At what annual interest rate must 137000 be invested so that it will grow to be 475009 in 14 years
This economics question tests your understanding of economic models and analysis. The step-by-step answer below applies the relevant framework and explains the reasoning.
ScanSolve AI Answer
Let's determine the annual interest rate.
Step 1: Identify the formula for Future Value (FV).
The formula to find the future value of a present sum is:
$$FV = PV \times (1 + r)^n$$
Where:
FV = Future Value ($475,009)
PV = Present Value ($137,000)
r = Annual interest rate (what we need to find)
n = Number of years (14)
Step 2: Substitute the given values into the formula.
$$475,009 = 137,000 \times (1 + r)^{14}$$
Step 3: Isolate the term $(1 + r)^{14}$.
Divide both sides by $137,000$:
$$\frac{475,009}{137,000} = (1 + r)^{14}$$
$$3.467218978 \approx (1 + r)^{14}$$
Step 4: Solve for $(1 + r)$ by taking the 14th root of both sides.
$$(1 + r) = (3.467218978)^{\frac{1}{14}}$$
$$(1 + r) \approx 1.09300$$
Step 5: Solve for $r$.
$$r = 1.09300 - 1$$
$$r = 0.09300$$
Step 6: Convert the interest rate to a percentage.
$$r = 0.09300 \times 100\%$$
$$r = 9.30\%$$
Step 7: State the final answer.
The annual interest rate must be approximately 9.30%.
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