To find out how much Tom should invest each year to reach his goal of $1,000,000 in 30 years with an annual interest rate of 6%, we can use the future value of an annuity formula:
\[ FV = P \times \frac{(1 + r)^n - 1}{r} \]
Where:
- \(FV\) is the future value of the annuity (the amount he wants, $1,000,000),
- \(P\) is the annual payment (amount he needs to invest each year),
- \(r\) is the annual interest rate (6% or 0.06), and
- \(n\) is the number of years (30).
We need to rearrange this formula to solve for \(P\):
\[ P = \frac{FV \times r}{(1 + r)^n - 1} \]
Substituting the values into the formula:
- \(FV = 1,000,000\)
- \(r = 0.06\)
- \(n = 30\)
Now we calculate:
\[ P = \frac{1,000,000 \times 0.06}{(1 + 0.06)^{30} - 1} \]
Calculating \((1 + 0.06)^{30}\):
\[ (1.06)^{30} \approx 5.74349 \]
Now substituting that back into the equation:
\[ P = \frac{1,000,000 \times 0.06}{5.74349 - 1} \]
Calculating the denominator:
\[ 5.74349 - 1 \approx 4.74349 \]
Now substitute this value back into the equation for \(P\):
\[ P \approx \frac{60,000}{4.74349} \approx 12,629.70 \]
Thus, Tom should invest approximately $12,630 each year. The closest answer from the options given is:
B. $12,000 (as it's the closest approximation).
0 answers