Multiplying and Dividing Fractions

1. Multiplying and Dividing Fractions By Alison Hebein http://ellerbruch.nmu.edu

2. Lesson Overview: • After going through this lesson you will: • Understand the concepts of multiplying and dividing fractions • Understand the algorithms of multiplying and dividing fractions • Be able to apply your new knowledge to solve multiplication and division problems involving fractions

3. A Quick Review: • You should remember that fractions have a numerator and a denominator. • The numerator tells how many parts we are talking about, and the denominator tells you how many parts the whole is divided into. So a fraction like 3/4 tells you that we are looking at three (3) parts of a whole that is divided into four (4) equal parts. • ¾--The top number (3) is the numerator, while the bottom number (4) is the denominator. • There are different types of fractions • Proper (Example: ½) • Improper (Example: 11/4) • Mixed Number (Example: 1 5/6 changes to 11/6) • When multiplying or dividing fractions, change a mixed number into an improper fraction, but when reducing, change an improper fraction back into a mixed number.

4. Why Multiply and Divide Fractions? • There are many reasons why we may need to multiply and divide fractions in real-life settings, such as: • To calculate a grade in a class • To calculate money while grocery shopping, running errands, etc. • To become better problem-solvers • To be able to get correct measurements while measuring things such as an area of a room • To ration portions of food equally among friends • To get through your math classes!

5. Multiplying Fractions: • What does 2 x 3 mean to you? One way to think of it is as “2 sets of 3.” For example, “Max bought 2 packages of three balloons.” • What does 2 ½ x ¾ mean? It is the same as when multiplying whole numbers: “2 ½ sets of ¾.” The first factor tells how much of the second factor you have or want. For instance, “Marga ate 2 ½ pieces of ¾ of a Hershey bar that was left.”

6. More on Multiplying Fractions: • When we see the word “of” in a problem involving fractions, it means we need to multiply. Here is an example: • There are 8 cars in Michael’s toy collection. 1/2 of the cars are red. How many red cars does Michael have? • This problem is asking “What is 1/2 of 8?” • A way to answer it is to put a multiplication sign in place of “of.” You then get 1/2 x 8 or 8 x ½ (remember that multiplication is commutative).

7. Multiplication Continued: • What do you think 2/3 of 15 means? • It means 2/3 x 15 • It could mean anything. It is helpful if you think of a situation such as: • Mike ate 2/3 of 15 cookies. • Susie took 2/3 of her 15 marbles to school. • The dog ran 2/3 of its 15 laps around the yard. • You are just about ready to learn the rules/algorithm for multiplying fractions!

8. Multiplying Fractions: • You may find that multiplying fractions is easier than adding or subtraction because you don’t need to find common denominators. • Instead, you multiply straight across. Multiply numerators together. Multiply denominators together.

9. Algorithm—Multiplication: • Set up the fractions side-by-side. • 1/2 X 3/4 • Multiply the numerators of the fractions and write the product as the numerator of the new fraction • ½ X ¾ = 3/-- (1X3=3) • Multiply the denominators of the fractions and write the answer as the denominator of the new fraction • ½ X ¾= 3/8 (2X4=8) • Remember to write your answers in lowest terms!

10. A Few Examples: • Proper Fraction: 2/3 X 4/5 • Answer: 8/15 (2X4=8, 3X5=15) • Improper Fraction: 9/2 X 3/7 • Answer: 27/14=1 13/27 (9X3=27, 7X2=14) • Mixed Number: 2 1/6 X 3/2 • Answer: 39/12=3 3/12=3 1/4 (13/6 X 3/2= 13X3, 6X2) • Whole Number: 5 X 2/7 • Answer: 10/7=1 3/7 (5X2=10, 1X7=7)

11. Dividing Fractions: • What does 8/2 mean? It means you are dividing 8 of something by 2 of something else. You could think of it as giving 8 pieces of candy to 2 friends. They would each get 4 pieces, right? • What does 2 ½ / ¼ mean? The same as dividing with whole numbers. For instance, “Jack split 2 ½ pizzas with ¼ of his brothers.”

12. Rules for Dividing Fractions • Change the "÷" sign to "x" and invert the fraction to the right of the sign.  • Multiply the numerators. • Multiply the denominators.  • Re-write your answer as a simplified or reduced fraction, if needed. • Example: ¼ / ½ changes to ¼ x 2/1 • 1/4 x 2/1=2/4=1/2

13. Why do We Invert in Division? • If you think about it, in the problem 1/2 / 1/3, we are dividing a fraction by a fraction, which looks like:

14. More on “Why do We Invert?”: • To make the problem easier, we want to get rid of the denominator (1/3). So, we multiply by its reciprocal (3/1) to get 1. Remember, though, if we multiply the denominator by a number, we must multiply the numerator by the same number. For more information, go to Ask Dr. Math • So, this is what we get:

15. Algorithm: Dividing Fractions • Set up the fractions side-by-side as you would when multiplying fractions. (3/4 / 1/2) • Now, you must invert the second number (called the divisor). Example: Change 1/2 to 2/1. • Next, multiply straight across as you would when multiplying fractions • Multiply numerators together • Multiply denominators • So, ¾ X ½ becomes ¾ X 2/1 , which equals 6/4 or 1 ½ in lowest terms

16. Some Examples: • Proper Fraction: 3/4 / 5/6 • Answer: 18/20=9/10 (3/4 x 6/5) • Improper: 8/3 / 2/4 • Answer: 32/6=16/3=5 1/3 (8/3 x 4/2) • Whole Number: 5 / 1/3 • Answer: 15 (5/1 x 3/1) • Mixed Number: 2 1/4 / 2/3 • Answer: 27/8=3 3/8 (9/4 x 3/2)

17. Sources: • http://www.helpwithfractions.com/dividing-fractions.html accessed 11/25/03 • http://mathforum.org/library/drmath/view/58170.html accessed 11/25/03 • http://school.discovery.com/homeworkhelp/webmath/fractions.html accessed 11/25/03 • Van De Walle, J.A. (2001). Elementary and middle school mathematics. New York: Longman.