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Multiplying and Mastering Equations With Three Digits on the Abacus


Tim Truzy is an expert instructor in the use of the abacus. He has taught adults and children in the use of the counting tool.

The abacus set at zero.

The abacus set at zero.

Mastering the Abacus

The abacus is an amazing tool for performing numerous types of arithmetic problems, including multiplication. In developing any skill, such as using the abacus, practice is required for mastery. In order to master the counting tool, a person should try to incorporate as many of the “learning” senses as possible. This includes consideration of visual aspects of the abacus, auditory cues and responses, and application of the sense of touch. If you were to watch long time masters of the abacus at work, you may see those experts moving invisible beads with their fingers while going through the calculation process. You may hear them mutter words associated with the abacus, such as payback, set, and clear. I’ve also worked with long-time users of the device who simply did the calculation quickly in their heads without a word or gesture. Nonetheless, to reach this level, time and dedication must be put into the effort to become proficient on a tool that has been around with mankind for ages.

Indeed, the abacus has a long history with humanity. The counting device is still a part of learning mathematics for specific reasons in areas of the western world and the globe. I have taught individuals to work math problems on the abacus, and they completely enjoyed learning about the counting device. Without question, the abacus will be with us for many years to come. This is because of the need for applying different approaches to learning math. Here are some other reasons why the abacus remains an important counting tool around the world:

Reasons Why the Abacus is Still Used Around the Globe

  • The abacus is durable. An abacus can be dropped and normally will continue to perform the job it was designed to do. In addition, an abacus doesn’t require electricity to function nor the internet. Everyone cannot afford calculators, and the abacus is a low cost functional alternative in poorer nations. Also, individuals with vision loss often can better grasp numerical concepts by using the counting tool.
  • The abacus has different varieties, vertical or horizontal. The counting tool can be portable or stationary. The abacus can also be a fun source of conversations.
  • The abacus can be used to help young children learn numerical concepts. The skills at correctly manipulating beads on the counting tool builds understanding of mathematical processes such as division, multiplication, subtraction, and addition. Finally, everyone does not learn the same way or at the same pace. Using the abacus for math offers an alternative to traditional pencil and paper methods.


Things to Know before Performing Multiplication on the Abacus

  • As with every skill, knowledge must be built upon in order to perform more and more complex tasks accurately and with confidence. The same is true with the abacus. These are skills which should be mastered before attempting multiplication of equations which have three digits on the abacus:
  • A person must understand how numbers are formed on the abacus. This includes setting numbers and clearing the counting tool. A person should also know how to put the abacus “at rest,” or set the device to zero, as shown in the first photo in this article.
  • A person should understand and be able to conduct addition problems on the abacus. A person should have performed subtraction equations on the abacus as well. These problems should have been of single digits, two-digits, and 3-digits or more.
  • Having understanding of the multiplication table is essential. For instance, a person should know the multiplication table through 9’s. (5 x 3, 6x 7, 8 x 9, etc.) A person should be familiar with terminology related to multiplication, such as “product.”
  • Terminology related to operating the abacus should be well understood. Terms such as “payback” should be understood with the skills to apply the concept in solving a problem. In addition, maintaining “balance” in relationship to the base-ten counting schemes should be firmly established in a person’s vocabulary and knowledge base. For example: 1 + 9 = 10, 2 + 8 = 10, 10 - 4 = 6, 3 + 7 = 10, etc.

Let's Start

In examining the abacus, we notice that there are at least thirteen rows of beads. To do multiplication, we must mentally think of the abacus as being divided down the middle of those rows, at about the seventh row of beads. This is because we will place one number on the left hand side of the counting tool and the other on the right hand side.

  1. Let’s start. Place 25 x 7 on the abacus.
  2. Place 25 on the furthest rows of beads.
  3. Now, let’s place the number 7.
  4. To do this, we know that there are three digits in the multiplication problem: 2, 5, and 7.
  5. For multiplication, we must give an extra row of beads “for the abacus.” Essentially, we think: three digits in the equation plus a row of beads "for the abacus."
  6. This means the 7 will be placed on the fourth row moving from the right. The importance of this act is that it gives the user of the counting tool some indication that the answer will be in the hundreds, the remaining three rows on the right. The problem should be set up like in the photo.
The abacus showing "25 X 7".

The abacus showing "25 X 7".

Here the abacus shows "7 times two tens".

Here the abacus shows "7 times two tens".

Now, let's solve the Equation

  1. Multiply: 7 times the first number, which is 2, or 2 tens. This gives us the answer of 14, or 14 tens, like shown in the picture. Do not clear the 7.
  2. Observe the answer before proceeding. You will see that the first product is placed beside of the 7. This result was predicted from the way the problem was set up. The first product is in the hundreds, tens, and ones columns. We still have the number 5 to calculate.
  3. Now, multiply: 7 times 5. This gives the answer of 35, or 3 tens and 5 ones, which can be added to 140. Your answer will be: 175 like shown in the photo. Now, bring the abacus to rest.
The product of "25 X 7" is shown on the abacus.

The product of "25 X 7" is shown on the abacus.

The abacus showing "9 X 50".

The abacus showing "9 X 50".

The Issue of Zero on the Abacus

When calculating problems with three digits in the equation where zero is part of a two-digit number, such as 80, 90, 40, etc., we still count over to the fourth row to set the second number. For example, 50 x 9, would still require the same procedure.

Let’s try it.

  1. Place 9 on the far left row.
  2. Now, place 50 on the fourth row from the right. The problem should be set up like in the photo.
  3. Multiply: 9 x 50.
  4. The answer would be: 450, which you would place on the third, second, and first rows of beads on the right hand side. The answer should look like the photo after clearing 9 and 50.
  5. These are the basic steps to working with equations which have three digits in a multiplication problem on the abacus. Now, since the work is done, the abacus can be brought to rest.
  6. Another issue with zero arises when the final product is less than 100. In these instances, we count the hundreds as zero. For example: 9 x 11 would be counted this way: (0) hundreds, 9 tens, and 9 ones. 3 x 12 would be counted this way: (0) hundreds, 3 tens, and 6 ones. Enjoy using the abacus and you may become a expert in using the counting tool in the future.
The abacus showing "450".

The abacus showing "450".

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© 2018 Tim Truzy


Tim Truzy (author) from U.S.A. on June 05, 2018:

Indeed, Manatita. The old methods we were taught are fabulous. But I enjoy watching my students grasp a concept; it is true, it's light is turned on.

I appreciate your kind comment.

I am sure you could master the abacus because you have dedication and you are a very smart individual.

Thanks again.

Much respect,



manatita44 from london on June 04, 2018:

Not really familiar with this Bro. I guess I studied the old way. I can do things easy in my head but struggle with new or different ideas.

I'm more old school, I'd say.

Interesting read though. Peace.

Tim Truzy (author) from U.S.A. on June 03, 2018:

This technique is an example of the many ways of using the abacus for multiplication. Basically, over the decades, I've redefined approaches and methods I had learned from various practitioners of the counting tool. I worked with people from India, Spain, and the Middle East and simplified some of their ideas for use with my students. I've also written articles on subtraction and addition regarding the abacus.

Thank you for reading, and your comment is appreciated.



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