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  # Solving Exponential Functions: Finding the Original Amount
  
  ## Algebra Solutions
  
  From Article/Worksheet
  
  Exponential functions tell the stories of explosive change. The two types of exponential functions are exponential growth and exponential decay. Four variables - percent change, time, the amount at the beginning of the time period, and the amount at the end of the time period - play roles in exponential functions. This article focuses on how to find the amount at the beginning of the time period, a.
  
  ### Exponential Growth
  
  Exponential growth: the change that occurs when an original amount is increased by a consistent rate over a period of time
  
  Exponential Growth in Real Life:
  
  • Values of home prices
  • Values of investments
  • Increased membership of a popular social networking site
  
  Here's an exponential growth function:
  
  y = a(1 + b)x
  
  • y: Final amount remaining over a period of time
  • a: The original amount
  • x: Time
  • The growth factor is (1 + b).
  • The variable, b, is percent change in decimal form.
  
  ### Exponential Decay
  
  Exponential decay: the change that occurs when an original amount is reduced by a consistent rate over a period of time
  
  Exponential Decay in Real Life:
  
  Here's an exponential decay function:
  
  y = a(1-b)x
  
  • y: Final amount remaining after the decay over a period of time
  • a: The original amount
  • x: Time
  • The decay factor is (1-b).
  • The variable, b, is percent decrease in decimal form.
  
  ### How to Solve for the Original Amount of an Exponential Function
  
  This function describes the exponential growth of the investment:
  
  120,000 = a(1 +.08)6
  
  • 120,000: Final amount remaining after 6 years
  • .08: Yearly growth rate
  • 6: The number of years for the investment to grow
  • a: The initial amount that your family invested
  
  Hint: Thanks to the symmetric property of equality, 120,000 = a(1 +.08)6 is the same as a(1 +.08)6 = 120,000. (Symmetric property of equality: If 10 + 5 = 15, then 15 = 10 +5.)
  
  If you prefer to rewrite the equation with the constant, 120,000, on the right of the equation, then do so.
  
  a(1 +.08)6 = 120,000
  
  Granted, the equation doesn't look like a linear equation (6a = \$120,000), but it's solvable. Stick with it!
  
  a(1 +.08)6 = 120,000
  
  Be careful: Do not solve this exponential equation by dividing 120,000 by 6. It's a tempting math no-no.
  
  1. Use Order of Operations to simplify.
  
  a(1 +.08)6 = 120,000
  
  a(1.08)6 = 120,000 (Parenthesis)
  
  a(1.586874323) = 120,000 (Exponent)
  
  2. Solve by Dividing
  
  a(1.586874323) = 120,000
  
  a(1.586874323)/(1.586874323) = 120,000/(1.586874323)
  
  1a = 75,620.35523
  
  a = 75,620.35523
  
  The original amount, or the amount that your family should invest, is approximately \$75,620.36.
  
  3. Freeze -you're not done yet. Use order of operations to check your answer.
  
  120,000 = a(1 +.08)6
  
  120,000 = 75,620.35523(1 +.08)6
  
  120,000 = 75,620.35523(1.08)6 (Parenthesis)
  
  120,000 = 75,620.35523(1.586874323) (Exponent)
  
  120,000 = 120,000 (Multiplication)
  
  ### Practice Exercises: Answers and Explanations
  
  1. 84 = a(1+.31)7
  Use Order of Operations to simplify.
  84 = a(1.31)7 (Parenthesis)
  84 = a(6.620626219) (Exponent)
  
  Divide to solve.
  84/6.620626219 = a(6.620626219)/6.620626219
  12.68762157 = 1a
  12.68762157 = a
  
  84 = 12.68762157(1.31)7 (Parenthesis)
  84 = 12.68762157(6.620626219) (Exponent)
  84 = 84 (Multiplication)
  
  2. a(1 -.65)3 = 56
  Use Order of Operations to simplify.
  a(.35)3 = 56 (Parenthesis)
  a(.042875) = 56 (Exponent)
  
  Divide to solve.
  a(.042875)/.042875 = 56/.042875
  a = 1,306.122449
  
  a(1 -.65)3 = 56
  1,306.122449(.35)3 = 56 (Parenthesis)
  1,306.122449(.042875) = 56 (Exponent)
  56 = 56 (Multiply)
  
  3. a(1 + .10)5 = 100,000
  Use Order of Operations to simplify.
  a(1.10)5 = 100,000 (Parenthesis)
  a(1.61051) = 100,000 (Exponent)
  
  Divide to solve.
  a(1.61051)/1.61051 = 100,000/1.61051
  a = 62,092.13231
  
  62,092.13231(1 + .10)5 = 100,000
  62,092.13231(1.10)5 = 100,000 (Parenthesis)
  62,092.13231(1.61051) = 100,000 (Exponent)
  100,000 = 100,000 (Multiply)
  
  4. 8,200 = a(1.20)15
  Use Order of Operations to simplify.
  8,200 = a(1.20)15 (Exponent)
  8,200 = a(15.40702157)
  
  Divide to solve.
  8,200/15.40702157 = a(15.40702157)/15.40702157
  532.2248665 = 1a
  532.2248665 = a
  
  8,200 = 532.2248665(1.20)15
  8,200 = 532.2248665(15.40702157) (Exponent)
  8,200 = 8200 (Well, 8,199.9999...Just a bit of a rounding error.) (Multiply.)
  
  5. a(1 -.33)2 = 1,000
  Use Order of Operations to simplify.
  a(.67)2 = 1,000 (Parenthesis)
  a(.4489) = 1,000 (Exponent)
  
  Divide to solve.
  a(.4489)/.4489 = 1,000/.4489
  1a = 2,227.667632
  a = 2,227.667632
  
  2,227.667632(1 -.33)2 = 1,000
  2,227.667632(.67)2 = 1,000 (Parenthesis)
  2,227.667632(.4489) = 1,000 (Exponent)
  1,000 = 1,000 (Multiply)
  
  6. a(.25)4 = 750
  Use Order of Operations to simplify.
  a(.00390625)= 750 (Exponent)
  
  Divide to solve.
  a(.00390625)/00390625= 750/.00390625
  1a = 192,000
  a = 192,000
  
  192,000(.25)4 = 750
  192,000(.00390625) = 750
  750 = 750
  From Article/Worksheet
  ##### More Tutorials from Jennifer
  Explore Mathematics
explanation: The document is an educational article that explains how to solve for the original amount in exponential functions, specifically focusing on exponential growth and decay. It provides a detailed example and several practice exercises with answers and explanations. The document contains both questions and answers.
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  Send to a Friend via Email
  
  ### Your suggestion is on its way!
  
  An email with a link to:
  
  was emailed to:
  
  Thanks for sharing About.com with others!
  
  Discuss in my forum
  
  # Solving Exponential Functions: Finding the Original Amount
  
  ## Algebra Solutions
  
  From Article/Worksheet
  
  Exponential functions tell the stories of explosive change. The two types of exponential functions are exponential growth and exponential decay. Four variables - percent change, time, the amount at the beginning of the time period, and the amount at the end of the time period - play roles in exponential functions. This article focuses on how to find the amount at the beginning of the time period, a.
  
  ### Exponential Growth
  
  Exponential growth: the change that occurs when an original amount is increased by a consistent rate over a period of time
  
  Exponential Growth in Real Life:
  
  • Values of home prices
  • Values of investments
  • Increased membership of a popular social networking site
  
  Here's an exponential growth function:
  
  y = a(1 + b)x
  
  • y: Final amount remaining over a period of time
  • a: The original amount
  • x: Time
  • The growth factor is (1 + b).
  • The variable, b, is percent change in decimal form.
  
  ### Exponential Decay
  
  Exponential decay: the change that occurs when an original amount is reduced by a consistent rate over a period of time
  
  Exponential Decay in Real Life:
  
  Here's an exponential decay function:
  
  y = a(1-b)x
  
  • y: Final amount remaining after the decay over a period of time
  • a: The original amount
  • x: Time
  • The decay factor is (1-b).
  • The variable, b, is percent decrease in decimal form.
  
  ### How to Solve for the Original Amount of an Exponential Function
  
  This function describes the exponential growth of the investment:
  
  120,000 = a(1 +.08)6
  
  • 120,000: Final amount remaining after 6 years
  • .08: Yearly growth rate
  • 6: The number of years for the investment to grow
  • a: The initial amount that your family invested
  
  Hint: Thanks to the symmetric property of equality, 120,000 = a(1 +.08)6 is the same as a(1 +.08)6 = 120,000. (Symmetric property of equality: If 10 + 5 = 15, then 15 = 10 +5.)
  
  If you prefer to rewrite the equation with the constant, 120,000, on the right of the equation, then do so.
  
  a(1 +.08)6 = 120,000
  
  Granted, the equation doesn't look like a linear equation (6a = \$120,000), but it's solvable. Stick with it!
  
  a(1 +.08)6 = 120,000
  
  Be careful: Do not solve this exponential equation by dividing 120,000 by 6. It's a tempting math no-no.
  
  1. Use Order of Operations to simplify.
  
  a(1 +.08)6 = 120,000
  
  a(1.08)6 = 120,000 (Parenthesis)
  
  a(1.586874323) = 120,000 (Exponent)
  
  2. Solve by Dividing
  
  a(1.586874323) = 120,000
  
  a(1.586874323)/(1.586874323) = 120,000/(1.586874323)
  
  1a = 75,620.35523
  
  a = 75,620.35523
  
  The original amount, or the amount that your family should invest, is approximately \$75,620.36.
  
  3. Freeze -you're not done yet. Use order of operations to check your answer.
  
  120,000 = a(1 +.08)6
  
  120,000 = 75,620.35523(1 +.08)6
  
  120,000 = 75,620.35523(1.08)6 (Parenthesis)
  
  120,000 = 75,620.35523(1.586874323) (Exponent)
  
  120,000 = 120,000 (Multiplication)
  
  ### Practice Exercises: Answers and Explanations
  
  1. 84 = a(1+.31)7
  Use Order of Operations to simplify.
  84 = a(1.31)7 (Parenthesis)
  84 = a(6.620626219) (Exponent)
  
  Divide to solve.
  84/6.620626219 = a(6.620626219)/6.620626219
  12.68762157 = 1a
  12.68762157 = a
  
  84 = 12.68762157(1.31)7 (Parenthesis)
  84 = 12.68762157(6.620626219) (Exponent)
  84 = 84 (Multiplication)
  
  2. a(1 -.65)3 = 56
  Use Order of Operations to simplify.
  a(.35)3 = 56 (Parenthesis)
  a(.042875) = 56 (Exponent)
  
  Divide to solve.
  a(.042875)/.042875 = 56/.042875
  a = 1,306.122449
  
  a(1 -.65)3 = 56
  1,306.122449(.35)3 = 56 (Parenthesis)
  1,306.122449(.042875) = 56 (Exponent)
  56 = 56 (Multiply)
  
  3. a(1 + .10)5 = 100,000
  Use Order of Operations to simplify.
  a(1.10)5 = 100,000 (Parenthesis)
  a(1.61051) = 100,000 (Exponent)
  
  Divide to solve.
  a(1.61051)/1.61051 = 100,000/1.61051
  a = 62,092.13231
  
  62,092.13231(1 + .10)5 = 100,000
  62,092.13231(1.10)5 = 100,000 (Parenthesis)
  62,092.13231(1.61051) = 100,000 (Exponent)
  100,000 = 100,000 (Multiply)
  
  4. 8,200 = a(1.20)15
  Use Order of Operations to simplify.
  8,200 = a(1.20)15 (Exponent)
  8,200 = a(15.40702157)
  
  Divide to solve.
  8,200/15.40702157 = a(15.40702157)/15.40702157
  532.2248665 = 1a
  532.2248665 = a
  
  8,200 = 532.2248665(1.20)15
  8,200 = 532.2248665(15.40702157) (Exponent)
  8,200 = 8200 (Well, 8,199.9999...Just a bit of a rounding error.) (Multiply.)
  
  5. a(1 -.33)2 = 1,000
  Use Order of Operations to simplify.
  a(.67)2 = 1,000 (Parenthesis)
  a(.4489) = 1,000 (Exponent)
  
  Divide to solve.
  a(.4489)/.4489 = 1,000/.4489
  1a = 2,227.667632
  a = 2,227.667632
  
  2,227.667632(1 -.33)2 = 1,000
  2,227.667632(.67)2 = 1,000 (Parenthesis)
  2,227.667632(.4489) = 1,000 (Exponent)
  1,000 = 1,000 (Multiply)
  
  6. a(.25)4 = 750
  Use Order of Operations to simplify.
  a(.00390625)= 750 (Exponent)
  
  Divide to solve.
  a(.00390625)/00390625= 750/.00390625
  1a = 192,000
  a = 192,000
  
  192,000(.25)4 = 750
  192,000(.00390625) = 750
  750 = 750
  From Article/Worksheet
  ##### More Tutorials from Jennifer
  Explore Mathematics
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is an educational article that explains how to solve for the original amount in exponential functions, specifically focusing on exponential growth and decay. It provides a detailed example and several practice exercises with answers and explanations. The document contains both questions and answers.
  
  Label: question-and-answer