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Recently, More Money 4U offered an annuity that pays 5.1% compounded monthly. If $1,718 is deposited into this annuity every​ month, how much is in the account after 8 ​years? How much of this is​ interest?

Sagot :

semsee45

Answer:

Account balance = $203,131.31

Interest = $38,203.31

Step-by-step explanation:

To determine how much is in the account after 8 years, we can use the Future Value of Annuity formula:

[tex]FV=\dfrac{PMT\left[\left(1+\dfrac{r}{n}\right)^{nt}-1\right]}{\dfrac{r}{n}}[/tex]

where:

  • FV = Future value
  • PMT = Regular deposit amount
  • r = Annual interest rate (decimal form)
  • n = number of times interest is compounded per year
  • t = time in years

In this case:

  • PMT = $1,718
  • r = 5.1% = 0.051
  • n = 12 (monthly)
  • t = 8 years

Substitute the values into the formula and solve for FV:

[tex]FV=\dfrac{1718\left[\left(1+\dfrac{0.051}{12}\right)^{12 \cdot 8}-1\right]}{\dfrac{0.051}{12}}[/tex]

[tex]FV=\dfrac{1718\left[\left(1.00425\right)^{96}-1\right]}{0.00425}[/tex]

[tex]FV=\$203131.31[/tex]

Therefore, the future value of the annuity after 8 years is $203,131.31.

To calculate how much of the balance is interest, subtract the monthly payment (PMT) multiplied by the number of payments (nt) from the future value (FV):

[tex]\textsf{Interest}=FV-PMT \cdot nt \\\\ \textsf{Interest}=203131.31-1718\cdot 12 \cdot 8 \\\\ \textsf{Interest}=203131.31-164928 \\\\ \textsf{Interest}=\$38203.31[/tex]

Therefore, the interest earned over the 8 years is $38,203.31.

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