Adding and Subtracting Fractions

Short Course 

This is a short course (slide show) in adding fractions.  Just click on the carousel arrow to start the show.    

Terminology

We already know that a fraction has at least two parts, a numerator (top number) and a denominator (bottom number).  You will also remember that a fraction may also include a whole number. A fraction with all three parts is called a mixed fraction.  But, where do we place the results of our calculation on our abaci?  Each part has a “reserved parking space” or section:

• A whole number is set in columns 9, 8, and 7 of the top abacus.  This is called the whole number section.

• The numerator is set in columns 5, 4, 3, 2 and 1 of the top abacus. This is called the numerator section.

• The denominator is set in columns 6, 5, 4, 3, 2 and 1 of the bottom abacus. This is called the denominator section.

Adding and Subtracting Fractions

In order to successfully add or subtract two fractions, we need to follow a series of steps. We break down each step.     

• Reduce both fractions to their simplest value.

• Determine the Lowest Common Denominator (LCD).

• Setup and convert the first fraction

• Add or subtract the whole number of the second fraction

• Setup, convert and add or subtract the second fraction of the second fraction

• Reduce the final fraction  

Now let’s break down each step into the detailed tasks that we need to take to complete the step.  Don’t get discouraged if you don’t completely understand or remember all of the details.  We will apply these steps to several examples which should help you to better understand the entire process.

Step 1: Reduce both fractions to their simplest value

First, we need to reduce both fractions to their simplest terms. When we add the two fractions, the calculations will be easier when we use smaller numbers.

Step 2: Determine the Lowest Common Denominator (LCD)

Next, we need to determine to lowest common denominator. To review, we can follow this process:  

• If both fractions have the same denominator, then both fractions have the same denominator in common, so we use this number.

• If the denominators are not the same, then we need to find the lowest number where both are factors. As we discussed in the previous step, the factors of a number are the smaller numbers we can multiply together to arrive at the larger number. There are several ways to do this, but we recommend following this procedure:

  • Start with the larger denominator. Multiply it by 2. Then divide the smaller denominator into this number.  Does it divide evenly (no remainder)?  If so, this new number is the lowest common denominator.

  • If it doesn’t divide evenly, then multiply the larger denominator by 3. Again, divide the smaller denominator into this number. Does it divide evenly (with no remainder)?  If so, this number is the lowest common denominator.

  • If it doesn’t divide evenly, repeat the steps multiplying by 4, 5, … until you find the number that is evenly divisible by both denominators.

  • The largest common denominator is the product of both denominators. If both denominators are prime numbers, then the only common denominator will be the product of the two numbers.

Refer to the examples in the Reducing Fractions and Lowest Common Denominator pages to refresh your skill  on reducing fractions and finding the LCD.

Step 3: Setup and convert the First Fraction

When both fractions are in their simplest form and we have determined the lowest common denominator, you are ready to set and convert the first fraction. 

• First, set the Lowest Common Denominator in right side of the bottom abacus.

• Then, set the whole number of the fraction in the whole number section of the top abacus.

• Next, set numerator of the fraction in the left side of the top abacus and the denominator in the left side of the bottom abacus.

This sets the first fraction. We are ready to convert it to the lowest common denominator:

• Divide the denominator (in the left side of the bottom abacus) into the Lowest Common Denominator.

• Multiply the result by the numerator (in the left side of the upper abacus). Set the result in the right side of the top abacus.  This has converted the first fraction to the Lowest Common Denominator.

• Clear the numerator and denominator from the left side of the top and bottom abaci.

The converted numerator is now set in the upper right columns of the top abacus.

Step 4: Add or Subtract the whole number of the Second Fraction

If the second fraction has a whole number, then add or subtract it to the number in the whole number section.

Step 5: Setup, Convert and add the Second Fraction

• Set the numerator of the second fraction in the left side of the top abacus. Then set the denominator in the left side of the bottom abacus. 

• Divide the denominator (in the left side of the bottom abacus) into the Lowest Common Denominator.

• Multiply the result of the division by the numerator.

• Add this number to the numerator of the first fraction, on the right side of the top abacus

Step 6: Reduce the Final Fraction

This is the final step.  Using the same process that you learned in the Reducing Fractions page, reduce the fraction into its simplest values.  If the numerator is larger than the denominator, add 1 to the whole number and reduce the numerator by the denominator.  Continue this process until the numerator is smaller than the denominator.  This is the sum of the two fractions.

Addition Examples

Now, let’s look at some more addition examples of this process.  Click on the link to show the steps to calculate the answer. 

• Example: 5/19 + 9/19

• Example: 4/15  + 3/20

• Example: 3/10 + 3/12

• Example: 2/3 + 3/4

• Example: 3/7 + 1 and 8/9

• Example: 3 and 4/5  + 1 and 1/2 

Subtraction Examples

Now, let’s look at some more subtraction examples of this process.  Click on the link to show the steps to calculate the answer. 

• Example: 6/11  - 2/11 

• Example: 2/3 -  1/6 

• Example: 3/5  - 1/4

• Example: 1 and 1/5  - 2/7 

• Example: 3 and 1/6  - 2 and 1/9 

• Example: 45 and 5/511  - 31 and 3/8 

Build Your Skill

Now you are ready to try some calculations on your own. Click on problems and their solutions to test your skill.