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3.1 Sets & Set Diagrams

3.1 Sets & Set Diagrams
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  • 9. Trivia Game Show

    A contestant on a game show must answer two trivia questions correctly to advance to the next qualifying round. Of the 10 contestants, 7 answer the first question correctly and 6 answer the second question correctly. Draw a set diagram that shows the information. (See Example 3.)

    • Worked-Out Solution

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     _____      _____     _  _    __   __    _  _   
    |  __ \\   |  ___||  | \| ||  \ \\/ //  | \| || 
    | |  \ ||  | ||__    |  ' ||   \ ` //   |  ' || 
    | |__/ ||  | ||__    | .  ||    | ||    | .  || 
    |_____//   |_____||  |_|\_||    |_||    |_|\_|| 
     -----`    `-----`   `-` -`     `-`'    `-` -`  
                                                    
    
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  • 10. Trivia Game Show

    A contestant on a game show must answer two trivia questions correctly to advance to the next qualifying round. Of the 10 contestants, 7 answer the first question correctly and 6 answer the second question correctly. How many of the contestants advance to the next qualifying round? (See Example 3.)

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                _____   __    __    _____    _    _   
      ____     |  ___|| \ \\ / //  |  ___|| | |  | || 
     |    \\   | ||__    \ \/ //   | ||__   | |/\| || 
     | [] ||   | ||__     \  //    | ||__   |  /\  || 
     |  __//   |_____||    \//     |_____|| |_// \_|| 
     |_|`-`    `-----`      `      `-----`  `-`   `-` 
     `-`                                              
    
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  • 11. Key Game Show

    A game show contestant has a chance to win a car, a boat, and an all-terrain vehicle (ATV). There are eight keys that start the engine of each prize, two of which start the engine of all three prizes. There is a 50% overlap for any 2 of the sets in the diagram. How many keys start the engine of exactly one prize? (See Example 4.)

    • Worked-Out Solution

      You are given the following information.

      • The red region has 8 keys.
      • The blue region has 8 keys.
      • The green region has 8 keys.
      • The intersection of blue and red has 4 keys.
      • The intersection of blue and green has 4 keys.
      • The intersection of green and red has 4 keys.

      The only way for this to occur is as follows.

      From the completed set diagram, you can see that 6 keys start exactly one prize.

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                _____    _    _      ___      ______  
      ____     |  ___|| | || | ||   / _ \\   /_____// 
     |    \\   | ||__   | || | ||  | / \ ||  `____ `  
     | [] ||   | ||__   | \\_/ ||  | \_/ ||  /___//   
     |  __//   |_____||  \____//    \___//   `__ `    
     |_|`-`    `-----`    `---`     `---`    /_//     
     `-`                                     `-`      
    
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  • 12. Key Game Show

    A game show contestant has a chance to win a car, a boat, and an all-terrain vehicle (ATV). There are eight keys that start the engine of each prize, two of which start the engine of all three prizes. There is a 50% overlap for any 2 of the sets in the diagram. How many keys start the engines of exactly two prizes? (See Example 4.)

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     __   _     _____    _    _     _____   __    __  
    | || | ||  |  ___|| | || | ||  |  ___|| \ \\ / // 
    | '--' ||  | ||__   | || | ||  | ||__    \ \/ //  
    | .--. ||  | ||__   | \\_/ ||  | ||__     \  //   
    |_|| |_||  |_____||  \____//   |_____||    \//    
    `-`  `-`   `-----`    `---`    `-----`      `     
                                                      
    
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  • 13. Key Game Show

    A game show contestant has a chance to win a car, a boat, and an all-terrain vehicle (ATV). There are eight keys that start the engine of each prize, two of which start the engine of all three prizes. There is a 50% overlap for any 2 of the sets in the diagram. How many keys start the engine of the car and the boat? (See Example 4.)

    • Worked-Out Solution

      You are given the following information.

      • The red region has 8 keys.
      • The blue region has 8 keys.
      • The green region has 8 keys.
      • The intersection of blue and red has 4 keys.
      • The intersection of blue and green has 4 keys.
      • The intersection of green and red has 4 keys.

      The only way for this to occur is as follows.

      From the completed set diagram, you can see that 4 keys start the engines of the car and the boat.

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      _____      ___     _____      _____             
     |__  //    / _ \\  |  __ \\   |  ___||   ____    
       / //    | / \ || | |  \ ||  | ||__    |    \\  
      / //__   | \_/ || | |__/ ||  | ||__    | [] ||  
     /_____||   \___//  |_____//   |_____||  |  __//  
     `-----`    `---`    -----`    `-----`   |_|`-`   
                                             `-`      
    
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  • 14. Key Game Show

    A game show contestant has a chance to win a car, a boat, and an all-terrain vehicle (ATV). There are eight keys that start the engine of each prize, two of which start the engine of all three prizes. There is a 50% overlap for any 2 of the sets in the diagram. How many keys start the engine of the car or the boat? (See Example 4.)

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     _____       ___      _  __     ___    __    __  
    |  __ \\    / _ \\   | |/ //   / _ \\  \ \\ / // 
    | |  \ ||  | / \ ||  | ' //   | / \ ||  \ \/ //  
    | |__/ ||  | \_/ ||  | . \\   | \_/ ||   \  //   
    |_____//    \___//   |_|\_\\   \___//     \//    
     -----`     `---`    `-` --`   `---`       `     
                                                     
    
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  • 15. Key Game Show

    A game show contestant has a chance to win a car, a boat, and an all-terrain vehicle (ATV). There are eight keys that start the engine of each prize, two of which start the engine of all three prizes. There is a 50% overlap for any 2 of the sets in the diagram. The contestant begins the game by choosing a key from a basket. How many keys are in the basket? (See Example 4.)

    • Worked-Out Solution

      You are given the following information.

      • The red region has 8 keys.
      • The blue region has 8 keys.
      • The green region has 8 keys.
      • The intersection of blue and red has 4 keys.
      • The intersection of blue and green has 4 keys.
      • The intersection of green and red has 4 keys.

      The only way for this to occur is as follows.

      From the completed set diagram, you can see that there are 14 keys altogether.

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      ____       ___     _    _     _____             
     |  _ \\    / _ \\  | |  | ||  |  ___||   ____    
     | |_| ||  / //\ \\ | |/\| ||  | ||__    |    \\  
     | .  //  |  ___  |||  /\  ||  | ||__    | [] ||  
     |_|\_\\  |_||  |_|||_// \_||  |_____||  |  __//  
     `-` --`  `-`   `-` `-`   `-`  `-----`   |_|`-`   
                                             `-`      
    
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  • 16. Taxes on Items Won

    In the United States, game show contestants must pay taxes on items they win. As a result, contestants often refuse some or all of their prizes. (See Example 4.)

    1. You win a $40,000 car, $15,000 boat, and $5,000 ATV in December. Suppose you will owe taxes in April equal to 25% of the value of your winnings. How much will you owe? What will you do?

    2. You decide to keep all of the prizes, and sell the boat to cover the taxes. Can you be certain this will cover the taxes? What other problems might you encounter?

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      ______   _    _    _____      _____    ______   
     /_   _// | || | || |  __ \\   |  ___|| |      \\ 
     `-| |,-  | || | || | |  \ ||  | ||__   |  --  // 
       | ||   | \\_/ || | |__/ ||  | ||__   |  --  \\ 
       |_||    \____//  |_____//   |_____|| |______// 
       `-`'     `---`    -----`    `-----`  `------`  
                                                      
    
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