In a previous grade, you worked with division of whole numbers and unit fractions. For example, 4 ÷ 1 over 5 1 5 can be represented using fraction squares.

Breaking 4 whole units into fifths reveals 20 fifths.

Likewise, you can also divide a unit fraction by a whole number. For example, 1 over 5 1 5 ÷ 4 can be represented using a fraction square. This division sentence is asking, “how many 4’s are in 1 over 5 1 5 ?”

From the model, you can see that there is 1 over 20 1 20 of 4 in 1 over 5 1 5 .

In this section, you will explore how to divide fractions by other fractions.

Problem 1:

Maria has 2 over 3 2 3 yard of ribbon. Each bow that she is making requires 1 over 6 1 6 of a yard. How many bows can Maria make?

Use the interactive below to see how to visually represent the division sentence 2 over 3 2 3 ÷ 1 over 6 1 6 .

Problem 2:

Brandon has 2 over 3 2 3 pound of chocolate. He has several containers that will each hold 1 over 2 1 2 pound of chocolate. How many containers can Brandon fill?

Use the interactive below to see how to visually represent the division sentence 2 over 3 2 3 ÷ 1 over 2 1 2 .

Use the two problems above to answer the questions that follow.

• For the first problem, you found that 2 over 3 2 3 ÷ 1 over 6 1 6 = 4. The reciprocal of the divisor, 1 over 6 1 6 , is 6. What is 2 over 3 2 3 × 6?

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  Check Your Answer

  2 over 3 2 3 × 6 = 4

• How does the product of 2 over 3 2 3 × 6 compare to the quotient of 2 over 3 2 3 ÷ 1 over 6 1 6 ?

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  Check Your Answer

  The quotient of 2 over 3 2 3 ÷ 1 over 6 1 6 is 4, which is the same as the product of 2 over 3 2 3 × 6.

• For the second problem, you found that 2 over 3 2 3 ÷ 1 over 2 1 2 = 11 over 3 1 3 = 4 over 3 4 3 . The reciprocal of the divisor, 1 over 2 1 2 , is 2.
  What is 2 over 3 2 3 × 2?

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  Check Your Answer

  2 over 3 2 3 × 2 = 4 over 3 4 3

• How does the product of 2 over 3 2 3 × 2 compare to the quotient of 2 over 3 2 3 ÷ 1 over 2 1 2 ?

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  Check Your Answer

  The quotient of 2 over 3 2 3 ÷ 1 over 2 1 2 is 4 over 3 4 3 , which is the same as the product of 2 over 3 2 3 × 2.

Pause and Reflect

How does the quotient of two fractions compare to the product of the dividend and the reciprocal of the divisor?

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Check Your Answer

The quotient of two fractions is the same as the product of the dividend and the reciprocal of the divisor.

Problem 3:

Alecia has 5 over 6 5 6 gallon of lemonade. She has containers that each hold 2 over 3 2 3 gallon. How many containers can Alecia fill?

Use the interactive below to see how to visually represent the division sentence 5 over 6 5 6 ÷ 2 over 3 2 3 .

Use the problem above to answer the questions that follow.

• In this problem, you found that 5 over 6 5 6 ÷ 2 over 3 2 3 = 1 1 over 4 1 4 = 5 over 4 5 4 . The reciprocal of the divisor, 2 over 3 2 3 , is 3 over 2 3 2 . What is 5 over 6 5 6 × 2 over 3 2 3 ?

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  Check Your Answer

  5 over 6 5 6 × 3 over 2 3 2 = 15 over 12 15 12 = 5 over 4 5 4 = 11 over 4 1 4

• How does the product of 5 over 6 5 6 × 3 over 2 3 2 compare to the quotient of 5 over 6 5 6 ÷ 2 over 3 2 3 ?

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  Check Your Answer

  The quotient of 5 over 6 5 6 ÷ 2 over 3 2 3 is 11 over 4 1 4 , which is the same as the product of 5 over 6 5 6 × 3 over 2 3 2 .

Pause and Reflect

If the numerator of the divisor is not 1, does the relationship between the quotient of two fractions and the product of the dividend and the reciprocal of the divisor change? Why or why not?

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Check Your Answer

No. When you divide a fraction by another fraction, you first have to break the dividend into additional parts (multiply by the denominator of the divisor) and then group the parts (divide by the numerator of the divisor).

Practice

1. What is the quotient of 3 over 4 3 4 and 5 over 12 5 12 ?

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   What is the reciprocal of the divisor, 5 over 12 5 12 ?
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   Check Your Answer

   3 over 4 3 4 ÷ 5 over 12 5 12 = 3 over 4 3 4 × 12 over 5 12 5 = 36 over 20 36 20 = 36 ÷ 4 over 20 ÷ 4 36 ÷ 4 20 ÷ 4 = 9 over 5 9 5 = 14 over 5 4 5
2. A rectangular piece of fabric has an area of 3 over 5 3 5 square yard and is 4 over 5 4 5 yard wide. What is the length of the piece of fabric?

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   The area of a rectangle is the product of the length and width of the rectangle. If you know the area, divide by the known dimension to determine the missing dimension.
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   Check Your Answer

   3 over 5 3 5 ÷ 4 over 5 4 5 = 3 over 5 3 5 × 5 over 4 5 4 = 15 over 20 15 20 = 15 ÷ 5 over 20 ÷ 5 15 ÷ 5 20 ÷ 5 = 3 over 4 3 4 yard
3. The flamingos at the Fort Worth Zoo are fed 5 over 8 5 8 of a barrel of grain each week. How many weeks will a shipment of 10 barrels of grain last?

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   To determine the number of 5 over 8 5 8 that there are in 10, divide 10 by 5 over 8 5 8 .
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   Check Your Answer

   10 ÷ 5 over 8 5 8 = 10 over 1 10 1 × 8 over 5 8 5 = 80 over 5 80 5 = 16 weeks