### 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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```            ______   _    _    __   __
___    /_   _// | \  / ||  \ \\/ //     ___
/   ||   -| ||-  |  \/  ||   \ ` //     /   ||
| [] ||   _| ||_  | .  . ||    | ||     | [] ||
\__ ||  /_____// |_|\/|_||    |_||      \__ ||
-|_||  `-----`  `-`  `-`     `-`'       -|_||
`-`                                     `-`
```
• 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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```   _____     ___      _____     ______   __   __
/ ___//   / _ \\   /  ___||  /_   _//  \ \\/ //
\___ \\  | / \ || | // __     -| ||-    \   //
/    //  | \_/ || | \\_\ ||   _| ||_    / . \\
/____//    \___//   \____//   /_____//  /_//\_\\
`-----`     `---`     `---`    `-----`   `-`  --`

```
• 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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```  _____    _    _    __   _    _    _      _____
|__  //  | || | || | || | || | || | ||   / ___//
/ //   | || | || | '--' || | || | ||   \___ \\
/ //__  | \\_/ || | .--. || | \\_/ ||   /    //
/_____||  \____//  |_|| |_||  \____//   /____//
`-----`    `---`   `-`  `-`    `---`   `-----`

```
• 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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```  _____    _    _     _  __     ___
/ ____|| | || | ||  | |/ //   / _ \\    ____
/ //---`' | || | ||  | ' //   / //\ \\  |    \\
\ \\___   | \\_/ ||  | . \\  |  ___  || | [] ||
\_____||  \____//   |_|\_\\ |_||  |_|| |  __//
`----`    `---`    `-` --` `-`   `-`  |_|`-`
`-`
```
• 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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```            ______  __    __    _____     ______
____     /_   _// \ \\ / //  |  ___||  /_   _//
|    \\    -| ||-   \ \/ //   | ||__      | ||
| [] ||    _| ||_    \  //    | ||__     _| ||
|  __//   /_____//    \//     |_____||  /__//
|_|`-`    `-----`      `      `-----`   `--`
`-`
```
• 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.)

These comments are not screened before publication. Constructive debate about the information on this page is welcome, but personal attacks are not. Please do not post comments that are commercial in nature or that violate copyright. Comments that we regard as obscene, defamatory, or intended to incite violence will be removed. If you find a comment offensive, you may flag it.

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

```
• 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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```  ____       ___     _____     __   __   _    _
|  _ \\    / _ \\  |  __ \\   \ \\/ // | \  / ||
| |_| ||  / //\ \\ | |  \ ||   \ ` //  |  \/  ||
| .  //  |  ___  ||| |__/ ||    | ||   | .  . ||
|_|\_\\  |_||  |_|||_____//     |_||   |_|\/|_||
`-` --`  `-`   `-`  -----`      `-`'   `-`  `-`

```
• 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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``` __   _      ___      ______    _____