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What lump sum must be invested at 12%, compounded monthly, for the investment to grow to [tex]$71,000 in 15 years?

The lump sum $[/tex] _______ invested at 12%, compounded monthly, grows to $71,000 in 15 years.

(Do not round until the final answer. Then round to the nearest cent as needed.)


Sagot :

To determine the lump sum that needs to be invested to grow to [tex]$71,000 in 15 years at an annual interest rate of 12%, compounded monthly, we will follow these steps: ### Step-by-Step Solution: 1. Identify the given values: - Future Value (FV) = $[/tex]71,000
- Annual Interest Rate (r) = 12% = 0.12
- Number of compounding periods per year (m) = 12 (monthly compounding)
- Number of years (t) = 15

2. Calculate the monthly interest rate:
- Monthly interest rate (i) = Annual Interest Rate / Number of compounding periods per year
[tex]\[ i = \frac{0.12}{12} = 0.01 \][/tex]

3. Compute the total number of compounding periods:
- Total Number of Periods (n) = Number of compounding periods per year * Number of years
[tex]\[ n = 12 \times 15 = 180 \][/tex]

4. Use the formula for compound interest to solve for the present value (PV):
The formula for compound interest is:
[tex]\[ FV = PV \times (1 + i)^n \][/tex]
Rearrange the formula to solve for the present value:
[tex]\[ PV = \frac{FV}{(1 + i)^n} \][/tex]
Substitute the given values into the formula:
[tex]\[ PV = \frac{71,000}{(1 + 0.01)^{180}} \][/tex]

5. Perform the calculation:
[tex]\[ PV = \frac{71,000}{(1.01)^{180}} \][/tex]
[tex]\[ PV = \frac{71,000}{6.0} \][/tex]

6. Result:
[tex]\[ PV \approx 11841.62 \][/tex]

So, the lump sum that must be invested at 12% compounded monthly for the investment to grow to [tex]$71,000 in 15 years is $[/tex]11841.62.