How much is the present value? How much is the interest after 3 years? What formula will be used to determine the maturity value?
Understand the Problem
The question is asking for the present value, interest after 3 years, and the formula for determining maturity value based on provided financial values and parameters.
Answer
The present value is approximately $23,558.06$, the interest after 3 years is $1,441.94$, and the formula is $F = P(1 + r)^t$.
Answer for screen readers
The present value is approximately $PV \approx 23,558.06$, the interest after 3 years is $I \approx 1,441.94$, and the formula used to determine the maturity value is $F = P(1 + r)^t$.
Steps to Solve
- Identify Given Information
We are given the future value $F = 25,000$, the interest rate $r = 2% = 0.02$, and the time $t = 3$ years.
- Calculate Present Value
The formula for the present value (PV) when compounding interest is:
$$ PV = \frac{F}{(1 + r)^t} $$
Substituting the values we have:
$$ PV = \frac{25000}{(1 + 0.02)^3} $$
- Perform Calculation for Present Value
Calculating $(1 + 0.02)^3$:
$$ (1 + 0.02)^3 = (1.02)^3 \approx 1.061208 $$
Now plug this back into the present value equation:
$$ PV = \frac{25000}{1.061208} \approx 23,558.06 $$
- Calculate Interest After 3 Years
To find interest, we use the formula:
$$ \text{Interest} = F - PV $$
Substituting the values:
$$ \text{Interest} = 25000 - 23,558.06 $$
- Perform Interest Calculation
Calculating the interest:
$$ \text{Interest} \approx 25000 - 23,558.06 \approx 1,441.94 $$
- Determine the Maturity Value Formula
The formula to determine the maturity value based on the formula for present value is:
$$ F = P (1 + r)^t $$
Where $P$ is the present value, $r$ is the interest rate, and $t$ is the time.
The present value is approximately $PV \approx 23,558.06$, the interest after 3 years is $I \approx 1,441.94$, and the formula used to determine the maturity value is $F = P(1 + r)^t$.
More Information
The present value is essential in finance to determine how much a future sum of money is worth today, considering a certain interest rate. Understanding these calculations is crucial for making informed financial decisions.
Tips
- Incorrect Compounding: Not applying the correct exponent to the rate and time when calculating present value.
- Substituting Incorrectly: Mishandling the substitution in the formulas can lead to errors in both present value and interest calculations.
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