Multiplying and Dividing Fractions Study Guide

Introduction to Multiplying and Dividing Fractions

If you ask your mother for one fried egg for breakfast and she gives you two fried eggs and you eat both of them, who is better in arithmetic, you or your mother?

—From "Arithmetic," by CARL SANDBURG, poet (1878–1967)

This fraction lesson focuses on multiplication and division with fractions and mixed numbers.

Fortunately, multiplying and dividing fractions is actually easier than adding and subtracting them. When you multiply, you can simply multiply both the top numbers and the bottom numbers. To divide fractions, you invert and multiply. Of course, there are extra steps when you get to multiplying and dividing mixed numbers. Read on.

Multiplying Fractions

Multiplication by a proper fraction is the same as finding a part of something. For instance, suppose a personal-size pizza is cut into 4 slices. Each slice represents of the pizza. If you eat of a slice, then you've eaten of of a pizza, or × of the pizza (of means multiply), which is the same as of the whole pizza.

Multiplying and Dividing Fractions

Multiplying Fractions by Fractions

To multiply fractions:

Multiply their top numbers together to get the top number of the answer.
Multiply their bottom numbers together to get the bottom number of the answer.

Example: ×

Multiply the top numbers:
Multiply the bottom numbers:

Example:

Multiply the top numbers:
Multiply the bottom numbers:
Reduce:

Cancellation Shortcut

Sometimes you can cancel before multiplying. Cancelling is a shortcut that speeds up multiplication because you're working with smaller numbers. Cancelling is similar to reducing: If there is a number that divides evenly into a top number and a bottom number, do that division before multiplying. By the way, if you forget to cancel, don't worry. You'll still get the right answer, but you'll have to reduce it.

Example:

1.	Cancel the 6 and the 9 by dividing 3 into both of them:
	6 ÷ 3 = 2 and 9 ÷ 3 = 3. Cross out the 6 and the 9.
2.	Cancel the 5 and the 20 by dividing 5 into both of them:
	5 ÷ 5 = 1 and 20 ÷ 5 = 4. Cross out the 5 and the 20.
3.	Multiply across the new top numbers and the new bottom numbers:

Tip

When multiplying three or more fractions, the cancelling shortcut can still be used between fractions that are not directly next to each other:

Nine and four cannot reduce and neither can five and 42. But you can reduce nine and 33 by dividing by three, and 42 and 21 can be reduced by dividing by 21:

Multiplying Fractions by Whole Numbers

To multiply a fraction by a whole number:

Rewrite the whole number as a fraction with a bottom number of 1.
Multiply as usual.

Example: 5 ×

1.	Rewrite 5 as a fraction:	5 =
2.	Multiply the fractions:	× =
3.	Optional: Change the product to a mixed number.	=

Have you noticed that multiplying any number by a proper fraction produces an answer that's smaller than that number? It's the opposite of the result you get from multiplying whole numbers. That's because multiplying by a proper fraction is the same as finding a part of something.

Multiplying with Mixed Numbers

To multiply with mixed numbers, change each mixed number to an improper fraction and multiply.

Example: ×

Change to an improper fraction:
Change to an improper fraction:
Multiply the fractions:
Notice that you can cancel the 14 and the 2 by dividing them by 2.
Optional: Change the improper fraction to a mixed number.

Dividing Fractions

Dividing means finding out how many times one amount can be found in a second amount, whether you're working with fractions or not. For instance, to find out how many -pound pieces a 2-pound chunk of cheese can be cut into, you have to divide 2 by . As you can see from the following picture, a 2-pound chunk of cheese can be cut into eight -pound pieces. (2 ÷ = 8)

Multiplying and Dividing Fractions

Dividing Fractions by Fractions

To divide one fraction by a second fraction, invert the second fraction (that is, flip the top and bottom numbers) and then multiply.

Example: ÷

1.	Invert the second fraction ():
2.	Change ÷ to × and multiply the first fraction by the new second fraction:

Another Format for Division

Sometimes fraction division is written in a different format. For example, ÷ can also be written as . Regardless of the format used, the solution is the same.

Reciprocal Fractions

Inverting a fraction, as we do for division, is the same as finding the fraction's reciprocal. For example, and are reciprocals. The product of a fraction and its reciprocal is 1. Thus, × = 1.

Tip

Remember, when dividing a number by a positive fraction that is less than one, the answer is going to be larger than the original number. When dividing by an improper fraction (which has a value greater than one), your answer will be smaller than the original number. Use these facts to make sure your answers make sense.

Have you noticed that dividing a number by a proper fraction gives an answer that's larger than that number? It's the opposite of the result you get when dividing by a whole number.

Dividing Fractions by Whole Numbers or Vice Versa

To divide a fraction by a whole number or vice versa, change the whole number to a fraction by putting it over 1, and then divide as usual.

Example: ÷ 2

1.	Change the whole number (2) into a fraction:	2 =
2.	Invert the second fraction ():
3.	Change ÷ to × and multiply the two fractions:	× =

Example: 2 ÷

1.	Change the whole number (2) into a fraction:	2 =
2.	Invert the second fraction ():
3.	Change ÷ to × and multiply the two fractions:	× =
4.	Optional: Change the improper fraction to a mixed number.	=

Did you notice that the order of division makes a difference? ÷ 2 is not the same as 2 ÷ . But then, the same is true of division with whole numbers; 4 ÷ 2 is not the same as 2 ÷ 4.

Dividing with Mixed Numbers

To divide with mixed numbers, change each mixed number to an improper fraction and then divide as usual.

Example:

1.	Change to an improper fraction::
2.	Rewrite the division problem:
3.	Invert and multiply:
4.	Optional: Change the improper fraction to a mixed number.

Tip

Buy a small bag of candy (or cookies or any other treat you like) as a reward for completing this lesson. Before you eat any of the bag's contents, empty the bag and count how many pieces of candy are in it. Write down this number. Then walk around and collect three friends or family members who want to share your candy. Now divide the candy equally among you and them. If the total number of candies you have is not divisible by four, you might have to cut some in half or quarters; this means you'll have to divide using fractions, which is great practice. Write down the equation that shows the fraction of candy that each of you received of the total amount.

Multiplying and Dividing Fractions Sample Questions

× 24
÷
÷ 2

Solutions to Sample Questions

Question 1

1. Multiply the top numbers:	2 × 3 = 6
2. Multiply the bottom numbers:	5 × 4 = 20
3. Reduce:

Question 2

1.	Cancel the 4 and the 22 by dividing 2 into both of them:
	4 ÷ 2 = 2 and 22 ÷ 2 = 11. Cross out the 4 and the 22.
2.	Cancel the 9 and the 15 by dividing 3 into both of them:
	9 ÷ 3 = 3 and 15 ÷ 3 = 5. Cross out the 9 and the 15.
3.	Multiply across the new top numbers and the new bottom numbers:

Question 3

1. Rewrite 24 as a fraction:
2. Multiply the fractions:
Cancel the 8 and the 24 by dividing both of them by 8;
then multiply across the new numbers.

Question 4

1. Change to an improper fraction:
2. Multiply the fractions:

Question 5

1. Invert the second fraction :
2. Change ÷ to × and multiply the first fraction by the new second fraction:
3. Optional: Change the improper fraction to a mixed number.