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4.4: Multiply and Divide Fractions (Part 2) - Mathematics


Find Reciprocals

The fractions (dfrac{2}{3}) and (dfrac{3}{2}) are related to each other in a special way. So are (− dfrac{10}{7}) and (− dfrac{7}{10}). Do you see how? Besides looking like upside-down versions of one another, if we were to multiply these pairs of fractions, the product would be 1.

[dfrac{2}{3} cdot dfrac{3}{2} = 1 quad and quad - dfrac{10}{7} left(- dfrac{7}{10} ight) = 1 ag{4.2.53} onumber ]

Such pairs of numbers are called reciprocals.

Definition: Reciprocal

The reciprocal of the fraction (dfrac{a}{b}) is (dfrac{b}{a}), where (a ≠ 0) and (b ≠ 0).

A number and its reciprocal have a product of (1).

[dfrac{a}{b} cdot dfrac{b}{a} = 1 ag{4.2.54}]

To find the reciprocal of a fraction, we invert the fraction. This means that we place the numerator in the denominator and the denominator in the numerator.

To get a positive result when multiplying two numbers, the numbers must have the same sign. So reciprocals must have the same sign.

To find the reciprocal, keep the same sign and invert the fraction. The number zero does not have a reciprocal. Why? A number and its reciprocal multiply to (1). Is there any number (r) so that (0 • r = 1)? No. So, the number (0) does not have a reciprocal.

Example (PageIndex{11}): reciprocal

Find the reciprocal of each number. Then check that the product of each number and its reciprocal is (1).

  1. (dfrac{4}{9})
  2. (− dfrac{1}{6})
  3. (− dfrac{14}{5})
  4. (7)

Solution

To find the reciprocals, we keep the sign and invert the fractions.

    Find the reciprocal of (dfrac{4}{9}).The reciprocal of (dfrac{4}{9}) is (dfrac{9}{4}).

    Check:

    Multiply the number and its reciprocal.(dfrac{4}{9} cdot dfrac{9}{4})
    Multiply numerators and denominators.(dfrac{36}{36} )
    Simplify.(1 ; checkmark )
      Find the reciprocal of (- dfrac{1}{6}).The reciprocal of (- dfrac{1}{6}) is (dfrac{6}{1}).
      Simplify.(-6 )
      Check.(- dfrac{1}{6} cdot (-6) = 1 ; checkmark )
        Find the reciprocal of (- dfrac{14}{5}).(- dfrac{5}{14} )
        Check.(- dfrac{14}{5} cdot left(- dfrac{5}{14} ight) = dfrac{70}{70} = 1 ; checkmark )
          Find the reciprocal of 7.
          Write 7 as a fraction.(dfrac{7}{1})
          Write the reciprocal of (dfrac{7}{1}).(dfrac{1}{7} )
          Check.(7 cdot left(dfrac{1}{7} ight) = 1 ; checkmark )

          Exercise (PageIndex{21})

          Find the reciprocal:

          1. (dfrac{5}{7})
          2. (− dfrac{1}{8})
          3. (− dfrac{11}{4})
          4. (14)
          Answer a

          (dfrac{7}{5})

          Answer b

          (-8)

          Answer c

          (-dfrac{4}{11})

          Answer d

          (dfrac{1}{14})

          Exercise (PageIndex{22})

          Find the reciprocal:

          1. (dfrac{3}{7})
          2. (− dfrac{1}{12})
          3. (− dfrac{14}{9})
          4. (21)
          Answer a

          (dfrac{7}{3})

          Answer b

          (-12)

          Answer c

          (-dfrac{9}{14})

          Answer d

          (dfrac{1}{21})

          In a previous chapter, we worked with opposites and absolute values. Table (PageIndex{1}) compares opposites, absolute values, and reciprocals.

          Table (PageIndex{1})
          OppositeAbsolute ValueReciprocal
          has opposite signis never negativehas same sign, fraction inverts

          Example (PageIndex{12}): fractions

          Fill in the chart for each fraction in the left column:

          NumberOppositeAbsolute ValueReciprocal
          (- dfrac{3}{8})
          (dfrac{1}{2})
          (dfrac{9}{5})
          (-5)

          Solution

          To find the opposite, change the sign. To find the absolute value, leave the positive numbers the same, but take the opposite of the negative numbers. To find the reciprocal, keep the sign the same and invert the fraction.

          NumberOppositeAbsolute ValueReciprocal
          (- dfrac{3}{8})(dfrac{3}{8})(dfrac{3}{8})(- dfrac{8}{3})
          (dfrac{1}{2})(- dfrac{1}{2})(dfrac{1}{2})(2)
          (dfrac{9}{5})(- dfrac{9}{5})(dfrac{9}{5})(dfrac{5}{9})
          (-5)(5)(5)(- dfrac{1}{5})

          Exercise (PageIndex{23})

          Fill in the chart for each number given:

          NumberOppositeAbsolute ValueReciprocal
          (- dfrac{5}{8})
          (dfrac{1}{4})
          (dfrac{8}{3})
          (-8)
          Answer
          NumberOppositeAbsolute ValueReciprocal
          (-dfrac{5}{8})(dfrac{5}{8})(dfrac{5}{8})(-dfrac{8}{5})
          (dfrac{1}{4})(-dfrac{1}{4})(dfrac{1}{4})(4)
          (dfrac{8}{3})(-dfrac{8}{3})(dfrac{8}{3})(dfrac{3}{8})
          (-8)(8)(8)(-dfrac{1}{8})

          Exercise (PageIndex{24})

          Fill in the chart for each number given:

          NumberOppositeAbsolute ValueReciprocal
          (- dfrac{4}{7})
          (dfrac{1}{8})
          (dfrac{9}{4})
          (-1)
          Answer
          NumberOppositeAbsolute ValueReciprocal
          (-dfrac{4}{7})(dfrac{4}{7})(dfrac{4}{7})(- dfrac{7}{4})
          (dfrac{1}{8})(-dfrac{1}{8})(dfrac{1}{8})(8)
          (dfrac{9}{4})(-dfrac{9}{4})(dfrac{9}{4})(dfrac{4}{9})
          (-1)(1)(1)(-dfrac{1}{1})

          Divide Fractions

          Why is (12 ÷ 3 = 4)? We previously modeled this with counters. How many groups of (3) counters can be made from a group of (12) counters?

          Figure (PageIndex{2})

          There are (4) groups of (3) counters. In other words, there are four (3)s in (12). So, (12 ÷ 3 = 4).

          What about dividing fractions? Suppose we want to find the quotient: (dfrac{1}{2} div dfrac{1}{6}). We need to figure out how many (dfrac{1}{6})s there are in (dfrac{1}{2}). We can use fraction tiles to model this division. We start by lining up the half and sixth fraction tiles as shown in Figure (PageIndex{3}). Notice, there are three (dfrac{1}{6}) tiles in (dfrac{1}{2}), so (dfrac{1}{2} div dfrac{1}{6} = 3).

          Figure (PageIndex{3})

          Example (PageIndex{13}): model

          Model: (dfrac{1}{4} div dfrac{1}{8}).

          Solution

          We want to determine how many (dfrac{1}{8})s are in (dfrac{1}{4}). Start with one (dfrac{1}{4}) tile. Line up (dfrac{1}{8}) tiles underneath the (dfrac{1}{4}) tile.

          Exercise (PageIndex{25})

          Model: (dfrac{1}{3} div dfrac{1}{6}).

          Answer

          Exercise (PageIndex{26})

          Model: (dfrac{1}{2} div dfrac{1}{4}).

          Answer

          Example (PageIndex{14}): model

          Model: (2 ÷ dfrac{1}{4}).

          Solution

          We are trying to determine how many (dfrac{1}{4})s there are in (2). We can model this as shown.

          Because there are eight (dfrac{1}{4})s in (2), (2 ÷ dfrac{1}{4} = 8).

          Exercise (PageIndex{27})

          Model: (2 ÷ dfrac{1}{3})

          Answer

          Exercise (PageIndex{28})

          Model: (3 ÷ dfrac{1}{2})

          Answer

          Let’s use money to model (2 ÷ dfrac{1}{4}) in another way. We often read (dfrac{1}{4}) as a ‘quarter’, and we know that a quarter is one-fourth of a dollar as shown in Figure (PageIndex{4}). So we can think of (2 ÷ dfrac{1}{4}) as, “How many quarters are there in two dollars?” One dollar is (4) quarters, so (2) dollars would be (8) quarters. So again, (2 ÷ dfrac{1}{4} = 8).

          Figure (PageIndex{4}):The U.S. coin called a quarter is worth one-fourth of a dollar.

          Using fraction tiles, we showed that (dfrac{1}{2} div dfrac{1}{6} = 3). Notice that (dfrac{1}{2} cdot dfrac{6}{1} = 3) also. How are (dfrac{1}{6}) and (dfrac{6}{1}) related? They are reciprocals. This leads us to the procedure for fraction division.

          Definition: Fraction Division

          If (a, b, c,) and (d) are numbers where (b ≠ 0), (c ≠ 0), and (d ≠ 0), then

          [dfrac{a}{b} div dfrac{c}{d} = dfrac{a}{b} cdot dfrac{d}{c} ]

          To divide fractions, multiply the first fraction by the reciprocal of the second.

          We need to say (b ≠ 0), (c ≠ 0) and (d ≠ 0) to be sure we don’t divide by zero.

          Example (PageIndex{15}): divide

          Divide, and write the answer in simplified form: (dfrac{2}{5} div left(- dfrac{3}{7} ight).

          Solution

          Multiply the first fraction by the reciprocal of the second.(dfrac{2}{5} left(- dfrac{7}{3} ight) )
          Multiply. The product is negative.(- dfrac{14}{15})

          Exercise (PageIndex{29})

          Divide, and write the answer in simplified form: (dfrac{3}{7} div left(− dfrac{2}{3} ight)).

          Answer

          (-dfrac{9}{14})

          Exercise (PageIndex{30})

          Divide, and write the answer in simplified form: (dfrac{2}{3} div left(− dfrac{7}{5} ight)).

          Answer

          (-dfrac{10}{21})

          Example (PageIndex{16}): divide

          Divide, and write the answer in simplified form: (dfrac{2}{3} div dfrac{n}{5}).

          Solution

          Multiply the first fraction by the reciprocal of the second.(dfrac{2}{3} div dfrac{5}{n} )
          Multiply.(dfrac{10}{3n})

          Exercise (PageIndex{31})

          Divide, and write the answer in simplified form: (dfrac{3}{5} div dfrac{p}{7}).

          Answer

          (dfrac{21}{5p})

          Exercise (PageIndex{32})

          Divide, and write the answer in simplified form: (dfrac{5}{8} div dfrac{q}{3}).

          Answer

          (dfrac{15}{8q})

          Example (PageIndex{17}): divide

          Divide, and write the answer in simplified form: (− dfrac{3}{4} div left(− dfrac{7}{8} ight)).

          Solution

          Multiply the first fraction by the reciprocal of the second.(- dfrac{3}{4} cdot left(- dfrac{8}{7} ight) )
          Multiply. Remember to determine the sign first.(dfrac{3 cdot 8}{4 cdot 7})
          Rewrite to show common factors.(dfrac{3 cdot cancel{4} cdot 2}{cancel{4} cdot 7} )
          Remove common factors and simplify.(dfrac{6}{7} )

          Exercise (PageIndex{33})

          Divide, and write the answer in simplified form: (− dfrac{2}{3} div left(− dfrac{5}{6} ight)).

          Answer

          (dfrac{4}{5})

          Exercise (PageIndex{34})

          Divide, and write the answer in simplified form: (− dfrac{5}{6} div left(− dfrac{2}{3} ight)).

          Answer

          (dfrac{5}{4})

          Example (PageIndex{18}): divide

          Divide, and write the answer in simplified form: (dfrac{7}{18} div dfrac{14}{27}).

          Solution

          Multiply the first fraction by the reciprocal of the second.(dfrac{7}{18} cdot dfrac{27}{14} )
          Multiply.(dfrac{7 cdot 27}{18 cdot 14} )
          Rewrite showing common factors.(dfrac{cancel{ extcolor{red}{7}} cdot cancel{ extcolor{red}{9}} cdot 3}{cancel{ extcolor{red}{9}} cdot cancel{ extcolor{red}{7}} cdot 2})
          Remove common factors.(dfrac{3}{2 cdot 2} )
          Simplify.(dfrac{3}{4} )

          Exercise (PageIndex{35})

          Divide, and write the answer in simplified form: (dfrac{7}{27} div dfrac{35}{36}).

          Answer

          (dfrac{4}{15})

          Exercise (PageIndex{36})

          Divide, and write the answer in simplified form: (dfrac{5}{14} div dfrac{15}{28}).

          Answer

          (dfrac{2}{3})

          Key Concepts

          • Equivalent Fractions Property
          • If (a, b, c) are numbers where (b eq 0, c eq 0), then (dfrac{a}{b} = dfrac{acdot c}{bcdot c}) and (dfrac{acdot c}{bcdot c} = dfrac{a}{b})
          • Simplify a fraction.
            1. Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers.
            2. Simplify, using the equivalent fractions property, by removing common factors.
            3. Multiply any remaining factors.
          • Fraction Multiplication
            • If (a, b, c,) and
          • Reciprocal
            • A number and its reciprocal have a product of 1. (frac{a}{b} cdot frac{b}{a} = 1)
            • OppositeAbsolute ValueReciprocal
              has opposite signis never negativehas same sign, fraction inverts
          • Fraction Division
            • If (a, b, c,) and (d) are numbers where (b eq 0), (c eq 0), and (d eq 0), then (dfrac{a}{b} div dfrac{c}{d} = dfrac{a}{b}cdot dfrac{d}{c})
            • To divide fractions, multiply the first fraction by the reciprocal of the second.

          Glossary

          reciprocal

          The reciprocal of the fraction (dfrac{a}{b}) is (dfrac{b}{a}) where (a eq 0) and (b eq 0).

          simplified fraction

          A fraction is considered simplified if there are no common factors in the numerator and denominator.

          Practice Makes Perfect

          Simplify Fractions

          In the following exercises, simplify each fraction. Do not convert any improper fractions to mixed numbers.

          1. (dfrac{7}{21})
          2. (dfrac{8}{24})
          3. (dfrac{15}{20})
          4. (dfrac{12}{18})
          5. (- dfrac{40}{88})
          6. (- dfrac{63}{99})
          7. (- dfrac{108}{63})
          8. (- dfrac{104}{48})
          9. (dfrac{120}{252})
          10. (dfrac{182}{294})
          11. (- dfrac{168}{192})
          12. (- dfrac{140}{224})
          13. (dfrac{11x}{11y})
          14. (dfrac{15a}{15b})
          15. (− dfrac{3x}{12y})
          16. (− dfrac{4x}{32y})
          17. (dfrac{14x^{2}}{21y})
          18. (dfrac{24a}{32b^{2}})

          Multiply Fractions

          In the following exercises, use a diagram to model.

          1. (dfrac{1}{2} cdot dfrac{2}{3})
          2. (dfrac{1}{2} cdot dfrac{5}{8})
          3. (dfrac{1}{3} cdot dfrac{5}{6})
          4. (dfrac{1}{3} cdot dfrac{2}{5})

          In the following exercises, multiply, and write the answer in simplified form.

          1. (dfrac{2}{5} cdot dfrac{1}{3})
          2. (dfrac{1}{2} cdot dfrac{3}{8})
          3. (dfrac{3}{4} cdot dfrac{9}{10})
          4. (dfrac{4}{5} cdot dfrac{2}{7})
          5. (− dfrac{2}{3} left(− dfrac{3}{8} ight))
          6. (− dfrac{3}{4} left(− dfrac{4}{9} ight))
          7. (- dfrac{5}{9} cdot dfrac{3}{10})
          8. (- dfrac{3}{8} cdot dfrac{4}{15})
          9. (− dfrac{7}{12} left(− dfrac{8}{21} ight))
          10. (dfrac{5}{12} left(− dfrac{8}{15} ight))
          11. (left(− dfrac{14}{15} ight) left(dfrac{9}{20} ight))
          12. (left(− dfrac{9}{10} ight) left(dfrac{25}{33} ight))
          13. (left(− dfrac{63}{84} ight) left(- dfrac{44}{90} ight))
          14. (left(− dfrac{33}{60} ight) left(- dfrac{40}{88} ight))
          15. (4 cdot dfrac{5}{11})
          16. (5 cdot dfrac{8}{3})
          17. (dfrac{3}{7} cdot 21n)
          18. (dfrac{5}{6} cdot 30m)
          19. (−28p left(− dfrac{1}{4} ight))
          20. (−51q left(− dfrac{1}{3} ight))
          21. (−8 left(dfrac{17}{4} ight))
          22. (dfrac{14}{5} (−15))
          23. (−1 left(− dfrac{3}{8} ight))
          24. ((−1) left(- dfrac{6}{7} ight))
          25. (left(dfrac{2}{3} ight)^{3})
          26. (left(dfrac{4}{5} ight)^{2})
          27. (left(dfrac{6}{5} ight)^{4})
          28. (left(dfrac{4}{7} ight)^{4})

          Find Reciprocals In the following exercises, find the reciprocal.

          1. (dfrac{3}{4})
          2. (dfrac{2}{3})
          3. (− dfrac{5}{17})
          4. (− dfrac{6}{19})
          5. (dfrac{11}{8})
          6. −13
          7. −19
          8. −1
          9. 1
          10. Fill in the chart.
            OppositeAbsolute ValueReciprocal
            (- dfrac{7}{11})
            (dfrac{4}{5})
            (dfrac{10}{7})
            (-8)
          11. Fill in the chart.
            OppositeAbsolute ValueReciprocal
            (- dfrac{3}{13})
            (dfrac{9}{14})
            (dfrac{15}{7})
            (-9)

          Divide Fractions

          In the following exercises, model each fraction division.

          1. (dfrac{1}{2} div dfrac{1}{4})
          2. (dfrac{1}{2} div dfrac{1}{8})
          3. (2 div dfrac{1}{5})
          4. (3 div dfrac{1}{4})

          In the following exercises, divide, and write the answer in simplified form.

          1. (dfrac{1}{2} div dfrac{1}{4})
          2. (dfrac{1}{2} div dfrac{1}{8})
          3. (dfrac{3}{4} div dfrac{2}{3})
          4. (dfrac{4}{5} div dfrac{3}{4})
          5. (- dfrac{4}{5} div dfrac{4}{7})
          6. (- dfrac{3}{4} div dfrac{3}{5})
          7. (− dfrac{7}{9} div left(- dfrac{7}{9} ight))
          8. (− dfrac{5}{6} div left(- dfrac{5}{6} ight))
          9. (dfrac{3}{4} div dfrac{x}{11})
          10. (dfrac{2}{5} div dfrac{y}{9})
          11. (dfrac{5}{8} div dfrac{a}{10})
          12. (dfrac{5}{6} div dfrac{c}{15})
          13. (dfrac{5}{18} div left(- dfrac{15}{24} ight))
          14. (dfrac{7}{18} div left(- dfrac{14}{27} ight))
          15. (dfrac{7p}{12} div dfrac{21p}{8})
          16. (dfrac{5q}{12} div dfrac{15q}{8})
          17. (dfrac{8u}{15} div dfrac{12v}{25})
          18. (dfrac{12r}{25} div dfrac{18s}{35})
          19. (-5 div dfrac{1}{2})
          20. (-3 div dfrac{1}{4})
          21. (dfrac{3}{4} div (-12))
          22. (dfrac{2}{5} div (-10))
          23. (−18 div left(− dfrac{9}{2} ight))
          24. (−15 div left(− dfrac{5}{3} ight))
          25. (dfrac{1}{2} div left(- dfrac{3}{4} ight) div dfrac{7}{8})
          26. (dfrac{11}{2} div dfrac{7}{8} cdot dfrac{2}{11})

          Everyday Math

          1. Baking A recipe for chocolate chip cookies calls for 3 4 cup brown sugar. Imelda wants to double the recipe.
            1. How much brown sugar will Imelda need? Show your calculation. Write your result as an improper fraction and as a mixed number.
            2. Measuring cups usually come in sets of (dfrac{1}{8}, dfrac{1}{4}, dfrac{1}{3}, dfrac{1}{2}), and 1 cup. Draw a diagram to show two different ways that Imelda could measure the brown sugar needed to double the recipe.
          2. Baking Nina is making 4 pans of fudge to serve after a music recital. For each pan, she needs 2 3 cup of condensed milk.
            1. How much condensed milk will Nina need? Show your calculation. Draw a diagram to show two different ways that Nina could measure the condensed milk she needs.
          3. Portions Don purchased a bulk package of candy that weighs 5 pounds. He wants to sell the candy in little bags that hold (dfrac{1}{4}) pound. How many little bags of candy can he fill from the bulk package?
          4. Portions Kristen has (dfrac{3}{4}) yards of ribbon. She wants to cut it into equal parts to make hair ribbons for her daughter’s 6 dolls. How long will each doll’s hair ribbon be?

          Writing Exercises

          1. Explain how you find the reciprocal of a fraction.
          2. Explain how you find the reciprocal of a negative fraction.
          3. Rafael wanted to order half a medium pizza at a restaurant. The waiter told him that a medium pizza could be cut into 6 or 8 slices. Would he prefer 3 out of 6 slices or 4 out of 8 slices? Rafael replied that since he wasn’t very hungry, he would prefer 3 out of 6 slices. Explain what is wrong with Rafael’s reasoning.
          4. Give an example from everyday life that demonstrates how (dfrac{1}{2} cdot dfrac{2}{3}) is (dfrac{1}{3}).

          Self Check

          (a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

          (b) After reviewing this checklist, what will you do to become confident for all objectives?


          Watch the video: Math 30-2: Rational Expressions: Multiplying and Dividing Rationals (October 2021).