Multiplying Fractions

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 and prime numbers.

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.

Multiplying Fractions

To multiply two fractions, we need to multiply the numerators of both fractions and then multiply the denominators of both fractions. For example, to multiply 1/4 by 1/2 we multiply the numerators 1 and 1 to get 1. Then we multiply the denominators, 4 and 2 to get 8. So the product of 1/4 times 1/2 = 1/8 . You can think of this in terms of our earlier discussion of cutting a pizza into slices. If you cut the pizza in to 4 slices (each slice is 1/4 ) of the pizza). If we cut each slice of the pizza in half again (multiplying by 1/2), then each slice will be 1/8 of the pizza.

But does this work for all fractions? The answer is, Yes. If you remember in our initial discussion of fractions, we said that every integer could be written as a fraction with 1 as the denominator. The number 2 can be written as 2/1 and 8 can be written as 8/1 . When an integer is written this way, it is called a rational number. When we multiply 2 and 8 as rational numbers, we are multiplying times . When we multiply the numerators 2 and 8 we get 16. When we multiply the denominators, 1 and 1, we get 1. So the product of the two fractions is 16/1 or 16. So, the product of two fractions is the product of their numerators and the product of their denominators.

But how do we multiply two fractions using the Double-a method? Two fractions can be successfully multiplied by following these steps:

Reduce both fractions to their simplest value.
Convert the first fraction to an improper fraction (if it is a mixed fraction)
Convert the second fraction to an improper fraction (if it is a mixed fraction)
Cancel common factors.
Multiply the numerators
Multiply the denominators
Reduce the final fraction

This will result in the final product 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: 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 5: Multiply the numerators

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

Step 6: Multiply the denominators

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

Step 7: 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.

Examples

Now, let’s look at some more examples of this process. In these examples, the multiplicand will have many digits, but the multiplier will always be a single digit. Click on the link to show the steps to calculate the answer.

Example: 3/4 times 2/5

Example: 14/15 times 5/7

Example: 4/5 times 3/5

Example: 2 and 1/4 times 2/27

Example: 3 and 1/6 times 2 and 1/9

Build Your Skill

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

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