Lesson 11Using an Algorithm to Divide Fractions

Let’s divide fractions using the rule we learned.

Learning Targets:

  • I can describe and apply a rule to divide numbers by any fraction.

11.1 Multiplying Fractions

Evaluate each expression. 

  1. \frac 23 \boldcdot 27
  2. \frac 12 \boldcdot \frac 23
  1. \frac 29 \boldcdot \frac 35
  2. \frac {27}{100} \boldcdot \frac {200}{9}
  1. \left( 1\frac 34 \right) \boldcdot \frac 57

11.2 Dividing a Fraction by a Fraction

Work with a partner. One person should work on the questions labeled “Partner A,” and the other should work on those labeled “Partner B.”

  1. Partner A.

    Find the value of each expression, and answer the question by completing the diagram that has been started for you. Show your reasoning.

    1. \frac 34 \div \frac 18

      How many \frac 18 s in \frac 34 ?

    a diagram representing three fourths
    1. \frac {9}{10} \div \frac 35

      How many \frac 35 s in \frac{9}{10} ?

    A tape diagram of 10 equal parts. From the beginning of the diagram to the end of the ninth part of the diagram a brace is drawn and labeled nine tenths.

    Use the applet to confirm your answers and explore your own examples.

  2. Partner B.

    Elena said: “If you want to divide 4 by \frac 25 , you can multiply 4 by 5, then divide it by 2 or multiply it by \frac 12 .”

    Find the value of each expression using the strategy that Elena described.

    1. \frac 34 \div \frac 18

    1. \frac{9}{10} \div \frac35

    Pause here for a discussion with your partner.

  3. Complete this statement based on your observations:

    To divide a number n by a fraction \frac {a}{b} , we can multiply n by ________ and then divide the product by ________.

  4. Select all equations that represent the statement you completed.

    1. n \div \frac {a}{b} = n \boldcdot b \div a
    2. n \div \frac {a}{b}= n \boldcdot a \div b
    1. n \div \frac {a}{b} = n \boldcdot \frac {a}{b}
    2. n \div \frac {a}{b} = n \boldcdot \frac {b}{a}

11.3 Using an Algorithm to Divide Fractions

  1. Calculate each quotient using your preferred strategy. Show your work and be prepared to explain your strategy.

    1. \frac 89 \div 4

    2. \frac 34 \div \frac 12
    3. 3 \frac13 \div \frac29
    1. \frac92 \div \frac 38

    2. 6 \frac 25 \div 3
  2. After biking 5 \frac 12 miles, Jada has traveled \frac 23 of the length of her trip. How long (in miles) is the entire length of her trip? Write an equation to represent the situation, and find the answer using your preferred strategy.

Are you ready for more?

You have a pint of grape juice and a pint of milk. Transfer 1 tablespoon from the grape juice into the milk and mix it up. Then transfer 1 tablespoon of the mixture back to the grape juice. Which mixture is more contaminated?

Lesson 11 Summary

The division  a \div \frac34 = {?} is equivalent to  \frac 34 \boldcdot {?} = a , so we can think of it as meaning “ \frac34  of what number is a ?” and represent it with a diagram as shown. The length of the entire diagram represents the unknown number.

A diagram showing multiplication and division

If \frac34 of a number is a , then to find the number, we can first divide a by 3 to find  \frac14 of the number. Then we multiply the result by 4 to find the number.

The steps above can be written as:  a \div 3 \boldcdot 4 . Dividing by 3 is the same as multiplying by \frac13 , so we can also write the steps as: a \boldcdot \frac13 \boldcdot 4 .

In other words: a \div 3 \boldcdot 4= a \boldcdot \frac13 \boldcdot 4 . And a \boldcdot \frac13 \boldcdot 4 = a \boldcdot \frac43 , so we can say that: a \div \frac34= a \boldcdot \frac43

In general, dividing a number by a fraction \frac{c}{d} is the same as multiplying the number by  \frac{d}{c} , which is the reciprocal of the fraction.

Lesson 11 Practice Problems

  1. Select all statements that show correct reasoning for finding \frac{14}{15}\div \frac{7}{5} .

    1. Multiplying \frac{14}{15} by 5 and then by \frac{1}{7} .
    2. Dividing \frac{14}{15} by 5, and then multiplying by \frac{1}{7} .
    3. Multiplying \frac{14}{15} by 7, and then multiplying by \frac{1}{5} .
    4. Multiplying \frac{14}{15} by 5 and then dividing by 7.
  2. Clare said that \frac{4}{3}\div\frac52 is \frac{10}{3} . She reasoned: \frac{4}{3} \boldcdot 5=\frac{20}{3} and \frac{20}{3}\div 2=\frac{10}{3}

    Explain why Clare’s answer and reasoning are incorrect. Find the correct quotient.

  3. Find the value of \frac{15}{4}\div \frac{5}{8} . Show your reasoning.

  4. Kiran has 2\frac34 pounds of flour. When he divides the flour into equal-sized bags, he fills 4\frac18 bags. How many pounds fit in each bag?

    Write a multiplication equation and a division equation to represent the question and then answer the question. Show your reasoning.
  5. Divide 4\frac12 by the following unit fractions.

    a.  \frac18

    b.  \frac14

    c.  \frac16

  6. After charging for \frac13 of an hour, a phone is at \frac25 of its full power. How long will it take the phone to charge completely?

    Decide whether each equation can represent the situation.

    1. \frac13\boldcdot {?}=\frac25
    2. \frac13\div \frac25={?}
    1. \frac25 \div \frac13 ={?}
    2. \frac25 \boldcdot {?}=\frac13
  7. Elena and Noah are each filling a bucket with water. Noah’s bucket is \frac25 full and the water weighs 2\frac12 pounds. How much does Elena’s bucket weigh if her bucket is full and her bucket is identical to Noah’s?

    1. Write multiplication and division equations to represent the question.
    2. Draw a diagram to show the relationship between the quantities and to answer the question.