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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
The text passage provides an explanation of exponential functions, including growth and decay, and includes several practice exercises with solutions that involve solving for the original amount in exponential equations.
multiple_questions_with_answers
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
• Share
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
user
The text passage provides an explanation of exponential functions, including growth and decay, and includes several practice exercises with solutions that involve solving for the original amount in exponential equations.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS