Final answer:
Using the annuity formula, it is estimated that Eileen will need approximately 6.1 years to accumulate $15,000 by annually investing the $1,980 she would otherwise pay in finance charges at a 9% interest rate.
Step-by-step explanation:
To calculate how long it would take Eileen to save for the outright purchase of a $15,000 car using the $1,980 she pays in finance charges annually, we can apply the formula for the future value of a series of cash flows (annuity). The formula is:
FV = Pmt * (((1 + r)^n - 1) / r), where:
- FV is the future value of the annuity, in this case, the price of the car which is $15,000.
- Pmt is the amount she will save annually, which is the $1,980 in finance charges no longer paid to the credit card company.
- r is the annual interest rate, which is 9% or 0.09 in decimal.
- n is the number of years Eileen needs to save.
Therefore, solving for n, we get:
15000 = 1980 * (((1 + 0.09)^n - 1) / 0.09)
This requires isolating 'n' and solving for it, which typically involves using logarithms. After some algebraic manipulations, the equation may look as complex as needing a financial calculator or software to find the exact number of years, 'n'
Given this, we estimate that Eileen will need approximately 6.1 years to save $15,000 if she invests $1,980 annually at 9% interest.