Recall that you can write a fraction greater than 1 as either a mixed number or an improper fraction. For example, the model below shows how 3 1 over 4 1 4 is equivalent to 13 over 4 13 4 .

Consider the problem below.

Click the image below to open an interactive to investigate using fraction circles to solve this pizza problem. The interactive will open in a new web browser tab or window. In the interactive, use the sliders to adjust the number of pizzas (including the numerator and denominator of the fraction) and the number of people sharing the pizzas. Use the check boxes to reveal solution steps in both using the model and the related expressions. Use the interactive to answer the questions that follow.

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Need additional directions?

Use what you noticed in the interactive to answer the following questions.

At the next student council meeting, there were 1 3 over 4 3 4 pizzas left. 5 student council officers agreed to share the pizzas. Marla wrote the division problem 13 over 4 3 4 ÷ 5 to represent the solution, and created the diagram below to show two solution methods.

Source: Anthony Or. Education Bureau, Hong Kong, Fraction Division2: p/q ÷ b (Partitive), Geogebratube

• In this diagram, each pizza was broken into fourths. Why do you think that is the case?

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  Check Your Answer

  The mixed number 13 over 4 3 4 includes a denominator of 4.

• How many fourths of a pizza remain after the meeting? How can you tell?

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  Check Your Answer

  7 fourths, or 7 over 4 7 4 of a pizza remain, which is 13 over 4 3 4 written as an improper fraction.

• How many parts is each fourth divided into? Why do you think that is the case?

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  Check Your Answer

  5 parts, because 5 people are sharing the pizzas.

• How does the model relate to the solution obtained by multiplying 7 over 4 7 4 by the reciprocal of 5?

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  Check Your Answer

  13 over 4 3 4 pizzas are separated into 7 over 4 7 4 . Then, each fourth is divided into 5 parts. Each of these parts is 1 over 20 1 20 of a whole pizza, and all of the parts are distributed evenly among the 5 student council officers. Each officer receives 7 parts, or 7 over 20 7 20 , of a whole pizza.

Pause and Reflect

In this section, you used models to divide mixed numbers. How can you use the algorithm, or procedure, that you developed for dividing fractions to divide mixed numbers or whole numbers?

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Check Your Answer

If you convert the mixed numbers or whole numbers to fractions (either as improper fractions or as a fraction with a denominator of 1), then you can multiply the dividend by the reciprocal of the divisor.

Practice

1. What is the quotient of 1 2 over 3 2 3 and 4 1 over 6 1 6 ?

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   Need a hint?

   Rewrite each mixed number as an improper fraction. Then, multiply the dividend by the reciprocal of the divisor.
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   Check Your Answer

   12 over 3 2 3 ÷ 4 1 over 6 1 6 = 5 over 3 5 3 ÷ 25 over 6 25 6 = 5 over 3 5 3 × 6 over 25 6 25 = 30 over 75 30 75 = 30 ÷ 15 over 75 ÷ 15 30 ÷ 15 75 ÷ 15 = 2 over 5 2 5
2. Jacob has 8 1 over 2 1 2 cups of milk. His favorite ice cream recipe calls for 13 over 4 3 4 cups of milk to make one quart of ice cream. How many quarts of ice cream can Jacob make?

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   Need a hint?

   Divide 81 over 2 1 2 by 13 over 4 3 4.
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   Check Your Answer

   81 over 2 1 2 ÷ 13 over 4 3 4 = 17 over 2 17 2 ÷ 7 over 4 7 4 = 17 over 2 17 2 × 4 over 7 4 7 = 68 over 14 68 14 = 68 ÷ 2 over 14 ÷ 2 68 ÷ 2 14 ÷ 2 = 34 over 7 34 7 = 4 6 over 7 6 7
   
   Jacob can make 4 6 over 7 6 7 quarts of ice cream.
3. A rectangular room is carpeted and has an area of 204 over 5 4 5 square yards. If the length of the room is 41 over 3 1 3 yards, what is the width of the room?

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   Need a hint?

   To calculate the area of a rectangle, use the formula A = lw. If you know the area and one dimension, divide to determine the missing dimension.
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   Check Your Answer

   204 over 5 4 5 ÷ 41 over 3 1 3 = 104 over 5 104 5 ÷ 13 over 3 13 3 = 104 over 5 104 5 × 3 over 13 3 13 = 312 over 65 312 65 = 312 ÷ 13 over 65 ÷ 13 312 ÷ 13 65 ÷ 13 = 24 over 5 24 5 = 44 over 5 4 5
   
   The width of the room is 44 over 5 4 5 yards.