 1098.

Remainder is 5.

__Practice Exercises__

**Division Problems with Divisors of More than One Digit**

Division involving divisors of more than one digit requires the ability of adding and subtracting two digit numbers mentally. Therefore, the study of this type of division is reserved for advanced students and is not included in this manual. Examples can be practiced by using the Demonstration mode of the "A is for Abacus" program (not included).

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