Lesson 11

Using an Algorithm to Divide Fractions

Let’s divide fractions using the rule we learned.

11.1: Multiplying Fractions

Evaluate each expression.

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

11.2: Dividing a Fraction by a Fraction

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

  1. Partner A:

    Find the value of each expression by completing the diagram.

    1. \(\frac 34 \div \frac 18\)

      How many \(\frac 18\)s in \(\frac 34\)?

      Fraction bar diagram. 4 equal parts. First 3 parts labeled the fraction 3 over 4. 

    2. \(\frac {9}{10} \div \frac 35\)

      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.

    Partner B:

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

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

    1. \(\frac 34 \div \frac 18\)
    2. \(\frac{9}{10} \div \frac35\)

  2. What do you notice about the diagrams and expressions? Discuss with your partner.

  3. Complete this sentence based on what you noticed:

    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.

    • \(n \div \frac {a}{b} = n \boldcdot b \div a\)
    • \(n \div \frac {a}{b}= n \boldcdot a \div b\)
    • \(n \div \frac {a}{b} = n \boldcdot \frac {a}{b}\)
    • \(n \div \frac {a}{b} = n \boldcdot \frac {b}{a}\)

11.3: Using an Algorithm to Divide Fractions

Calculate each quotient. Show your thinking and be prepared to explain your reasoning.

  1. \(\frac 89 \div 4\)
  2. \(\frac 34 \div \frac 12\)
  3. \(3 \frac13 \div \frac29\)
  4. \(\frac92 \div \frac 38\)
  5. \(6 \frac 25 \div 3\)
  6. 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 then find the answer.


Suppose you have a pint of grape juice and a pint of milk. You pour 1 tablespoon of the grape juice into the milk and mix it up. Then you pour 1 tablespoon of this mixture back into the grape juice. Which liquid is more contaminated?

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.

Fraction bar diagram. 4 equal parts. 3 parts shaded. 

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: \(\displaystyle a \div \frac34= a \boldcdot \frac43\)

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.

Glossary Entries

  • reciprocal

    Dividing 1 by a number gives the reciprocal of that number. For example, the reciprocal of 12 is \(\frac{1}{12}\), and the reciprocal of \(\frac25\) is \(\frac52\).