Concept explainers
(a)
To find : the time necessary for P dollars to triple when investment compounded continuously.
(a)
Answer to Problem 17E
The completed table for time t (in years) necessary for P dollars to triple when it is invested at an interest rate r compounded continuously is given below
r | ||||||
t (in years) | 54.93 | 27.47 | 18.31 | 13.73 | 10.99 | 9.16 |
Explanation of Solution
Given information : Amount invested is P dollars, annual rate of interest are 2%, 4%, 6%, 8%, 10% &12%compounded continuously
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 continuous 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:
Logarithmic property:
Calculation:
Description | Steps |
For rate of interest | |
Use symmetric property of equality which states that if a = b then b = a to rewrite the equation |
Calculation (Continued):
Description | Steps |
Divide by | |
Simplifying fraction on both sides | |
Take natural logarithm on both sides | |
Apply the logarithmic rule | |
Divide by 0.02 on both sides | |
Simplify fraction on both sides of the equation | |
It would take time of 54.93 years for P dollars to triple when it is invested at interest rate | |
For rate of interest | |
Use symmetric property of equality which states that if a = b then b = a to rewrite the equation | |
Divide by | |
Simplifying fraction on both sides | |
Take natural logarithm on both sides | |
Apply the logarithmic rule | |
Divide by 0.04 on both sides | |
Simplify fraction on both sides of the equation | |
It would take time of 27.47 years for P dollars to triple when it is invested at interest rate | |
For rate of interest | |
Use symmetric property of equality which states that if a = b then b = a to rewrite the equation | |
Divide by | |
Simplifying fraction on both sides | |
Take natural logarithm on both sides | |
Apply the logarithmic rule | |
Divide by 0.06 on both sides | |
Simplify fraction on both sides of the equation | |
It would take time of 18.31 years for P dollars to triple when it is invested at interest rate | |
For rate of interest |
Calculation (Continued):
Description | Steps |
Use symmetric property of equality which states that if a = b then b = a to rewrite the equation | |
Divide by | |
Simplifying fraction on both sides | |
Take natural logarithm on both sides | |
Apply the logarithmic rule | |
Divide by 0.08 on both sides | |
Simplify fraction on both sides of the equation | |
It would take time of 13.73 years for P dollars to triple when it is invested at interest rate | |
For rate of interest | |
Use symmetric property of equality which states that if a = b then b = a to rewrite the equation | |
Divide by | |
Simplifying fraction on both sides | |
Take natural logarithm on both sides | |
Apply the logarithmic rule | |
Divide by 0.1 on both sides | |
Simplify fraction on both sides of the equation | |
It would take time of 10.99 years for P dollars to triple when it is invested at interest rate | |
For rate of interest | |
Use symmetric property of equality which states that if a = b then b = a to rewrite the equation | |
Divide by | |
Simplifying fraction on both sides | |
Take natural logarithm on both sides | |
Apply the logarithmic rule | |
Divide by 0.12 on both sides | |
Simplify fraction on both sides of the equation | |
It would take time of 9.16 years for P dollars to triple when it is invested at interest rate |
Conclusion:
It would take time of 54.93 years for P dollars to triple when it is invested at interest rate
It would take time of 27.47 years for P dollars to triple when it is invested at interest rate
It would take time of 18.31 years for P dollars to triple when it is invested at interest rate
It would take time of 13.73 years for P dollars to triple when it is invested at interest rate
It would take time of 10.99 years for P dollars to triple when it is invested at interest rate
It would take time of 9.16 years for P dollars to triple when it is invested at interest rate
We can notice that as the rate increases, the time taken for the invested amount to triple will decrease.
(b)
To find : the time necessary for P dollars to triple (when investment compounded annually).
(b)
Answer to Problem 17E
The completed table for time t (in years) necessary for P dollars to triple when it is invested at an interest rate r compounded annually is given below:
r | ||||||
t (in years) | 55.48 | 28.01 | 18.85 | 14.27 | 11.53 | 9.69 |
Explanation of Solution
Given information : Amount invested is P dollars, annual rate of interest are 2%, 4%, 6%, 8%, 10% & 12% compounded annually
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:
Logarithmic property:
Calculation:
Description | Steps |
When | |
Use symmetric property of equality which states that if a = b then b = a to rewrite the equation | |
Simplify the expression in the left side of the equation | |
Divide by | |
Simplifying fraction on both sides | |
Take natural logarithm on both sides | |
Apply the logarithmic rule | |
Divide by ln(1.02) on both sides | |
Simplify fraction on both sides of the equation gives time for r = 2% | |
When | |
Use symmetric property of equality which states that if a = b then b = a to rewrite the equation | |
Simplify the expression in the left side of the equation | |
Divide by | |
Simplifying fraction on both sides | |
Take natural logarithm on both sides | |
Apply the logarithmic rule | |
Divide by ln(1.04) on both sides | |
Simplify fraction on both sides of the equation gives time for r = 4% |
Calculation (Continued):
Description | Steps |
When | |
Use symmetric property of equality which states that if a = b then b = a to rewrite the equation | |
Simplify the expression in the left side of the equation | |
Divide by | |
Simplifying fraction on both sides | |
Take natural logarithm on both sides | |
Apply the logarithmic rule | |
Divide by ln(1.06) on both sides | |
Simplify fraction on both sides of the equation gives time for r = 6% | |
When | |
Use symmetric property of equality which states that if a = b then b = a to rewrite the equation | |
Simplify the expression in the left side of the equation | |
Divide by | |
Simplifying fraction on both sides | |
Take natural logarithm on both sides | |
Apply the logarithmic rule | |
Divide by ln(1.08) on both sides | |
Simplify fraction on both sides of the equation gives time for r = 8% |
Calculation (Continued):
Description | Steps |
When | |
Use symmetric property of equality which states that if a = b then b = a to rewrite the equation | |
Simplify the expression in the left side of the equation | |
Divide by | |
Simplifying fraction on both sides | |
Take natural logarithm on both sides | |
Apply the logarithmic rule | |
Divide by ln(1.1) on both sides | |
Simplify fraction on both sides of the equation gives time for r = 10% | |
When | |
Use symmetric property of equality which states that if a = b then b = a to rewrite the equation | |
Simplify the expression in the left side of the equation | |
Divide by | |
Simplifying fraction on both sides | |
Take natural logarithm on both sides | |
Apply the logarithmic rule | |
Divide by ln(1.12) on both sides | |
Simplify fraction on both sides of the equation gives time for r =12% |
Conclusion:
It would take time of 55.48 years for P dollars to triple when it is invested at interest rate
It would take time of 28.01 years for P dollars to triple when it is invested at interest rate
It would take time of 18.85 years for P dollars to triple when it is invested at interest rate
It would take time of 14.27 years for P dollars to triple when it is invested at interest rate
It would take time of 11.53 years for P dollars to triple when it is invested at interest rate
It would take time of 9.69 years for P dollars to triple when it is invested at interest rate
We can notice that as the rate increases, the time taken for the invested amount to triple will decrease.
Chapter 3 Solutions
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