Multiplying and Dividing Fractions

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Multiplying and Dividing Fractions

• By
• Alison Hebein
• http//ellerbruch.nmu.edu

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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

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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.

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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!

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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.

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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 Michaels 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).

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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!

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Multiplying Fractions

• You may find that multiplying fractions is easier
  than adding or subtraction because you dont need
  to find common denominators.
• Instead, you multiply straight across. Multiply
  numerators together. Multiply denominators
  together.

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AlgorithmMultiplication

• 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/-- (1X33)
• Multiply the denominators of the fractions and
  write the answer as the denominator of the new
  fraction
• ½ X ¾ 3/8 (2X48)
• Remember to write your answers in lowest terms!

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A Few Examples

• Proper Fraction 2/3 X 4/5
• Answer 8/15 (2X48, 3X515)
• Improper Fraction 9/2 X 3/7
• Answer 27/141 13/27 (9X327, 7X214)
• Mixed Number 2 1/6 X 3/2
• Answer 39/123 3/123 1/4 (13/6 X 3/2
• 13X3, 6X2)
• Whole Number 5 X 2/7
• Answer 10/71 3/7 (5X210, 1X77)

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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.

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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 2/12/41/2

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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

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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

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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

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Some Examples

• Proper Fraction 3/4 / 5/6
• Answer 18/209/10 (3/4 x 6/5)
• Improper 8/3 / 2/4
• Answer 32/616/35 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 27/83 3/8 (9/4 x 3/2)

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Sources

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