Final answer:
Rachel's mother has invested $20 monthly at 3.50% interest compounded monthly. The future value on Rachel's 20th birthday, the total investment, and the interest earned are calculated using the future value formula for an annuity, multiplying the monthly amount by the total number of months, and subtracting the total investment from the future value, respectively, all rounded to the nearest cent.
Step-by-step explanation:
When calculating the final amount in a savings account where deposits are made regularly, such as Rachel's mother investing $20 at the beginning of each month, we use the future value formula for an annuity. The formula to calculate the future value of an annuity with monthly compounding interest is:
FV = P * ((1 + r/n)(nt) - 1) / (r/n)
Where:
- FV is the future value of the annuity.
- P is the periodic payment amount.
- r is the annual interest rate (decimal).
- n is the number of times that interest is compounded per year.
- t is the time the money is invested for in years.
In Rachel's case, we have:
- P = $20
- r = 3.50% or 0.035
- n = 12 (monthly compounding)
- t = 20 years
Using these values and calculating without a deposit on the 20th birthday, we get:
FV = 20 * ((1 + 0.035/12)(12*20) - 1) / (0.035/12)
After solving the equation, we would round to the nearest cent to find the exact amount Rachel will have in the account on her 20th birthday. The total investment by Rachel's mother would be the monthly investment amount multiplied by the total number of months, which is:
Total Investment = Monthly Investment * Number of Months
Total Investment = $20 * (12 months * 20 years) = $4,800
The interest earned can be calculated as the difference between the future value and the total investment:
Interest Earned = FV - Total Investment
The exact value requires the future value calculation to be completed first, after which it would be rounded to the nearest cent as well.