## Direct Proportion Calculator

A compound interest calculator is a tool used to compute the amount of interest earned or paid on an initial principal over a specified period of time with interest being compounded at regular intervals. This calculator typically takes into account the principal amount, the annual interest rate, the number of compounding periods per year, and the total time the money is invested or borrowed.

### Variables Required

**Principal (P)**: The initial amount of money invested or loaned.**Annual Interest Rate (r)**: The annual interest rate (in decimal form, so 5% becomes 0.05).**Number of Compounding Periods per Year (n)**: How often the interest is compounded per year (e.g., annually, semiannually, quarterly, monthly, daily).**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 \( A – P = $1161.60 – $1000 = $161.60 \).