Now that you’ve investigated addition and subtraction with decimals and fractions, let’s take a closer look at multiplication and division. You will start in this section with decimals, and then use a similar model to multiply and divide fractions and mixed numbers in the next section.

Consider the following problem.

• Write an expression that you can use to determine the amount of oil that Rachel started with.

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  2.2 × 2.5

• Anu, one student, wrote the following solution. Describe what she did to solve the problem.
  
  
  

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  Anu multiplied 22 and 25 as though there were no decimal point present. Then, she noticed that each factor has 1 digit to the right of the decimal for a total of 2 digits to the right of the decimal. This information told Anu that the product must also have 2 digits to the right of the decimal, so she knew her product would be 5.50.

• How would you represent 2.2 and 2.5 as improper fractions with denominators of 10?

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  2.2 is 22 over 10 2 10 , which is the same as 22 over 10 22 10 .
  
  2.5 is 25 over 10 5 10 , which is the same as 25 over 10 25 10 .

The interactive below uses blocks to multiply decimals. When the blocks are combined, they will form a rectangle; the area of the rectangle is the product of the two decimals or the answer to Rachel’s problem.

The represents 1 unit, the represents one tenth of a unit (0.1), and the represents one hundredth of a unit (0.01).

• The first decimal is the length of the rectangle, and the second decimal is the width. Represent each decimal by clicking on the appropriate block and moving it under the decimal.
• Once you have represented the decimals with the blocks, use the information from the decimals to create a rectangle. You will use the green blocks to fill in the missing pieces of the rectangle.

• Is the answer the same as what we found earlier in Anu's solution?

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  Yes, it is still 5.5 ounces.

• Adjust the numerators to create and represent two more multiplication problems. Record those problems on a piece of paper.

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  See a sample problem

  

Pause and Reflect

Based on what you saw in the interactive, why do you think that the product has the same number of digits to the right of the decimal as the total number of digits to the right of the decimal in the two factors?

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If only one factor has 1 digit to the right of the decimal (tenths), then the product will have tenths. If both factors have 1 digit to the right of the decimal (tenths), then the product will have hundredths. This pattern geometrically confirms the relationship you saw earlier with Anu’s solution.

Practice

1. Mario makes $8.50 per hour at his family’s pizza restaurant. According to his last paycheck, he worked 22.38 hours. What is Mario’s gross pay, before any withholdings?

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   Need a hint?

   Set up and solve a multiplication problem.
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   Mario's gross pay will be $190.23.
2. There are 32 servings in a 14.5-ounce bottle of salad dressing. How many ounces is one serving?

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   Since this question is asking you to determine a rate, use division.
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3. The Nguyen family drove 75.6 miles to the beach in Galveston. It took them 1.2 hours to complete the drive. What was the Nguyen family’s average speed, in miles per hour?

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   Need a hint?

   Since this question is asking you to determine a rate, use division.
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4. Breanna and Brian are hiking down Guadalupe Peak. Their change in elevation is −225 feet per hour. If they started at an elevation of 7,600 feet, what will their elevation be in 4.5 hours?

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   Determine how far Breanna and Brian have traveled downward, and then determine their new elevation.
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   7,600 − 225(4.5) = 7,600 − 1,012.5 = 6,587.5
   
   In 4.5 hours, Breanna and Brian will be at an elevation of 6,587.5 feet.