The tImpanist is for an orchestra deposits $250 at the beginning of each month in the IRA. The IRA has an annual interest rate of 5.5% compounded monthly find the value of the already in dollars after 12 years answer to the nurse sent
The Correct Answer and Explanation is:
To solve this problem, we need to calculate the future value of a series of monthly deposits into an IRA, using the formula for the future value of an annuity compounded monthly.
Formula:
The formula for the future value of an annuity compounded monthly is: A=P×(1+rn)nt−1rnA = P \times \frac{{(1 + \frac{r}{n})^{nt} – 1}}{{\frac{r}{n}}}A=P×nr(1+nr)nt−1
Where:
- AAA is the future value of the investment/annuity
- PPP is the monthly deposit (250 dollars)
- rrr is the annual interest rate (5.5% or 0.055)
- nnn is the number of times the interest is compounded per year (12, since it’s compounded monthly)
- ttt is the number of years (12 years in this case)
Step-by-Step Calculation:
- Values:
- Monthly deposit P=250P = 250P=250
- Annual interest rate r=0.055r = 0.055r=0.055
- Number of compounding periods per year n=12n = 12n=12
- Number of years t=12t = 12t=12
- Plugging into the formula:
A=250×(1+0.05512)12×12−10.05512A = 250 \times \frac{{(1 + \frac{0.055}{12})^{12 \times 12} – 1}}{{\frac{0.055}{12}}}A=250×120.055(1+120.055)12×12−1
- Simplifying the terms:
- 0.05512=0.00458333\frac{0.055}{12} = 0.00458333120.055=0.00458333
- 1+0.00458333=1.004583331 + 0.00458333 = 1.004583331+0.00458333=1.00458333
- 12×12=14412 \times 12 = 14412×12=144
Now calculate the power: (1.00458333)144=1.938653(1.00458333)^{144} = 1.938653(1.00458333)144=1.938653
- Substitute the result back into the formula:
A=250×1.938653−10.00458333A = 250 \times \frac{{1.938653 – 1}}{{0.00458333}}A=250×0.004583331.938653−1 A=250×0.9386530.00458333A = 250 \times \frac{{0.938653}}{{0.00458333}}A=250×0.004583330.938653 A=250×205.5A = 250 \times 205.5A=250×205.5 A=51,375A = 51,375A=51,375
Final Answer:
The value of the IRA after 12 years is $51,375.
Explanation:
In this case, the IRA grows due to the interest being compounded monthly, meaning every month the interest earned from the previous month is added to the balance, and future interest is calculated on this larger amount. Over the course of 12 years, even with regular monthly contributions of $250, the power of compounding helps to grow the investment significantly. The calculation assumes that the interest rate remains constant at 5.5% annually, and that the deposits are made at the beginning of each month.
