Answer The initial investment is P ≈ $ 296004 _ Explanation Given information : Amount available for retirement is $500,000, rate of interest is 3.5% and duration of investment is 15 years Concept Involved: Solving for a variable means getting the variable alone in one side of the equation by undoing whatever operation is done to it. Formula Used: For periodic compounding, after t years, the balance A in an account with principal P, number of times interest applied per time period n and annual interest rate r (in decimal form) is given by the formula: A = P ( 1 + r n ) n t Calculation: Description Steps Substitute the given information in the formula A = P ( 1 + r n ) n t {A=500000, r = 0.035, t = 15, & n = 12 because compounded monthly} 500000 = P ( 1 + 0.035 12 ) 12 ⋅ 15 Use symmetric property of equality which states that if a = b then b = a to rewrite the equation P ( 1 + 0.035 12 ) 12 ⋅ 15 = 500000 Simplifying the left side of the equation P ( 1.00291 6 ¯ ) 180 = 500000 Calculation (Continued): Description Steps Divide by ( 1.00291 6 ¯ ) 180 on both sides P ( 1.00291 6 ¯ ) 180 ( 1.00291 6 ¯ ) 180 = 500000 ( 1.00291 6 ¯ ) 180 Simplify fraction on both sides P = $ 296004 Conclusion: The principal $296004 should be invested at rate of 3.5%, compounded monthly, so that $500,000 will be available for retirement in 15 years

4th Edition

ISBN: 9781337516853

Author: Larson

Publisher: CENGAGE CO

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Question

Chapter 3.5, Problem 14E

To determine

To find : the principal P.

Expert Solution & Answer

Answer to Problem 14E

The initial investment is P≈$296004_

Explanation of Solution

Given information : Amount available for retirement is $500,000, rate of interest is 3.5% and duration of investment is 15 years

Concept Involved:

Solving for a variable means getting the variable alone in one side of the equation by undoing whatever operation is done to it.

Formula Used:

For periodic compounding, after t years, the balance A in an account with principal P, number of times interest applied per time period n and annual interest rate r (in decimal form) is given by the formula: A = P(1+rn)nt

Calculation:

Description	Steps
Substitute the given information in the formula A = P(1+rn)nt {A=500000, r = 0.035, t = 15, & n = 12 because compounded monthly}	500000 =P(1+0.03512)12⋅15
Use symmetric property of equality which states that if a = b then b = a to rewrite the equation	P(1+ 0.035 12)12⋅15=500000
Simplifying the left side of the equation	P(1.002916¯)180=500000

Calculation (Continued):

Description	Steps
Divide by (1.002916¯)180 on both sides	P( 1.00291 6 ¯ )180( 1.00291 6 ¯ )180=500000( 1.00291 6 ¯ )180
Simplify fraction on both sides	P=$296004

Conclusion:

The principal $296004 should be invested at rate of 3.5%, compounded monthly, so that $500,000 will be available for retirement in 15 years

Chapter 3.5, Problem 13E

Chapter 3.5, Problem 15E

Chapter 3 Solutions

EBK PRECALCULUS W/LIMITS

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