Multiplying and Dividing Fractions

Single Abacus Approach

1 and 1/15  divided by  2 and 2/3  =  2/5

Some Important Terms

In order to understand the process of multiplying and dividing fractions, there are few terms that we need to clarify: mixed fractions, improper fractions, common factors, prime numbers and reciprocal of a fraction.  

Mixed Fractions and Improper Fractions    

Any improper fraction can be converted back to a mixed fraction by reversing these steps.  To convert back to a mixed fraction, first divide the denominator into the numerator: 252 / 8 = 31 with a remainder of 3.  The 31 becomes the whole number and the remaining fraction is 3/8.


In the last example on the Subtracting Fractions page, we subtracted 31 and 3/8 from from 45 and 5/11 .  Both of these fractions have a whole number and a fraction. These fractions and any fraction that has both whole number component and a fractional component are called a mixed fraction.


When we multiply or divide two fractions, we can’t separate the whole number from the fraction. If we did that, the result of our calculation would be invalid except when the denominators of both fractions are the same. Instead when we have a mixed fraction, we multiply the denominator by the whole number and then add the numerator.  This is called an improper fraction. For example, let’s convert the mixed number 31 and 3/8 to an improper fraction. First, multiply the whole number and the denominator: 31  times 8. = 248.  Then add the numerator: 248 + 3 = 251.  The improper fraction is  251/8. 

Common Factors and Prime Numbers

We will also need to find the common factors between two numbers. As you know, we calculate a product by multiplying two numbers.  The two numbers are called factors of the product. For example, let’s take two numbers, 6 and 9.  Multiplying a combination of 1, 2, 3 and 6 will give you 6.  We can calculate the number 9 with a combination of 1, 3 and 9.  What are the common factors between 6 and 9?  To find the common factors, we are looking for the factors that are the same for both numbers.  In this example, 1 and 3 are the common factors.  The number 1 will always be a common factor since 1 times a number is the number.  So, we don’t use the number 1.  That leaves us with the number 3 as the only common factor. We will use common factors to reduce our fractions to simplify our multiplying and dividing of fractions. Sometimes there aren’t any common factors so the fractions can’t be reduced.


Some numbers have a special name because their factors are limited. When the only factors for a number are 1 and the number, the number is called a prime number. For example, 1 and 19 are the only factors for 19.  This makes 19 a prime number.

Reciprocal Fractions

There is one last term we need to understand. When we divide fractions, we need to use the reciprocal of a fraction. What is this? The reciprocal of a fraction is when we flip the numerator and the denominator. The numerator becomes the denominator, and the denominator becomes the numerator.  For example, the reciprocal of 251/8  is 8/251 . In this example, 251 becomes the denominator and 8 becomes the numerator. 

Multiplying and Dividing Fractions

Dividing fractions is very similar to multiplying fractions. It uses the same process that we used when we multiplied two fractions with one major difference. The difference is that when we divide two fractions, we will multiply by the reciprocal of the second fraction.

 

Let’s look at why we need to do this. When we multiply two numbers, 3 and 4, the product is 12. We write this calculation as 3 times 4 = 12.  If we knew that the product of two numbers is 12 and one of the factors was 3, we would find the answer by dividing 12 by 3, or 12 / 3.  The answer of course is 4,  4 = 12 / 3.  Instead, if we knew that one of the factors was 4 we would divide 12 by 4 or 12 / 4 to find the other factor, 3.

 

Using rational numbers,  3/1 times 4/1   =  12/1.  If we only know one of the factors say, 3/1 , then to find the other factor (4/1), we would divide 12 by 4 .  We can also write  12/4 as  12/1 times 1/4 .   So we could write our calculation as  12/1  times 1/4  = 3/1  or 3.  Notice that dividing by 4 is the same thing as multiplying by its reciprocal,1/4 . 

 

Let’s do another calculation: How do we divide 1/4  by 1/2 , or  1/4 ÷ 1/2 ?  If our pizza is cut into 4 parts but we only want to eat half or 1/2 of it, we will have 1/2 of the pizza remaining.  Using the same technique as the previous calculation, 1/4 ÷ 1/2  = 2/1 times 2/1   = 2/4  which reduces to 1/2.


As you can see, this technique works when we divide two integers and when we divide two fractions. So, in general, when we want to divide two fractions, we first switch the numerator and denominator of the second fraction (use the reciprocal of the second fraction) and then multiply the numerators and denominators of both fractions.  


But how do we multiply two fractions using the Double-a method?  We will use the same procedure we used to multiply two fractions but add one step:    

This will result in the final quotient of the two fractions.  

Step 1: Reduce both fractions to their simplest value

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

Step 2: Convert the first fraction to an improper fraction (if it is a mixed fraction)

As we discussed in the Some Important Terms section, all mixed fractions  (whole number and fraction) need to be converted to an improper fraction.  This is a fraction where the numerator is larger than the denominator.  For example, a mixed fraction is 3 and 1/2. In order to multiply this fraction, we need to change this to an improper fraction by multiplying the whole number, 3, times  the denominator, 2 giving 6.  Then add the numerator 1 to 6 giving 7.  The improper fraction of 3 and 1/2 is 7/2. 

If the first fraction doesn't have a whole number, then we can multiply it without any changes. 

Step 3: Convert the second fraction to an improper fraction (if it is a mixed fraction)

The second fraction, also, needs to be converted to an improper fraction. Multiply the whole number by the denominator of the fraction and add the numerator.  Of course, if the second fraction doesn't have a whole number, we can multiply it without any changes. 

Step 4: (Dividing Fractions ONLY) Flip (switch) the numerator and the denominator of the second fraction

This is the additional step we use ONLY when  we divide fractions.  Leaving one column empty, enter the denominator of the second fraction next to the numerator of the first fraction in the top abacus.  Then, in the bottom abacus, enter the numerator of the second fraction (again leaving one column empty) next to the denominator of the first fraction.  The second fraction has been flipped (entered the reciprocal).  

Step 5: Cancel Common Factors

Using the factors of the numerators, look for any common factors in the denominators.  Any common factors can be cancelled and removed from the fraction.  This reduces the fractions and simplifies the calculation. 

Step 6: Multiply the numerators

Now, multiply the numerators of both fractions.  This product will be set in the right side of the top abacus. 

Step 7: Multiply the denominators

Then, multiply the denominators of both fractions.  This product will be set in the right side of the bottom abacus. 

Step 8: Reduce the Final Fraction

This is the final step.  Using the same process that you learned in 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 product of the two fractions.

Multiplication Examples

Now, let’s look at some more examples of this process multiplying fractions. In these multiplication examples, the multiplicand and multiplier may be simple or mixed fractions.  Click on the link to show the steps to calculate the answer. 

Division Examples

These are some examples of using this process to divide fractions. In these examples, the divisor and dividend may be simple fractions or mixed fractions. Click on the link to show the steps to calculate the answer. 

Build Your Skill

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