You have used what you know about addition, subtraction, multiplication, and division in order to write mathematical expressions from verbal problem situations. In this section, you will use that skill as part of the four-step problem-solving model.

Armando received a new media player for his birthday. His favorite website allows him to download songs for $0.95 each and movies for $9.95 each. He downloaded 20 songs and some movies, earned a 20% discount, and his bill for the transaction was $62.96 without sales tax. How many movies did Armando download? Justify your solution.

Use the problem solving model to solve Armando’s problem. For each section of the problem-solving model, select one of the two options by clicking on the option that you think is best. Click on the first step to begin.

Pause and Reflect

How can writing an expression or equation help you make a plan when solving problems?

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Check Your Answer

Once you have an expression or equation that represents the situation, you can use other strategies such as the order of operations or working backwards to determine a solution to the problem.

Why is it important to always check your answer for reasonableness?

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Check Your Answer

Sometimes, you can do all the calculations correctly, but you chose to use the wrong numbers or the wrong operations. Checking your answer to be sure it makes sense helps you identify these potential errors.

Practice Problems

Use the problem situation below to answer questions 1 – 4.

Madison is building a rectangular pen for her pet goat. The goat has a house that is a square with side lengths of 3 feet. The pen will have dimensions of 91 over 2 1 2 feet by 7.25 feet. What will be the area inside the pen and outside the house?

1. Step 1: Understand the Problem
   Draw a diagram representing the goat pen. Be sure to label the appropriate dimensions.

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   Need a hint?

   What shape is the goat’s house? What shape is the pen?
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   Check Your Answer

   
2. Step 2: Make a Plan
   Write an expression equation that you can use to determine the area enclosed inside the pen, but outside the house.

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   Use the area formulas for a square and a rectangle. Subtract the area of the house from the area of the pen.
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   Check Your Answer

   Area of House = 3 ft × 3 ft
   Area of Pen = 91 over 2 1 2 ft × 7.25 ft
   
   Area of Region = Area of Pen − Area of House
   Area of Region = 91 over 2 1 2 ft × 7.25 ft − 3 ft × 3 ft
3. Step 3: Do the Plan
   Calculate the area of the region inside the pen but outside the house.

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   Need a hint?

   Remember to use the order of operations.
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   Check Your Answer

   Area of Region = 91 over 2 1 2 ft × 7.25 ft − 3 ft × 3 ft
   
   Area of Region = 687 over 8 7 8 ft² − 9 ft²
   
   Area of Region = 597 over 8 7 8 ft²
4. Step 4: Check for Reasonableness
   Use rounding to check your answer for reasonableness.

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   Need a hint?

   Round each dimension to the nearest foot, and use the area formulas to estimate the area.
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   Check Your Answer

   Area of House = 3 ft × 3 ft
   Area of Pen ≈ 10 ft × 7 ft
   Area of Region = Area of Pen − Area of House
   Area of Region ≈ 10 ft × 7 ft − 3 ft × 3 ft
   Area of Region ≈ 70 ft² − 9 ft²
   Area of Region ≈ 61 ft²
   
   The estimate of 61 square feet is close to the calculation of 597 over 8 7 8 square feet, so your answer is reasonable.