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6.4 Savings & Retirement Plans

6.4 Savings & Retirement Plans
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  • 1. Savings Account

    You deposit $1000 into a savings account. The savings account earns 5%, compounded monthly. Find the balance in the account after each time period. (See Example 1.)

    1. 10 years
    2. 20 years
    • Worked-Out Solution

      Use the formula for compound interest.

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      _  _     _____    __   __    _____    _    _   
     | \| ||  |  ___||  \ \\/ //  |  ___|| | \  / || 
     |  ' ||  | ||__     \   //   | ||__   |  \/  || 
     | .  ||  | ||__     / . \\   | ||__   | .  . || 
     |_|\_||  |_____||  /_//\_\\  |_____|| |_|\/|_|| 
     `-` -`   `-----`   `-`  --`  `-----`  `-`  `-`  
                                                     
    
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  • 2. Savings Account

    Suppose that you deposit $1000 into a savings account. The savings account earns 6%, compounded monthly. Find the balance in the account after each time period. (See Example 1.)

    1. 10 years
    2. 20 years
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     _    _     _____    _    _     ______    _  __  
    | |  | ||  |  ___|| | \  / ||  /_   _//  | |/ // 
    | |/\| ||  | ||__   |  \/  ||   -| ||-   | ' //  
    |  /\  ||  | ||__   | .  . ||   _| ||_   | . \\  
    |_// \_||  |_____|| |_|\/|_||  /_____//  |_|\_\\ 
    `-`   `-`  `-----`  `-`  `-`   `-----`   `-` --` 
                                                     
    
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  • 3. Savings Account

    You deposit $1000 into a savings account. The savings account earns 5.5%, compounded monthly. Your friend deposits $700 into a savings account that earns 7.5%, compounded monthly. Which account has the greater balance after 15 years? (See Example 1.)

    • Worked-Out Solution

      Use the formula for compound interest.

      Your Account:

      Your Friend's Account:

      Your account has a greater balance after 15 years.

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      ____     __   __    _  _    _    _     ____    
     |  _ \\   \ \\/ //  | \| || | || | ||  |  _ \\  
     | |_| ||   \ ` //   |  ' || | || | ||  | |_| || 
     | .  //     | ||    | .  || | \\_/ ||  | .  //  
     |_|\_\\     |_||    |_|\_||  \____//   |_|\_\\  
     `-` --`     `-`'    `-` -`    `---`    `-` --`  
                                                     
    
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  • 4. Savings Account

    Suppose that you deposit $1000 into a savings account. The savings account earns 6.5%, compounded monthly. Your friend deposits $600 into a savings account that earns 8%, compounded monthly. Which account has the greater balance after 40 years? (See Example 1.)

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     _    _    __   __    ____       ___     _    _   
    | \  / ||  \ \\/ //  |  _ \\    / _ \\  | \  / || 
    |  \/  ||   \ ` //   | |_| ||  / //\ \\ |  \/  || 
    | .  . ||    | ||    | .  //  |  ___  ||| .  . || 
    |_|\/|_||    |_||    |_|\_\\  |_||  |_|||_|\/|_|| 
    `-`  `-`     `-`'    `-` --`  `-`   `-` `-`  `-`  
                                                      
    
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  • 5. Purchase of Manhattan

    According to legend, in 1626, Peter Minuit purchased Manhattan Island from Native Americans for $24 worth of trade goods. Suppose the $24 had been deposited into a savings account earning 7%, compounded annually. How much would be in the account in 2014? (See Example 2.)

    • Worked-Out Solution

      From 1626 to 2014 is 388 years. Use the formula for compound interest.

      So, about $6 trillion would be in the account in 2014.

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      _____      ___     ______     _____    _    _   
     / ____||   / _ \\  |      \\  |  ___|| | |  | || 
    / //---`'  / //\ \\ |  --  //  | ||__   | |/\| || 
    \ \\___   |  ___  |||  --  \\  | ||__   |  /\  || 
     \_____|| |_||  |_|||______//  |_____|| |_// \_|| 
      `----`  `-`   `-` `------`   `-----`  `-`   `-` 
                                                      
    
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  • 6. Gift for the Future

    You deposit $3000 into a savings account that earns 5%, compounded annually, for future generations of your family. How much will be in the account after 200 years? (See Example 2.)

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      ______    _____    ______    __   __    ____    
     /_   _//  |  ___|| |      \\  \ \\/ //  |  _ \\  
     `-| |,-   | ||__   |  --  //   \ ` //   | |_| || 
       | ||    | ||__   |  --  \\    | ||    | .  //  
       |_||    |_____|| |______//    |_||    |_|\_\\  
       `-`'    `-----`  `------`     `-`'    `-` --`  
                                                      
    
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  • 7. Investment by an Ancestor

    Suppose that 350 years ago, 1 of your ancestors deposited $1 into a savings account earning 6%, compounded annually. How much would be in the savings account today? (See Example 2.)

    • Worked-Out Solution

      Use the formula for compound interest.

      How is that possible? The point is that 6% annual interest is a high rate, especially during the past 350 years. During the first 250 of those years, the rate of inflation was low.

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      ______     ___      _  _     _____    ______   
     /_   _//   / _ \\   | \| ||  |  ___|| |      \\ 
       | ||    | / \ ||  |  ' ||  | ||__   |  --  // 
      _| ||    | \_/ ||  | .  ||  | ||__   |  --  \\ 
     /__//      \___//   |_|\_||  |_____|| |______// 
     `--`       `---`    `-` -`   `-----`  `------`  
                                                     
    
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  • 8. Compounding a Penny

    Suppose that 500 years ago, the equivalent of 1 penny had been deposited into a savings account earning 8%, compounded annually. How much would be in the savings account today? (See Example 2.)

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