Direct Proportion Calculator

Variables Required

1. Principal (P): The initial amount of money invested or loaned.
2. Annual Interest Rate (r): The annual interest rate (in decimal form, so 5% becomes 0.05).
3. Number of Compounding Periods per Year (n): How often the interest is compounded per year (e.g., annually, semiannually, quarterly, monthly, daily).
4. Time (t): The number of years the money is invested or borrowed for.

Formulas

1.  Simple Interest:

\[
A = P(1 + rt)
\]
– Where \( A \) is the amount of money accumulated after n years, including interest.

A=P(1+rn)ntA = P \left(1 + \frac{r}{n}\right)^{nt}

2.  Compound Interest:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
– Where \( A \) is the amount of money accumulated after n years, including interest.

3. Future Value of an Annuity (for regular investments):

\[
FV = P \left( \frac{(1 + r/n)^{nt} – 1}{r/n} \right)
\]
– Where \( FV \) is the future value of the annuity.

4. Present Value of an Annuity (for calculating the value of a series of future payments):

\[
PV = P \left( \frac{1 – (1 + r/n)^{-nt}}{r/n} \right)
\]
– Where \( PV \) is the present value of the annuity.

5. Continuous Compounding:

\[
A = Pe^{rt}
\]
– Where \( e \) is the base of the natural logarithm (approximately equal to 2.71828).

Step-by-Step Guide to Calculate Compound Interest

Calculating compound interest involves a few key steps. Here’s a step-by-step guide:

Formula:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]

Where:

– \( A \) = the amount of money accumulated after n years, including interest.
– \( P \) = the principal amount (the initial amount of money).
– \( r \) = the annual interest rate (in decimal form, so 5% becomes 0.05).
– \( n \) = the number of times interest is compounded per year.
– \( t \) = the number of years the money is invested or borrowed for.

Steps:

1. Determine the Principal (P):
– This is the initial amount of money you invest or borrow.

2. Determine the Annual Interest Rate (r):
– This is the yearly interest rate, expressed as a decimal. For example, if the annual interest rate is 5%, you would use 0.05 in the formula.

3. Determine the Number of Compounding Periods per Year (n):
– This is how many times the interest is compounded in one year. Common compounding periods include annually (n=1), semiannually (n=2), quarterly (n=4), monthly (n=12), weekly (n=52), and daily (n=365).

4. Determine the Time Period in Years (t):
– This is the total number of years the money is invested or borrowed for.

5. Plug the Values into the Formula:
– Substitute the values of \( P \), \( r \), \( n \), and \( t \) into the compound interest formula.

6. Calculate the Compound Interest:
– Perform the calculations inside the parentheses first.
– Raise the result to the power of \( nt \).
– Multiply the result by the principal amount \( P \).

Example Calculation:

Let’s say you invest $1,000 at an annual interest rate of 5% compounded monthly for 3 years.

1. Principal (P): $1,000
2. Annual Interest Rate (r): 0.05
3. Number of Compounding Periods per Year (n): 12
4. Time Period in Years (t): 3

Step-by-Step Calculation:

1. Substitute the Values into the Formula:

\[
A = 1000 \left(1 + \frac{0.05}{12}\right)^{12 \times 3}
\]

2. Calculate the Interest Rate per Compounding Period:

\[
\frac{0.05}{12} = 0.0041667
\]

3. Add 1 to the Interest Rate per Compounding Period:

\[
1 + 0.0041667 = 1.0041667
\]

4. Calculate the Total Number of Compounding Periods:

\[
12 \times 3 = 36
\]

5. Raise the Result to the Power of the Total Number of Compounding Periods:

\[
(1.0041667)^{36} \approx 1.1616
\]

6. Multiply by the Principal:

\[
1000 \times 1.1616 \approx 1161.60
\]

Result:

After 3 years, the investment will grow to approximately $1,161.60. The compound interest earned is \( A – P = $1161.60 – $1000 = $161.60 \).