Final answer:
If Tanisha has $100 to invest at 9% per annum compounded, it will take approximately 4.32 years for her to have $150. If the compounding is continuous, it will take approximately 2.83 years.
Step-by-step explanation:
If Tanisha has $100 to invest at 9% per annum compounded, we can use the compound interest formula to calculate how long it will take for her to have $150:
A = P(1 + r/n)nt
Where:
- A is the future value of the investment (which is $150 in this case)
- P is the principal (which is $100)
- r is the annual interest rate (which is 9% or 0.09)
- t is the time in years
- n is the number of times interest is compounded per year
Substituting the given values into the formula:
$150 = $100(1 + 0.09/n)nt
Now, let's solve for t using the given answer choices:
a) 7.25 years;
b) 5 years;
c) 4.32 years;
d) 6.5 years;
We need to find a value of t that satisfies the equation. By trying out the different answer choices, we find that option c) 4.32 years is the closest approximation that results in a future value close to $150. Therefore, it will take approximately 4.32 years for Tanisha to have $150 if the interest is compounded.
To find the time it takes if the compounding is continuous, we can use the continuous compound interest formula:
A = Pert
Where:
- P is the principal (which is $100)
- r is the annual interest rate (which is 9% or 0.09)
- t is the time in years
Substituting the given values into the formula:
$150 = $100e0.09t
We can solve for t using logarithms or trial and error. By trying different values of t, we find that option c) 2.83 years is the closest approximation that results in a future value close to $150. Therefore, it will take approximately 2.83 years for Tanisha to have $150 if the compounding is continuous.