Multiplying Fractions

Multiplying fractions is a piece of cake! For example, if you take 1/4 of a cake, and then decide to eat only half of it, you’ve eaten (1/4) × (1/2) = 1/8 of the cake. Pretty tasty math in my opinion!

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In this article you’ll learn all you need to know about multiplying fractions, including techniques for dealing with mixed fractions.

multiplying fractions, piece of cake - magoosh

Multiplying fractions is like taking a slice of a slice of the cake.

Image by TesaPhotography

Multiplying Fractions — The Rules

There are a few simple rules for multiplying two fractions.

  1. Multiply the tops together, and place on the top of the resulting fraction.
  2. Multiply the bottoms together, and place on the bottom of the result.
  3. Simplify the result if possible.

These rules are often summarized by the phrase: “Multiply across the top and across the bottom.”

I’m using the words top and bottom, but you should also know the mathematical terms:

  • Top top number of a fraction is the numerator.
  • The bottom number of a fraction is the denominator

Let’s see a couple of examples.

Examples — Multiplying Simple Fractions

  1. Multiply:

    multiplying fractions, example 1 - magoosh

    First multiply the numerators to get: 4 × 7 = 28.

    Then multiply the denominators to get: 5 × 9 = 45.

    Finally, put your answer together: multiplying fractions, example 1 answer - magoosh.

    There are no common factors (aside from 1) between 28 and 45, so this fraction is already in lowest terms. That means that we are done! The answer is 28/45.

  2. Multiply:

    multiplying fractions, example 2 - magoosh

    You can do all the work in a single line of mathematics. Notice how the tops remain on top, and the bottoms remain on bottom throughout.

    multiplying fractions, example 2 answer - magoosh

    Now we should simplify the result. You could find the common factors between 126 and 60 and reduce the result directly…

    However, I want to show you an easier way that takes out much of the guess work!

    multiplying fractions, simplify - magoosh

    Yes please — I’ll take the easy way!

    Image by suetot
     
    The trick is to reduce the fraction before you even multiply the tops and bottoms!

    Let me show you how it works in this example. The method starts the same way, by collecting the numerators and denominators. But then, look for any cancellations that may be done before multiplying it all out.

    We can see in this example that 9 and 15 have a common factor of 3. Also, 14 and 4 have a common factor of 2. Make sure you divide those common factors out first, and then you can multiply across the top and bottom.

    multiplying fractions, simplifying fractions - magoosh

    Thus, the final answer is 21/10. (Note, if you reduced the original answer of 126/60 down completely, you’d get 21/10 that way as well.)

More than Two Factors

The cancelling shortcut is great for products involving more than two factors. In fact, these problems may become very easy if you just look for factors that cancel first.

Multiply (1/2) × (2/3) × (3/4) × (4/5).

It looks at first like a lot of work, but don’t forget to cancel common factors!

multiplying fractions, cancel common factors - magoosh

Special Cases

There are a few special cases to consider. For example, cases including whole numbers or mixed fractions.

Multiplying Fractions and Whole Numbers

How would you multiply a fraction and a whole number?

The key is to rewrite the whole number as a fraction.

Any whole number n can be written as a fraction whose denominator is 1.

multiplying fractions, whole numbers - magoosh

Then you can use the methods of multiplying fractions to proceed.

Now let’s see how this helps in the following example.

There are 60 students in Ms. Burke’s class. 3/5 of them are girls. How many total girls are in the class?

First we must translate the word problem into mathematical language. Any time you see a phrase that mentions a fraction of something, think multiplication. So we need to find the value of (3/5) × 60.

multiplying fractions, working with whole numbers - magoosh

Look for any common factors. 60 and 5 both have 5 as a factor.

multiplying fractions, finding factors - magoosh

Thus, there are 36 girls in Ms. Burke’s class.

Multiplying Mixed Fractions

A mixed fraction is a number like 7½, consisting of a whole part (7) and a fractional part (½).

In order to multiply two mixed fractions (or a mixed fraction with a simple fraction), you must first convert each mixed fraction into a simple fraction.

Remember the rule to convert mixed fractions:

  1. Multiply the denominator to the whole part.
  2. Add the result to the numerator. This is your new numerator.
  3. Keep the old denominator.

For example, to convert 7½, first multiply 2 × 7 = 14. Then add the 1 from the numerator to get 15. Finally, keep the old denominator to obtain a simple fraction of 15/2.

Fractions that have a larger numerator than denominator are often called improper. That terminology makes them seem bad in some sense — but improper fractions are just as “good” as any other fraction!

Anyway, once you have converted each mixed fraction to a simple one, then you may use the methods outlined above to multiply. Let’s see how this works!

Multiplying fractions, answer to example - magoosh

Convert each mixed fraction to an equivalent simple fraction, then multiply.

multiplying fractions, converting mixed to simple - magoosh

For this one, there were no common factors to cancel out. The result is also improper, so you may need to convert this back into a mixed fraction, depending on the nature of the problem.

Converting back to a mixed fraction is just like division with a remainder. The quotient is the whole part, and the remainder is the new numerator.

multiplying fractions, converting back to a mixed fraction

A Time-Saving Trick for Mixed Fractions

Sometimes, if the mixed fractions have small enough denominators, you can use a neat trick to save work. Now I’m all about saving work — I hope you are too!

The key concept is that a mixed fraction is just shorthand for addition of the whole part and the fractional part. For example, 7½ = 7 + ½.

So you can use the Distributive Property to multiply fractions, especially when only one of the factors is mixed.

Multiply: 7½ × 6/7, and simplify.

multiplying fractions, mixed Fraction trick - magoosh

Now work on each term as a separate multiplication. Don’t forget to treat 7 as a fraction 7/1.

multiplying fractions, fraction result - magoosh

What if There’s a Negative?

The rules for multiplying negative fractions are exactly the same as for multiplying negatives in general.

  • Positive times positive is positive. For instance, 3/4 × 7/2 = 21/8.
  • Negative times positive or positive times negative is negative. For example 2/3 × (–3/4) = –6/12 = –1/2.
  • Negative times negative is positive. For instance, (–4/3) × (–7/5) = 28/15.

But what if there is a negative symbol on just the numerator or just the denominator? Does it make a difference?

The answer is no. Whether the negative symbol occurs on the top, on the bottom, or to the left of the entire fraction, these all mean the same thing!

multiplying fractions, negative fractions - magoosh

Practice Problems

Now it’s your turn! Here are seven problems to test your skills at multiplying fractions. Solutions will be given at the end.

  1. 1/3 × 4/5
  2. –3/7 × 2/11
  3. 9/10 × 35/36
  4. 3/8 × 10
  5. 5/3 × 7/2 × 6/7
  6. 6 × 4¾
  7. –11/3 × (–3/11)
multiplying fractions, books - magoosh

Ready to see the answers?

Image by By Andrey_Popov
 

Solutions

  1. 1/3 × 4/5 = (1 × 4) / (3 × 5) = 4/15. No further simplification is possible because 4 and 15 do not have any common factors (aside from 1).
  2. –3/7 × 2/11 = –(3 × 2) / (7 × 11) = –6/77. Again, there are no nontrivial common factors, so the answer does not reduce. The answer is negative because the problem had the form negative times positive.
  3. multiplying fractions, fraction multiplication solution - magoosh
  4. 3/8 × 10/1 = (3 × 10) / (8 × 1). You can cancel the common factor of 2 from the 10 in the numerator and the 8 in the denominator. (3 × 5) / (4 × 1) = 15/4.
  5. Look for the common factors and cancel before you multiply.
    multiplying fractions, multiplying factors - magoosh
  6. You can use either the method of converting mixed fractions, or the method of distribution. For comparison, I’ll show you both methods.
    1. Converting: First convert 4¾ = 19/4. Then multiply by 6 = 6/1.

      multiplying fractions, mixed fraction problem - magoosh

      Then you may need to convert this into a mixed fraction. Divide 57 by 2. You get a quotient of 28 with remainder 1.

      Therefore, the final answer is 28½.

    2. Distribution: Think of 4¾ as 4 + ¾

      multiplying fractions, mixed fraction problem 2 - magoosh

      Here, you could convert 9/2 = 4½. Then adding that to 24, you get 28½, just as above.

  7. First observe that a negative times a negative gives a positive. Therefore, we have:

    (–11/3) × (–3/11) = (11 × 3)/(3 × 11). Both the 11 and the 3 will cancel, leaving us with 1/1. In other words, the final answer is 1.

    Note: –11/3 and –3/11 are reciprocals. The reciprocal of a fraction is basically what you get when you switch the places of the numerator and denominator. Reciprocals always multiply to yield 1.

Summary

Hopefully now you have a better understanding of how to multiply fractions.

There are three main steps.

  1. Multiply numerators (tops).
  2. Multiply denominators (bottoms).
  3. Simplify the result if possible.

In addition, you should be aware of these special situations.

  1. You can express a whole number n as a fraction n/1.
  2. When there are mixed fractions involved, you can first convert the mixed fraction to a simple (improper) fraction, and then do the multiplication. Or, you might use distribution, thinking of the mixed fraction as the sum of its whole part and its fractional part.
  3. If any of the fractions are negative, simply follow the same rules as for multiplying positive and negative numbers in general. In other words:
    • Positive × positive = positive
    • Positive × negative = negative
    • Negative × positive = negative
    • Negative × negative = positive

If you’d like to see more articles about fractions, then check out these informative posts: Adding Fractions: Everything You Need to Know, What Are Rational Numbers?.

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Author

  • Shaun Ault

    Shaun earned his Ph. D. in mathematics from The Ohio State University in 2008 (Go Bucks!!). He received his BA in Mathematics with a minor in computer science from Oberlin College in 2002. In addition, Shaun earned a B. Mus. from the Oberlin Conservatory in the same year, with a major in music composition. Shaun still loves music -- almost as much as math! -- and he (thinks he) can play piano, guitar, and bass. Shaun has taught and tutored students in mathematics for about a decade, and hopes his experience can help you to succeed!