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1.3 Using Percent

1.3 Using Percent
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  • 17. Carat Weight of a Diamond Ring

    The graph shows the price ranges for five weights of diamond rings at a jewelry store. The jewelry store had a diamond ring listed at $11,499. What is the carat weight of the diamond ring? Explain your reasoning. (See Example 5 and Example 6.)

    • Worked-Out Solution

      The price range for a 0.30-carat diamond ring is

      $0 through $971

      The price range for a 0.50-carat diamond ring is

      $971 through ($971 + $1055) or $971 through $2026.

      The price range for a 1.00-carat diamond ring is

      $2026 through ($2026 + $4866) or $2026 through $6892.

      The price range for a 2.00-carat diamond ring is

      $6892 through ($6892 + $9200) or $6892 through $16,092.

      Because $11,499 falls in the range $6892 through $16,092, the carat weight of the diamond ring is 2.00 carats.

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     __   _     ______    _____     _____     ____    
    | || | ||  /_   _//  /  ___||  |  ___||  |  _ \\  
    | '--' ||   -| ||-  | // __    | ||__    | |_| || 
    | .--. ||   _| ||_  | \\_\ ||  | ||__    | .  //  
    |_|| |_||  /_____//  \____//   |_____||  |_|\_\\  
    `-`  `-`   `-----`    `---`    `-----`   `-` --`  
                                                      
    
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    Ron Larson (author)7 years ago |
    This type of graph is called a waterfall graph. The numbers on the graph (like $9200) show the increase in price from the previous price.
    0
  • 18. Prices of Diamond Rings

    The graph shows the price ranges for five weights of diamond rings at a jewelry store. What is the price range of a 1.00-carat diamond ring? (See Example 5 and Example 6.)

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      ____     __   __    _____    __   __    _____   
     |  _ \\   \ \\/ //  |__  //   \ \\/ //  /  ___|| 
     | |_| ||   \ ` //     / //     \ ` //  | // __   
     | .  //     | ||     / //__     | ||   | \\_\ || 
     |_|\_\\     |_||    /_____||    |_||    \____//  
     `-` --`     `-`'    `-----`     `-`'     `---`   
                                                      
    
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  • 19. Comparing Prices for Diamond Rings

    The graph shows the price ranges for five weights of diamond rings at a jewelry store. How much more does a 2.00-carat diamond ring cost than a 1.00-carat diamond ring? (See Example 5 and Example 6.)

    • Worked-Out Solution

      The lower-end price for a 1.00-carat diamond ring is

      971 + 1055 = $2026

      The upper-end price for a 1.00-carat diamond ring is

      92026 + 4866 = $6892

      The upper-end price for a 2.00-carat diamond ring is

      6892 + 9200 = $16,092

      So, at the one extreme, a 2.00-carat diamond ring could cost

      16,092 - 2026 = $14,066

      more than a 1.00-carat diamond ring. Or, at the other extreme, the two rings could be about equal in price.

      Comparing jewelry prices is difficult. There are several categories of jewelry stores. Stores like Kay, Jared, and Zales tend to be located in middle-class malls and cater to a typical middle-class consumer. Discount jewelers cater to bargin shoppers. Designer stores like Cartier, Tiffany, and Bvlgari cater to wealthy customers.

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                _____    ______    _    _      _____  
        ___    |  ___|| |      \\ | || | ||   / ___// 
       /   ||  | ||__   |  --  // | || | ||   \___ \\ 
      | [] ||  | ||__   |  --  \\ | \\_/ ||   /    // 
       \__ ||  |_____|| |______//  \____//   /____//  
        -|_||  `-----`  `------`    `---`   `-----`   
         `-`                                          
    
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  • 20. Comparing Prices for Diamond Rings

    The graph shows the price ranges for five weights of diamond rings at a jewelry store. Reword the statement so it uses a percent that is greater than 200%. (See Example 5 and Example 6.)

    A 0.50-carat diamond ring costs $2026, which is about 108.7% more than the cost of a 0.30-carat diamond ring.

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     _    _    _    _     _  _      ___      ______  
    | |  | || | || | ||  | \| ||   / _ \\   /_____// 
    | |/\| || | || | ||  |  ' ||  | / \ ||  `____ `  
    |  /\  || | \\_/ ||  | .  ||  | \_/ ||  /___//   
    |_// \_||  \____//   |_|\_||   \___//   `__ `    
    `-`   `-`   `---`    `-` -`    `---`    /_//     
                                            `-`      
    
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  • 21. Comparing Carat Sizes of Diamond Rings

    Is it true that a 0.50-carat diamond is more than 60% larger than a 0.30-carat diamond? Explain your reasoning. (See Example 5 and Example 6.)

    • Worked-Out Solution

      A 0.50-carat diamond weighs 0.20 carat more than a 0.30-carat diamond.

      So, it is true that a 0.50-carat diamond is more than 60% heavier than a 0.30-carat diamond. On the other hand, the word "larger" can be misleading. It is not true that the 0.50-carat diamond has a diameter that is more that 60% greater. As diamonds increase in carat weight, their diameter roughly increases as the cube root of the weight (see the following comparisons). When comparing widths (what you see), a 0.50-carat diamond has a diameter (5.2 mm) that is only 21% greater than the diameter (4.3 mm) of a 0.30-carat diamond.

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     __   __     ___      _____     _____     ______  
     \ \\/ //   / _ \\   / ____||  |  ___||  /_____// 
      \   //   | / \ || / //---`'  | ||__    `____ `  
      / . \\   | \_/ || \ \\___    | ||__    /___//   
     /_//\_\\   \___//   \_____||  |_____||  `__ `    
     `-`  --`   `---`     `----`   `-----`   /_//     
                                             `-`      
    
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  • 22. Comparing Prices for Diamond Rings

    The graph shows the price ranges for five weights of diamond rings at a jewelry store. Use percent to describe the change in maximum price. (See Example 5 and Example 6.)

    1. from a 0.50-carat diamond ring to a 1.00-carat diamond ring

    2. from a 1.00-carat diamond ring to a 2.00-carat diamond ring

    3. from a 2.00-carat diamond ring to a 3.00-carat diamond ring

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     __   _     _____    __   __     ___      _____   
    | || | ||  |  ___||  \ \\/ //   / _ \\   / ____|| 
    | '--' ||  | ||__     \   //   / //\ \\ / //---`' 
    | .--. ||  | ||__     / . \\  |  ___  ||\ \\___   
    |_|| |_||  |_____||  /_//\_\\ |_||  |_|| \_____|| 
    `-`  `-`   `-----`   `-`  --` `-`   `-`   `----`  
                                                      
    
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  • 23. Comparing Weights of Diamonds

    Use percent to describe the change in carat weight. (See Example 5 and Example 6.)

    1. from a 0.50-carat diamond ring to a 1.00-carat diamond ring

    2. from a 1.00-carat diamond ring to a 2.00-carat diamond ring

    3. from a 2.00-carat diamond ring to a 3.00-carat diamond ring

    • Worked-Out Solution
      1. A 1.00-carat diamond weighs 0.50 carat more than a 0.50-carat diamond.

        Increase in weight divided by weight of 0.50 carat diamond = 0.5 divided by 0.5 = 1

        So, to change from a 0.50-carat diamond to a 1.00-carat diamond is an increase of 100%.

      2. A 2.00-carat diamond weighs 1.00 carat more than a 1.00-carat diamond.

        Increase in weight divided by Weight of 1.00 carat diamond = 1.00 divided by 1.00 = 1.00

        So, to change from a 1.00-carat diamond to a 2.00-carat diamond is an increase of 100%.

      3. A 3.00-carat diamond weighs 1.00 carat more than a 2.00-carat diamond.

        Increase in weight divided by Weight of 2.00 carat diamond = 1.00 divided by 2.00 = 0.50

      So, to change from a 2.00-carat diamond to a 3.00-carat diamond is an increase of 50%.

      Although diamonds are not particularly rare, extracting diamonds is laborious. Mines move many tons of dirt per carat of diamond found. Large gem-quality diamonds are relatively rare. Only about 1 in 1 million diamonds are quality 1-carat stones, only 1 in 5 million are 2-carat, and 1 in 15 million are 3-carat.

      Most of the natural diamonds mined and sold are controlled by a company called De Beers. To maintain the high prices of diamonds, De Beers creates an artificial scarcity by stockpiling diamonds and selling them in small amounts.

      In 2004, astronomer Travis Metcalfe of the Harvard-Smithsonian Center for Astrophysics and colleagues discovered a diamond star that is 10 billion trillion trillion carats! "The cosmic diamond is a chunk of crystallised carbon, 4,000 km across, some 50 light-years from the Earth in the constellation Centaurus. It's the compressed heart of an old star that was once bright like our Sun but has since faded and shrunk." (source: BBC News)

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      ______    ______    _  __    ______    ____    
     /_____//  /_   _//  | |/ //  /_   _//  |  _ \\  
     `____ `    -| ||-   | ' //    -| ||-   | |_| || 
     /___//     _| ||_   | . \\    _| ||_   | .  //  
     `__ `     /_____//  |_|\_\\  /_____//  |_|\_\\  
     /_//      `-----`   `-` --`  `-----`   `-` --`  
     `-`                                             
    
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  • 24. Relationship between Weight and Price of a Diamond

    The graph shows the price ranges for five weights of diamond rings at a jewelry store. Describe the relationship between carat weight and price of a diamond ring. (See Example 5 and Example 6.)

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