# Adding and Subtracting Fractions With the Abacus in Easy Steps

## Adding and Subtracting Fractions with the Abacus

The abacus can be used to perform any number of mathematical operations. This includes problems concerning addition, subtraction, division, and multiplication. Indeed, the abacus can be a trusted ally when solving equations with whole numbers, fractions, or mixed numbers. With appropriate training and practice, working with addition and subtraction problems pertaining to fractions will be easy.

Of course, we know fractions are parts of a whole. These values can be represented on the abacus just like with a pen and paper or on a computer. As a counselor with Teacher of the Visually Impaired (TVI) training, I’ve worked with my students on utilizing the fascinating counting tool for solving equations involving fractions and other types of arithmetic. I have many years of experience working with the fabulous abacus, and I have received extensive training on using the counting device from masters. Below I’ve provided simple techniques for finding solutions for math related to the addition and/or subtraction of fractions.

If you require more information on working with the abacus, visit my articles on this site about the wondrous counting tool mankind has used for centuries.

## Knowledge You Should have before Working with Fractions on the Abacus

- Primarily, a person should be experienced enough with the counting tool to place any representation of a whole number on the device with the only limitation being the availability of the columns of beads. Second, mentally dividing the abacus to perform division and multiplication should present no difficulty at this point. Furthermore, concepts concerning the operation of the abacus should be understood thoroughly. Those terms include: set (place), one for the abacus, and clear. The concepts of “keeping balance” and “pay-back” should present no problems for the person using an abacus by this time.
- Coincidentally, issues concerning the function of “0” in multiplication and division related to the abacus must be thoroughly comprehended before working with fractions. A person should have successfully used the abacus to perform division, addition, multiplication, and subtraction problems with whole numbers. In essence, a person should be comfortable with carrying out the various steps to find solutions for these mathematical operations. Finally, the concepts associated with fractions should be recognized and their importance comprehended. Those terms and concepts include: denominator, numerator, and the significance of the dividing line. A person should understand the importance and process for finding a common denominator.

## Poll

### Have you set a fraction on the abacus before today?

## Three Crucial Points to Remember when Working with Fractions on the Abacus

- To begin with, we have mentally divided the abacus. Therefore, you can think of all of the rows of beads not involved in the equation as representing the “dividing line” of the fractions we are working with to solve the problem.
- Next, the numerator of a fraction is set on the far left. The denominator is placed on the furthest right row of beads. This is demonstrated in the photo showing 3/4 above.
- Be aware: when placing the numerator on the furthest left column of beads, the first digit is representative of the highest value of ten in the number. For instance, the number 3 takes up one column on the left. 35 would be shown with the first two rows of beads, moving from left to right. 357 would be set using the first three columns moving from left to right on the counting tool, and so on. Now, let’s perform an addition problem using simple fractions.

## Let's solve an Addition Equation involving Fractions

- Since we already have the fraction 3/4 set on the abacus, we can begin with it for this equation. Our equation is: ¾ + 1/5.
- Find a common denominator for these fractions. That number is 20.
- We know: 5 times the denominator 4 in the fraction ¾ = 20. Therefore, we multiply 5 times the numerator 3 in ¾ to get the answer of 15/20.
- You may wish to place this fraction on the abacus: 15/20.
- Now, we know four times the denominator 5 in the fraction 1/5 = 20. Therefore, we multiply the numerator 1 by 4 for the answer of 4.
- Add the numerators: 4 + 15. The answer is 19 in the numerator, and we also have 20 as a denominator.
- Set 19 on the left side of the counting device.
- The solution is 19/20.
- Essentially: you should have 19 on the tens and ones columns on the left-hand side; you should be showing 20 on the right-hand side of the counting tool.
- It should look like the photo below.
- After you have examined the result, bring the abacus to rest. Let's try subtracting simple fractions.

## Let's Perform a Subtraction Problem Using the Abacus for Fractions

- Our subtraction problem is: 2/3 – 2/5.
- Start by finding the common denominator for these fractions. In this case, we know that number is 15.
- Now, place the fraction 2/3 on the abacus.
- We know: 5 x 3 = 15. Therefore, we multiply the numerator by 5 for the answer of 10.
- Now, set 10/15 on the abacus. This is the number we will be subtracting 2/5 from after we convert it to a fraction with a common denominator.
- We know: 3 x 5 = 15. Therefore, we multiply the numerator by 3 for the product of 6.
- Our fractions now have common denominators. We can solve the equation.
- Subtract: 10 – 6 on the left-hand side of the abacus.
- Your answer is 4.
- Our final result is: 4/15.
- After you have reviewed the answer to the equation, bring the abacus to rest.

## Adding and Subtracting Mixed Numbers and Complex Fractions on the Abacus

Not only can you use the abacus for solving equations involving simple fractions, but the amazing counting device is useful for working with complex fractions as well as mixed numbers. A complex fraction is one in which the numerator, the denominator, or both consist of a fraction. Convert these fractions to simple fractions by finding common denominators and simplifying them. This process may be necessary when adding or subtracting mixed numbers during an equation, too.

A mixed number is an integer with a proper fraction. To perform addition and/or subtraction on the abacus, we must convert a mixed number into an improper fraction. An improper fraction is one in which the numerator is greater than the denominator, such as in 7/6.

Once the improper fraction is placed on the counting tool, you can proceed with solving a subtraction or addition equation. Let’s do this with the mixed number: 3 ½.

## Converting a Mixed Number into an Improper Fraction

- Begin by multiplying the whole number and the denominator: 3 x 2, for the product: 6.
- Next, add the numerator and the product: 6 + 1. This will give you the answer of 7.
- Place the 7 on the far left of the abacus. This is your new numerator.
- Place the denominator, 2, on the far right. Your answer should look like the photo below.
- Now, you will be able to work with an addition or subtraction problem involving the improper fraction: 7/2.
- After you have studied the result, bring your abacus to rest.
- Congratulations. You have used the abacus to perform subtraction and addition for fractions.

## Poll

### Which of these tasks is easiest for you on the abacus when working with fractions?

## How to Use the Abacus to Introduce Children to Fractions

Although the Latin word abacus means “flat surface,” the counting tool has many forms. It may be used horizontally, like the Cranmer abacus shown in all of the photos in this article. Yet, some abaci may stand vertically. There are digital abaci as well. The history of the counting tool is debatable, but many researchers suggest the abacus was first used in China or Babylon. Regardless of the design or origin of the counting tool, the abacus can be helpful in assisting young children who are still developing numerical concepts and understanding about fractions. Below is a simple way to introduce children to fractions with the abacus:

- First, tell the child you will be exploring what fractions are. Explain what fractions are in terms the child can comprehend.
- Next, have the child count the number of columns of beads on the abacus. In the case of the abacus used in this article, the number would be 13 columns of beads.
- Now, explain the thirteen columns of beads represent one complete set. Let the child ask questions at this point.
- Now, have the child cover a few rows with his hands. Explain this represents part of the whole.
- For example, if the young person covers two rows of beads, explain that 2 out of 13 columns of beads have been covered.
- Enhance comprehension by using different examples. For example, try the same thing with money, i.e., four quarters make a dollar etc. The child must develop the skills to relate the knowledge of fractions to various situations.
- Conclude your simple lesson by explaining how this is the basic underlying concept of fractions. In time and with practice, the young person will be able to apply his/her knowledge to working with fractions on the amazing abacus.

## Comments

I didn't know you could handle fractions on an abacus.

Tim this is so great. My boy is of course with in a top % of his class in math. His mom is a real drill Sargent. I will get an abacus today. How fun will that be. Is there an abacus game(s)?

Tim, I have not been around an abacus very often.

When we studied math in school we simply had to memorize everything. I can still divide or multiply in my head, but fractions take a bit more thought. I understand what you were teaching and you explained it quite well. Thank you for an informative article.

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