Multiply Using the Russian Peasant Method

Russian peasant multiplication is an interesting way to multiply numbers that uses a process of halving and doubling. Like standard multiplication and division, Russian peasant multiplication is an algorithm; however, it allows you to multiply any two whole numbers using only multiplication and division by 2. Although Russian peasant multiplication is not as quick as the multiplication method that is now standard, it's still fun to try.

Steps

Trying the First Example

1. Think of the numbers you want to multiply. Choose two numbers that you want to multiply, whether to solve a particular problem or just to try out the Russian peasant method of multiplication.
   • For example, try multiplying 146 x 37.
2. Make two columns. Using a piece of paper and pen, divide the piece of paper into two columns by drawing a line down the middle of the paper. Write one of the numbers you want to multiply at the top the each column.
   • In this example, write “146” at the top of the left column, and “37” at the top of the right column.
3. Halve the number in the left column repeatedly. Multiply and Divide Integers the number at the top of the left column by 2 continually until you get to 1. Ignore any remainder each time you halve the number. Write each halved number down in the left column, in order. In this example:
   • Start by halving 146 (146 ÷ 2 = 73). Write “73” in the left column below “146.”
   • Next, halve 73 (73 ÷ 2 = 36 with a remainder of 1). Write “36” in the left column below “73,” ignoring the remainder.
   • Next, halve 36 (36 ÷ 2 = 18). Write “18” in the left column below “36.”
   • Next, halve 18 (18 ÷ 2 = 9). Write “9” in the left column below “18.”
   • Next, halve 9 (9 ÷ 2 = 4 with a remainder of 1). Write “4” in the left column below “18,” ignoring the remainder.
   • Next, halve 4 (4 ÷ 2 = 2). Write “2” in the left column below “4.”
   • Finally, halve 2 (2÷ 2 = 1). Write “1” in the left column below “2.”
4. Double the number in the right column until the columns are the same length. Multiply the number in the second column by 2 until there are the same amount of numbers here as there are in the first column. In this example:
   • Each column should have 8 numbers in it. This is because it took seven steps of dividing the original number in the right column to reach 1.
   • In the right column, start by doubling 37 (37 x 2 = 74). Write “74” in the right column below “37.”
   • Next, double 74 (74 x 2 = 148). Write “148” in the right column below “74.”
   • Next, double 148 (148 x 2 = 296). Write “296” in the right column below “148.”
   • Next, double 296 (296 x 2 = 592). Write “592” in the right column below “296.”
   • Next, double 592 (592 x 2 = 1184). Write “1184” in the right column below “592.”
   • Next, double 1184 (1184 x 2 = 2368). Write “2368” in the right column below “1184.”
   • Finally, double 2368 (2368 x 2 = 4736). Write “4736” in the right column below “2368.”
5. Cross out rows with an even number in the left column. Using your pen, strike through those horizontal rows which begin with an even number in the left column. In this example:
   • There are 8 rows. You will strike through 5 of them.
   • Strike through the rows beginning with 146, 36, 18, 4, and 2 in the left column, since these are even numbers. Working left to right, this means striking through the first row (146, 37), the third row (36, 148), the fourth row (18, 296), the sixth row (4, 1184), and the seventh row (2, 2368).
   • Please note that you should strike through even numbers, even if they begin with an odd numeral. For example, you should strike through the row beginning with 146 since it is an even number, even though 146 begins with an odd numeral, 1. Likewise, you should strike though 36, since it is an even number, even though 36 begins with an odd numeral, 3.
   • If you prefer, you can just strike through the numbers in the right side that fall into the rows that begin with an even number on the left side (as in the picture above). In this example, this means striking through the numbers on the right hand side of the first, third, fourth, sixth, and seventh rows: 37, 148, 296, 1184, and 2368.
6. Find the sum of the remaining numbers in the right column. Add the numbers in the right column that you did not strike through. The sum of these numbers is equal to the product you would get from multiplying the original numbers using the standard method. In this example:
   • The remaining numbers in the right column are 74, 592, 4736.
   • Add these numbers to get the sum of 5402 (74 + 592 + 4736 = 5402).
   • This number is the product of multiplying the original numbers in this example, 146 and 37 (146 x 37 = 5402).

Troubleshooting

1. Multiply by hand to check your solution. You can use the standard method of multiplying by hand in order to make sure that you got the right answer using the Russian peasant method of multiplication. Using the previous example (146 x 37):
   • Write “146,” and then “37” below it so that the numbers line up flush on the right side.
   • Draw a line below these numbers to create a work area. You will do your calculations below this line.
   • Multiply 7 x 6 (= 42). Write “2” down in the work area and carry the “4.”
   • Multiply 7 x 4 (= 28). Add the carried “4” to this sum to get a new sum, 32 (4 + 28 = 32). Write down “2” in the work area to the left of “2” you wrote down in your work area, and carry the “3.”
   • Multiply 7 x 1 (= 7). Add the carried “3” to this sum to get a new sum, 10 (7 + 3 = 10). Write down “10” in your work area, to the left of the “22” that is already there.
   • You should now have the number “1022” written down in your work area.
   • Start working on a new row in your work area, directly below “1022.”
   • Write “0” on the right of this new row, directly below the rightmost “2” of “1022.”
   • Multiply 3 x 6 (= 18). Write “8” to the left of the “0,” and carry the “1.”
   • Multiply 3 x 4 (= 12). Add the carried “1” to this sum to get a new sum, 13 (12 + 1 = 13). Write down “3” in your work area to the left of the “8,” and carry the “1.”
   • Multiply 3 x 1 (= 3). Add the carried “1” to this sum to get a new sum, 4 (3 + 1 = 4).
   • You should now have the number “4380” written in the second row of your work area.
   • Add the two rows in your work area (1022 + 4380). You should get the sum 5402 (1022 + 4380 = 5402).
   • This sum, which is the same number derived by the Russian peasant method, verifies that the method produced the correct answer.
2. Verify your solution with a calculator. You can also use an electronic calculator in order to make sure that you got the right answer using the Russian peasant method of multiplication. Using the previous example (146 x 37):
   • Enter 146 in the calculator by pressing the corresponding digits, 1, 4, 6.
   • Press the multiplication (“x”) key.
   • Enter 37 in the calculator by pressing the corresponding digits, 3, 7.
   • Press the equals (“=”) key.
   • The display should show “5402,” which is the same number derived by the Russian peasant method, verifying that the method produced the correct answer.
3. Start over if you made a mistake. If you check your answer and it is not correct, start looking for where you might have made a mistake, and try multiplying the numbers again using the Russian peasant method.
   • First, try checking your work again using the standard multiplication method or a calculator, just to be sure.
   • Then, work through the problem again using the Russian peasant method.
   • This time, double-check to make sure you are halving and doubling the numbers correctly.
   • In addition, make sure you correctly add the numbers that remain in the right-hand column after you cross out the other numbers.
   • Check all of your work twice to make sure that you get the correct answer.
4. Ask someone for help if you are struggling. If you can’t seem to get the right answer, or if you aren’t sure how to check your work, ask some for help (such as a friend, sibling, parent, or teacher). Ask this person to try the problem out, too, and then you compare answers. The person might also be able to explain how to use the Russian peasant method.
5. Understand why the method works. The Russian peasant method works because it converts the problem into binary (base 2) multiplication, rather than base 10 (which standard multiplication uses). It does this by halving and doubling the numbers you are trying to multiply (since halving and doubling convert all numbers into multiples of 2, or into factors of numbers that are divisible by 2).
6. Know when the Russian peasant method of multiplication is useful. You might be asked to try the Russian peasant method of multiplication for a school assignment, or just try it out for fun. However, the method has advantages that might make it turn out to be a handy way to multiply in other situations.
   • You don’t need to know or memorize multiplication tables in order to multiply using the Russian peasant method (like you do in order to use standard long multiplication). As long as you can double and halve, you can multiply any two numbers using the Russian peasant method.
   • You can use bits of material in order to multiply using the Russian peasant method. This can be useful when you need to multiply and you don’t have a calculator or a pencil and paper handy. For example, you could multiply the example above (146 x 37) by using lots of beans or other material. Each time you need to halve or double a number in the process, just add or take away the respective number of beans.
   • The Russian peasant method does normally take longer than standard long multiplication, however.
   • In addition, the Russian peasant method of multiplication can become very unwieldy if you are trying to multiply large numbers, since this will involve many multiples of 2 or numbers that are factors of numbers divisible by 2, and potentially very large ones.

Video

Template:Video:Multiply Using the Russian Peasant Method

Related Articles

• Multiply Using the Line Method
• Multiply by 11 in Your Head
• Multiply Large Numbers Quickly With the Box Method
• Multiply Like the Ancient Egyptians

Sources and Citations

See also

• Teach the Multiplication Tables to Your Child
• Do Long Multiplication
• Do Long Division
• Do Double Digit Multiplication
• Do Short Division
• Multiply
• Divide Square Roots
• Multiply Mixed Numbers
• Divide a Whole Number by a Decimal
• Find the Greatest Common Divisor of Two Integers

Read more ...