Barnett Bird
01/06/2024 · Junior High School
How much would you need to deposit in an account now in order to have \( \$ 1000 \) in the account in 10 years? Assume the account earns \( 8 \% \) interest compounded monthly.
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
To find out how much you need to deposit now in order to have \( \$1000 \) in 10 years with an interest rate of \( 8\% \) compounded monthly, we can use the formula for the present value of a future amount:
\[
PV = \frac{FV}{(1 + r/n)^{nt}}
\]
Where:
- \( PV \) is the present value (the amount to deposit now),
- \( FV \) is the future value (the amount you want in the future, which is \( \$1000 \)),
- \( r \) is the annual interest rate (as a decimal, so \( 0.08 \)),
- \( n \) is the number of times the interest is compounded per year (monthly compounding means \( n = 12 \)),
- \( t \) is the number of years the money is invested (which is \( 10 \) years).
Now, we can plug in the values:
- \( FV = 1000 \)
- \( r = 0.08 \)
- \( n = 12 \)
- \( t = 10 \)
Now, we calculate \( PV \):
\[
PV = \frac{1000}{(1 + \frac{0.08}{12})^{12 \times 10}}
\]
First, calculate \( \frac{0.08}{12} \):
\[
\frac{0.08}{12} = 0.00666667
\]
Now, calculate \( 1 + 0.00666667 \):
\[
1 + 0.00666667 = 1.00666667
\]
Next, calculate \( 12 \times 10 = 120 \), and then raise \( 1.00666667 \) to the power of \( 120 \):
\[
(1.00666667)^{120} \approx 2.21964
\]
Now, substitute this back into the present value formula:
\[
PV = \frac{1000}{2.21964} \approx 450.50
\]
Thus, the amount you need to deposit now in order to have \( \$1000 \) in 10 years is approximately \( \boxed{450.50} \).
Quick Answer
You need to deposit approximately \( \$450.50 \) now to have \( \$1000 \) in 10 years with an 8% annual interest rate compounded monthly.
Answered by UpStudy AI and reviewed by a Professional Tutor
UpStudy ThothAI
Self-Developed and Ever-Improving
Thoth AI product is constantly being upgraded and optimized.
Covers All Major Subjects
Capable of handling homework in math, chemistry, biology, physics, and more.
Instant and Accurate
Provides immediate and precise solutions and guidance.
Try Now
Ask Tutors
Ask AI
10x
Fastest way to Get Answers & Solutions
By text
Enter your question here…
By image
Re-Upload
Submit