In a multiplication problem, the top number to be multiplied is called the __multiplicand__, the
second number is called the __multiplier__ and the answer is called the __product__.

When we write a number, say 1,234, we refer to 1 as the leading digit (or the leftmost digit from the digit 4).

__General Procedure__

1. Count the number of digits in the multiplier. Move the beads for the multiplicand to the crosspiece with its unit digit placed to the unit answer column equal to the number of digits in the multiplier.

** For example**, if there is one digit in the multiplier, then the unit digit of the
multiplicand would be placed on column 7. If there are two digits in the multiplier,
then the unit digit of the multiplicand would be placed in column 6, and so on.

2. Move the beads for the multiplier to the crosspiece so that its unit digit is __two columns
to the left of the leftmost digit of the multiplicand__.

3. Starting from the __right__, multiply the unit digit of the multiplicand by the unit digit of
the multiplier. Place the product on the abacus with its unit digit on the unit answer
column (column 8).

Multiply the unit digit of the multiplicand by the 10's digit of the multiplier (if there is one) and add this product to the first one, placing its unit digit one column to the left of the first product's unit digit. Continue until all digits of the multiplier have been multiplied. (Before adding the last product on the abacus, clear the unit digit of the multiplicand from the abacus.)

4. Next, multiply the 10's digit of the multiplicand by the unit digit of the multiplier and add the product one column to the left of the unit answer column. (Each time a new digit of the multiplicand is multiplied, the unit digit of the product is moved one column to the left.) Multiply the 10's digit of the multiplicand by the 10's digit of the multiplier (if there is one) and add this product one column to the left of the last product's unit digit. Continue until all digits of the multiplier have been multiplied. (Before adding the last product on the abacus, clear the 10's digit of the multiplicand from the abacus.)

5. Continue in the same fashion with the 100's digit of the multiplicand (if there is one) until all digits of the multiplicand have been multiplied.

**Important:** Always remember to clear the current digit of the multiplicand from the
abacus before adding the product of the current digit of the multiplicand and the leftmost
digit of the multiplier.

When a multiplication problem is completed, none of the digits of the multiplicand will be left on the abacus and the multiplier will remain where it was. The answer will be read from left to right ending with its unit digit on column 8.

**Example 9.** (One digit times one digit)

Multiply: 3 x 2

Enter multiplicand (3) in col 7.

**Since there is one digit in the multiplier, the multiplicand is placed one column to
the left of the unit answer column (or column 7).**

Enter multiplier (2) in col 5.

**The multiplier is placed two columns to the left of the multiplicand (column 5).**

Multiply 3 x ? = 6; clear col 7,

**Multiply the digit of the multiplicand (3) by the digit of the multiplier (2) which
equals 6. Clear the multiplicand from column 7 before placing the answer on the
abacus.**

and add 6 on col 8.

Product is 6.

**Example 10.** Multiply: 7 x 3

Enter multiplicand (7) in col 7.

Enter multiplier (3) in col 5.

Multiply 7 x 3 = 21; clear col 7.

and add 21 in cols 7-8.

Product is 21.

**Abacus Shortcut: The number of bead movements in the above example can be
reduced by simply subtracting 5 beads in column 7 leaving 2 beads (which
represents the tens digit of the product) and then adding 1 bead in column 8 (which
represents the unit digit of the product) instead of removing the multiplicand from
column 7 and then placing the answer in columns 7 and 8.**

**Example 11.** (One digit times two digits)

Multiply: 4 x 15

Enter multiplicand (4) in col 6.

**There are two digits in the multiplier so the beads for the multiplicand (4) are
placed two columns to the left of the unit answer column (column 6).**

Enter multiplier (15) in cols 3-4.

**The multiplier is placed two columns to the left of the multiplicand (columns 3 and
4).**

Multiply 4 x 5 = 20

Add 20 in cols 7-8.

**Multiply the multiplicand (4) by the right digit of the multiplier (5) and place the
product (20) in columns 7-8.**

Multiply 4 x 1 = 4; clear col 6.

**Multiply the multiplicand (4) by the left digit of the multiplier (1). Clear the
multiplicand from col 6.**

and add 4 in col 7.

Add 5,

and subtract 1.

Product is 60.

**Add the product (4) in col 7. To add 4 to 2 in col 7, use complementary numbers.
Add 5 and subtract 1 (the complement of 4 relative to 5).**

**Example 12.** (Two digits times one digit)

Multiply: 25 x 3

Enter multiplicand (25) in cols 6-7

**The multiplier has one digit so the beads for the multiplicand are moved to the
crosspiece in columns 6-7.**

Enter multiplier (3) in col 4.

**Place the beads for the multiplier two columns to the left of the multiplicand.**

Multiply 5 x 3 = 15.

Clear col 7 and add 15 in cols 7-8.

**Multiply the right digit of the multiplicand (5) by the multiplier (3), clear the
multiplicand from col 7 and place the product (15) in columns 7-8.**

Multiply 2 x 3 = 6;

clear col 6 and add 6 in col 7.

Product is 75.

**Multiply the left digit of the multiplicand (2) by the multiplier (3). Clear col 6 and
add the product (6) in col 7.**

**Example 13.** (Two digits times two digits)

Multiply: 43 x26

Enter multiplicand (43) in cols 5-6.

**The multiplier has two digits so the beads for the multiplicand are moved to the
crosspiece in columns 5-6.**

Enter multiplier (26) in cols 2-3.

**Place the beads for the multiplier (26) in columns 2-3.**

Multiply 3 x 6 = 18;

add 18 in cols 7-8.**Multiply the unit digit of the multiplicand (3) by the unit digit of the multiplier (6)
and place the product (18) in columns 7-8.**

Multiply 3 x 2 = 6;

clear col 6 and add 6 in col 7.

**Multiply the unit digit of the multiplicand (3) by the 10's digit of the multiplier (2).
Clear column 6 and add the product (6) to column 7.**

Multiply 4 x 6 = 24;

add 24 in cols 6-7.

**Multiply the 10's digit of the multiplicand (4) by the unit digit of the multiplier (6)
and add the product (24) in columns 6-7. When adding the 24, first add 2 beads in
column 6. When 4 is added to the 7 in col 7, complementary numbers are used, so
subtract 6 in column 7 and add one bead in column 6.**

Multiply 4 x 2 = 8;

clear col 5 and add 8 in col 6.

Product is 1118.

**Multiply the 10's digit of the multiplicand (4) by the 10's digit of the multiplier (2).
Clear column 5 and add the product (8) in column 6.**

__Practice Exercises__

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