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4.4: Multiply and Divide Fractions (Part 2)


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?


          Finding Fractional Parts with Division

          In this 4th grade lesson, students learn the connection between division and finding a fractional part of a quantity. For example, to find 2/3 of 9 apples, we first find 1/3 of 9 apples using division, and then double our result.

          Mom's 24 brownies are divided into 6 equal
          parts. Each part is 1/6th of the whole. How
          many pieces are in each part?

          1. Write a division sentence and a fractional part sentence.

          2. Write a fractional part sentence for each division sentence.

          a . 30 ÷ 5 = _____

          3. Find a part. Also write a division sentence.

          Divide these ten
          fish into 5 groups.

          a. Marsha got $18 from her mom. She put into her savings $6, which was one-__________ part of it.

          Division sentence: _______ ÷_____ = _____

          b. Mariana spent one-fourth of her $80 savings, or 4.4: Multiply and Divide Fractions (Part 2),[nobr][H1toH2]

          THE MEANING OF MULTIPLYING FRACTIONS

          In the previous Lesson we simply stated the rule for multiplying fractions. In this Lesson we want to understand where that rule comes from. It comes from what multiplying by a fraction means .

          First, in Lesson 15 we saw what "the third part", or "one third" of a number means. "One third of 15," for example, is 5.

          In symbols, "One third of 15" will be written as multiplication:

          That is what multiplication by a fraction means.


          1. What does it mean to multiply a number by a fraction?
          1
          2
          × 8, .15 × 20
          It means to take that part of the number.

          For, according to the meaning of multiplication, we are to repeatedly add the multiplicand as many times as there are 1's in the multiplier. In the multiplier ½ there is one half of 1. Therefore we are to add the multiplicand 8 one half a time. We are to take one half of 8.

          Also, although "½ × 8" looks like multiplication, there is nothing to multiply. "½ × 8" is a symbolic abbreviation for "One half of 8." And to calculate it, we have to divide. (Lesson 15.) We can now begin to see why we have the cancelation rules.

          This is another use for fractions apart from numbers we need for measuring: Multiplication by a fraction signifies a part of the multiplicand.

          And so the symbolic statement, "4 = ½ × 8," expresses the ratio of 4 to 8: "4 is one half of 8."

          For the most general definition of multiplication, see Section 3.

          Example 1. Calculate × 21 "Two thirds of 21." (We may read " × 21" as "Two thirds of 21" rather than "Two thirds times 21.")

          One third of 21 is 7 -- "3 goes into 21 seven ( 7 ) times." 2 × 7 = 14.

          If the problem were just to evaluate Two thirds of 21, the student should not have to resort to writing × 21.

          Simply say, " One third of 21 is 7. So two thirds are 14." (Lesson 15.) The point of this Lesson is to explain what it means to multiply by a fraction.

          Problem . × 32. What does that mean?

          To see the answer, pass your mouse over the colored area.
          To cover the answer again, click "Refresh" ("Reload").
          Do the problem yourself first!

          "One eighth of 32 is 4. So five eighths are five times 4: 20."

          Example 2. Calculate × 5. "Three fourths of 5."

          Solution . Although 5 is not exactly divisible by 4, we can still take its fourth part -- by dividing by 4:

          "4 goes into 5 one ( 1 ) time with 1 left over."

          One fourth of 5 is 1 therefore, three fourths are 3 × 1 = 3 .

          Alternatively, we can multiply first:

          "4 goes into 15 three ( 3 ) times (12) with 3 left over."

          We may take a part first or multiply first .

          Three fourths of 5 = 3 × One fourth of 5 = One fourth of 3 × 5.

          Example 3. You are going on a a trip of four miles, and you have gone two thirds of the way. How far have you gone?

          Solution . We must take two thirds of 4.

          "Therefore two thirds are 2 × 1 = 2 ."

          Example 4. How much is a fifth of 3?

          Solution . While we could write × 3 = , we know that to find a fifth of a number, we divide by 5. And 3 ÷ 5 is . Lesson 11, Example 17.

          Therefore, we could know immediately:

          Example 5. How much is a fourth of 9 gallons?

          "4 goes into 9 two times with 1 left over."

          Example 6. Calculator problem. Tim and his business partner invested $71,000 in a property. Tim invested $51,000, and his partner, $20,000.

          They had to sell the property at a loss for $48,000. If each one receives the same fraction that they invested, how much will each one recieve?

          Solution . First, what fraction of the $71,000 did Tim invest? 51,000 is what fraction of 71,000? It is of it. (Lesson 20. Note that we may omit the final 0's.)

          We are to find that same fraction of 48,000:

          Tim's share will be $34,479. (Lesson 12.) Therefore his partner's share to make up the difference will be

          Example 7 How much money is 64 quarters?

          Answer . 64 quarters would be 64 × $.25. But according to the order property of multiplication,

          Now . 25 is the decimal for ¼. Therefore we can evaluate 64 quarters by taking one quarter of 64. And we can do that by taking half of half. (Lesson 16.)

          Half of 64 is 32. Half of 32 is 16. Therefore 64 quarters are $16.

          Example 8. A slot machine at a casino paid 93 quarters. How much money is that?

          Answer . To find a quarter of 93, divide 93 by 4. We can easily do that mentally by decomposing 93 into multiples of 4. For example:

          On dividing each term by 4, we have

          93 quarters, then, are $23.25.

          Example 9. A recipe calls for 3 cups of flour and 4 cups of milk. Proportionally, how much milk should you use if

          a) you use 1½ cups of flour? b) you use 2 cups of flour?

          c) you use 2½ cups of flour?

          a) 1½ cups flour are half of 3 cups. Therefore you should use half as
          a) much milk. You should use 2 cups.

          b) 2 cups flour are two thirds of 3 cups. That is the ratio of 2 cups to 3. b) Therefore you should use two thirds as much milk.

          c) What ratio has 2½ cups of flour to the original 3 cups?

          On expressing 2½ as the improper fraction , then on cross-multiplying:

          2½ cups are five sixths of 3 cups.

          Therefore, you should use five sixths of 4 cups of milk.


          2. How do we multiply a whole number by a mixed number?
          2½ × 8
          Multiply by the whole number of the mixed number, then multiply by the fraction. It is not necessary to change to an improper fraction.

          In Lesson 16, Question 3, we saw this as a mixed number of times.

          Answer . 2 ½ × 8 = 2 × 8 + ½ × 8
          ( "Two times 8 + Half of 8")
          Answer . 2½ × 8 = 16 + 4
          Answer . 2½ × 8 = 20.

          In multiplication, when one of the numbers is a whole number, it is not necessary to change to an improper fraction.

          Answer. "5 times 7 is 35. A third of 7 is 2 ." (Lesson 20, Problem 16.)

          Example 12. Mental calculation. What is the price of 12 items at $3 . 25 each?

          Answer . 12 × $3 . 25 is equal to $3 . 25 × 12, or, 3¼ × 12:

          Example 13. Multiplying by numbers ending in 5. Calculate mentally: 75 × 6.

          Answer . Rather than 75 × 6, let us do

          Now, by replacing 75 with 7 . 5, we divided by 10. (Lesson 4,
          Question 5.) Therefore, to name the right answer we must multiply 45 by 10:

          Answer . 3 . 5 × 16 = 48 + 8 = 56. Therefore,


          3. How can we express a fraction as a percent?
          Multiply it times 100%.

          That is how to change any number to a percent. (Lesson 4.)

          Example 15. Express as a percent. Express as a percent.

          Solution . 100% is the whole. Therefore, take one eleventh of 100%:

          "11 goes into 100 nine (9) times (99) with 1 left over."

          We will go into this again in Lesson 30, Question 3.

          For the most frequent percent equivalents, see Lesson 24.

          The order of taking a part and multiplying

          3
          4
          × 5 "Three fourths of 5,"

          we may either take the fourth part first, or multiply by 3 first. That is,

          Three fourths of 5 = 3 × One fourth of 5 = One fourth of 3 × 5.

          In both figures, each 5 has been divided into fourths.

          The upper figure shows 3 × One fourth of 5, that is, Three fourths of 5.

          The bottom figure shows one fourth of three 5's. And they are equal.

          Therefore to multiply a whole number by a fraction, we may either take the part first or multiply first.


          10 Simple Fraction Problems and How to Solve Them

          Below are ten examples of fractional equations and guidance on how to solve them. If you’re working with fractions in an exam setting, always be sure to show your method.

          1. How to Convert a Mixed Fraction to an Improper Fraction

          As discussed, a mixed fraction consists of a whole number followed by a fractional number. In this example, we’ll use the mixed fraction of seven and four-fifths, written numerically as 7⅘.

          When asked to convert a mixed fraction to an improper fraction:

          • First, multiply the whole number by the denominator of the fractional part.
          • Take the resulting figure and add it to the fraction’s numerator.
          • Take this final figure as your new numerator and place it over the original denominator. This gives you your improper fraction.

          Using our mixed fraction of 7⅘:

          • Whole number multiplied by fractional denominator: 7 x 5 = 35
          • Add the result to the fractional numerator: 35 + 4 = 39
          • Place it over the original denominator: 39/5

          Therefore, the correct answer is: 7⅘ = 39/5

          2. How to Convert a Fraction to a Decimal

          Since both are used to identify values of less than one, a decimal is just a different way of representing a fraction.

          The method used to convert a fraction to a decimal is a simple division: you just divide the numerator by the denominator.

          Take the fraction 3/10. Divide the numerator by the denominator to get the decimal figure:

          The easiest way to remember how to work out fractions as decimals is to think of the line separating the numerator and the denominator as a division symbol.

          3. How to Convert a Fraction to a Percentage

          There are three easy ways to convert a fraction to a percentage. We’ll cover them all here using the same fraction of 7/20.

          Divide the numerator by the denominator, then multiply the resulting figure by 100 to get the percentage conversion:

          Multiply the numerator by 100, then divide the resulting figure by the denominator:

          Method Three:

          Divide the numerator by the denominator and move the decimal point of your answer two places to the right:

          Moving the decimal point gives you the conversion of 35%.

          When converting a fraction to a percentage, always remember to include the % sign in your answer.

          4. How to Add Fractions

          The process for adding fractions is straightforward provided the denominators are the same.

          As a basic example, take 1/6 + 3/6. In this case, you have equal denominators, so simply add the numerators of both fractions, sticking with the lower figure of 6:

          When adding fractions where the lower figures don’t tally, you’ll first need to find the lowest common denominator. This is the lowest number wholly divisible by both existing denominators.

          The lowest figure divisible by both 4 and 3 is 12. This is your common denominator.

          You now need to find equivalent fractions using 12 as your bottom figure.

          To turn 4 into 12, you multiply it by 3, so you must also multiply the numerator by 3 to keep the fraction equivalent:

          Your equivalent fraction to 1/4 is therefore 3/12

          Follow the same method for the second fraction:

          Your equivalent fraction to 2/3 is 8/12

          Now simply add the numerators together and place the answer over 12:

          The correct answer to the equation 1/4 + 2/3 is: 11/12

          5. How to Subtract Fractions

          As with addition, subtracting fractions is easy when the denominators are the same. It’s simply a matter of subtracting the second numerator from the first, keeping the bottom number the same.

          Take the equation 4/7 – 3/7. You have a common denominator, so just subtract 3 from 4:

          Now, let’s look at subtracting fractions with different denominators.

          First, find the lowest common denominator in this case, 15.

          Now, find your equivalent fractions:

          4/5 becomes 12/15 (both sides are multiplied by 3)

          2/3 becomes 10/15 (both sides are multiplied by 5)

          You can now subtract your numerators:

          The answer to the equation 4/5 – 2/5 is: 2/15

          6. How to Divide Fractions

          To divide one fraction by another, you first need to turn the dividing fraction into a reciprocal by switching the denominator and the numerator.

          Taking the example of 1/2 ÷ 1/5, the latter fraction as a reciprocal is 5/1.

          Now multiply your first fraction by your reciprocal:

          To do this, multiply both your numerators and denominators:

          The answer to the equation 1/2 ÷ 1/5 is: 5/2 or 2½

          7. How to Multiply Fractions

          The process of how to work out fractions as multiplications of each other is a simple one:

          • Multiply your numerators
          • Multiply your denominators
          • Write your new numerator over your new denominator

          Using an example equation of 1/2 x 1/6:

          The answer to 1/2 x 1/6 is: 1/12

          8. How to Simplify a Fraction

          To simplify a fraction is to reduce it to its most basic form. Essentially, to find the lowest equivalent fraction possible.

          First, find the greatest common factor. This is the highest whole number by which both numerator and denominator are divisible.

          To do this, write down all the factors for both parts of your fraction, as shown below using the example of 32/48:

          • Factors of 32: 1, 2, 4, 8, 16, 32
          • Factors of 48: 1, 2, 3, 4, 8, 12, 16, 24, 48

          The greatest common factor here is: 16

          Now divide both numerator and denominator by this number to find your simplified fraction:

          Therefore, 32/48 simplified is: 2/3

          When completing any form of fractional equation, always simplify your answer to the lowest possible form.

          9. How to Work out Fractions of Quantities

          When presented with a quantity and asked to work out a fractional portion, simply divide the given amount by the fraction’s denominator, then multiply this figure by the numerator.

          You have 55 sweets, you want to give your neighbor two-fifths of them to take home. How many sweets would she take?

          Divide the given amount by the fraction’s denominator: 55 ÷ 5 = 11

          Multiply this figure by the numerator: 11 x 2 = 22

          Therefore, the correct answer is: 22 sweets

          10. How to Determine Equivalent Fractions

          To determine if one fraction is equivalent to another, either multiply or divide both parts of one fraction by the same whole number.

          If your answers are also both whole numbers, then the fraction keeps its value and is equivalent.

          To work out if 12/15 is equivalent to 4/5, divide both 12 and 15 by a whole number:

          Since you do not have a whole figure as your answer here, move on to the next primary number:

          This shows that 12/15 and 4/5 are equivalent fractions.

          You can also do this in reverse, multiplying both parts of the lower fraction:

          Essentially, if one fraction is a simplified version of another, then they are equivalent.


          Part 4. Adding fractions

          To add fractions, we again need to find a common denominator. Let's look at the following example.

          We need to add 2/7 and 3/9. The common denominator is 7 times 9 = 63. The next step would be to replace each fraction's own denominator with the common one.

          For the first fraction, 63 divided by 7 = 9 and 9 times 2 = 18. The result is 18/63. For the second one, 63 divided by 9 = 7 and 7 times 3 = 21. The result is 21/63.

          Next, we add the numerators. 18 plus 21 = 39, which leaves us with the sum of 39/63.

          As a useful habit, always check if the resulting fraction can be further simplified.

          We know that 39 is evenly divisible by 3. 63 is also evenly divisible by 3. Since both numerator and denominator are divided by the same number, the fraction will remain the same. 39 divided by 3 = 13 and 63 divided by 3 = 21. Our final result is 13/21.

          Fraction addition calculation 2/7 + 3/9 = 39/63 = 13/21

          What if we need to add mixed numbers? To add mixed numbers, we first add the whole numbers together and then the fractions.

          For example, to add 1 and a half to 2 and a half, add 1 and 2 = 3, then add 1/2 and 1/2 = 1. Finally, add 3 and 1 = 4. Let's have some practice and remember how to simplify results.

          What is the result of 4/6 + 2/9?


          4.4: Multiply and Divide Fractions (Part 2)

          A fraction is simply a part of a whole thing. The example below is of a circle divided into four pieces. Each segment represents 1/4 of the circle.

          In each of the circles below, the same area is represented, but the area is divided into different numbers of equal parts.

          This diagram demonstrates that the fractions 1/2, 2/4 and 4/8 represent the same quantity.

          The fractions 1/3, 2/6 and 3/9 are equivalent. You can determine fractions of equivalent value by multiplying both the numerator and the denominator of the fraction by the same number.

          1 x7=7 thus7 =1
          3 x 7 21 21 3

          A similar rule holds when dividing the numerators and denominators of fractions. This is necessary to reduce fractions to their lowest form.

          5 divided by 5=1
          15 divided by 5 3

          When a fraction has a larger numerator than denominator then the fraction is larger than one. The diagram below illustrates an example of improper fractions.

          Adding and Subtracting Fractions

          Whenever you are adding or subtracting fractions, you have to ensure that the denominators of the fractions are the same. For example:

          By multiplying both the denominator and the numerator of 1/2 by 4, you will be able to add the fractions together. 1/2 becomes 4/8.

          When you are adding and subtracting fractions, you also maintain the same denominator, and add or subtract the numerator.

          3 -1 =2 =1
          4 4 4 2
          3 +12 =15 =5
          18 18 18 6
          5 -3 =2 =1
          10 10 10 5
          7 +5 =12 =14 =11
          8 8 8 8 2

          If you are having difficulty with the subject matter, there is always a great deal of support available to you from other members through the Forum. Simply do a search for the material you need help with (math, algebra, integers, etc.) Members are very supportive and helpful. Don't hesitate to post any questions, or provide answers to others seeking assistance.

          Adding and Subtracting Fractions Practice Tests

          Work through the following tests to make sure you are comfortable with the material.


          When multiplying fractions, there is no need to find a common denominator. Simply multiply the two top numbers and then multiply the two bottom numbers. Multiplying two fractions together (other than improper) will result in a fraction that is smaller than the original numbers.

          4 x3 =12 =3
          5 4 20 5
          1 x1 =1
          2 5 10
          3 x7 =21 =7
          4 18 72 24
          3 x4 =12 =11
          2 5 10 5

          Division with fractions is very similar to multiplying with fractions.

          12 divided by 12 = 1 12 goes into 12 once
          12 divided by 6 = 2 6 goes into 12 twice
          12 divided by 4 = 3 4 goes into 12 three times
          12 divided by 3 = 4 3 goes into 12 four times
          12 divided by 2 = 6 2 goes into 12 six times
          12 divided by 1 = 12 1 goes into 12 twelve times
          12 divided by 1/2 = 24 1/2 goes into 12 twenty four times

          This is logical when you think about the statement on the right. Whenever you are dividing by a fraction you have to multiply one fraction by the reciprocal of the other. That is, when you divide one fraction by another, you have to multiply one fraction by the inverse of the other. For example:

          1 divided by6 =1 x7 =7
          2 7 2 6 12

          3 divided by4 =3 x5 =15
          4 5 4 4 16

          13 divided by4 =7 x5 =35 = 23
          4 5 4 4 16 16

          Whenever dividing mixed fractions (1 1/2, 2 3/4 etc) you must use improper fractions (3/2, 11/4 etc).

          Multiplying and Dividing Fractions Practice Tests

          Work through the following tests to make sure you are comfortable with the material.


          Reducing Fractions

          Reducing fractions is just the opposite of converting fractions. What we do is factor out the multiples common to both the numerator and denominator if we can. We factor out (divide out) the greatest common factor of both the numerator and denominator. This will result in a equivalent fraction of lowest terms or what we call a reduced fraction.

          So here is how to reduce fractions.
          As shown below, the fraction 2/4 is being divided by the greatest common factor of 2 and 4 which is 2. The resulting quotient is the fraction 1/2 which is in the lowest terms since its numerator and denominator is divisible by itself and 1.

          As seen above, all we do is divide straight across. Begin by finding all the factors of 2 and 4. If we remember our greatest common factor lesson we know that all the factors of 2 are 1, 2 and the factors of 4 are 1, 2 and 4. The greatest common factor shared by both 2 and 4 is 2. So we divided 2/4 by 2/2 and reduced it to 1/2. We did the same thing to 4/8. We find the greatest common factor of 4 and 8 which is 4 and we divided 4 out of the fraction and reduced it to 1/2.

          Below are several problems you can practice on.

          Remember, the most important thing is to find a factor that will divide BOTH THE NUMERATOR AND DENOMINATOR if possible. The reason we say “if possible” is because we could run into fractions with prime numbers in the numerator or denominator that can’t be factored any further such as those shown below.


          How to Use the Fractions Calculator

          You can use the fractions calculator without remembering all those arithmetic functions!

          In the box for Fraction One, enter the numerator for the first fraction. In the box for Fraction Two, enter the numerator and denominator of the second fraction.

          In the Operation menu, choose which function to perform – do you want to add (+), subtract (-), multiply (*) or divide (÷) the fractions? Choose your preference from the menu. Then hit the Perform Fractional Math button.

          The automatically simplified answer will be listed in the Result box.



          There are 5 similar strips. The shaded portion of the first strip is more than the shaded portion of the second strip. Therefore, 1 > 1/2.

          The shaded portion of the second ship is more than the shaded portion of the third strip. Therefore, 1/2 > 1/3

          Similarly, 1/3 > 1/4, 1/4 > 1/5, 1/5 > 1/6, 1/6 > 1/7, 1/7 > 1/8, 1/8 > 1/9, 1/9 > 1/10

          Let us consider some other example.

          Similarly, from the above example we can say, 2/3 > 2/4, 2/4 > 2/5, 2/5 > 2/6

          From the above two examples we conclude that, if two fractions have same numerator the fraction having smaller denominator is greater than the other.


          From the shaded portion in the circles above, it is clear that 2/4 > 1/4 and 3/4 > 2/4.

          From the shaded portions in the above figures, we can say that
          2/5 (numerator) 3
          So, 7/5 > 3/5

          Example 2. Which is smaller, 4/11 or 4/13?

          Solution. Both the fractions have same numerator but (denominator) 11 < (denominator) 13
          So, 4/13 < 4/11


          How to multiply fractions?

          Even though our fraction calculator allows you perform this calculation let&rsquos answer on how to multiply fractions .

          ■ How to multiply a fraction by another fraction?

          In order to multiply a fraction by another you have to multiply their numerators and multiply their denominators as shown in this example:

          ■ How to multiply a fraction by a whole number?

          In case you have to multiply a whole number with a fraction you need to consider that the integer number can be written as a fraction having the denominator 1, and then simply multiply two fractions by following the rule explained above. For instance:

          ■ How to multiply mixed numbers?

          In order to multiply mixed numbers the best way to do it is by converting the mixed number into a common fraction and then perform the calculation by following the standard multiplication rule. For instance:

          - In case of a multiplication between an integer number and a mixed number:

          - In case of a multiplication between a simple fraction and a mixed number you only have to convert the mixed number into a common fraction and then apply the standard multiplication rule. For instance:


          Teaching Fraction Operations in Your Classroom

          I hope this article offered a useful overview of the main concepts needed for success with fraction operations.

          You can find resources for teaching fractions with all three vehicles in our online store.

          Or level up your inquiry-based math skills by enrolling in an online workshop. These are real-time sessions, conducted by a live facilitator. We offer separate sessions on each vehicle for elementary and for middle school teachers, so you can focus on the techniques and standards that are most important to your students.

          Finally, if you’d like to incorporate this type of learning right away, download our Fractions Essentials Bundle. It has everything you need to get started, from interactive Google Slides activities, to lesson plans, answer keys, and more!

          About the Author

          Jeff Lisciandrello is the founder of Room to Discover and an education consultant specializing in student-centered learning. His 3-Bridges Design for Learning helps schools explore innovative practices within traditional settings. He enjoys helping educators embrace inquiry-based and personalized approaches to instruction. You can connect with him via Twitter @EdTechJeff


          Watch the video: Multiplying mixed numbers and fractions lesson (December 2021).