Interest represents additional fees paid for borrowing or depositing money or assets. It's typically paid on a monthly, quarterly, or annual schedule. The amount of interest earned depends on factors such as the duration of the investment, the initial amount invested or borrowed, and the interest rate.

The formula for monthly compound interest helps calculate the interest earned each month. This formula helps find out how much interest accrues each month on an initial amount of money based on the rate of interest. Unlike simple interest, where interest is only calculated on the principal amount, compound interest considers interest earned in previous periods as well. Essentially, with compound interest, you earn interest not just on your original investment but also on the interest it generates over time. Let's go deeper into the monthly compound interest formula with some practical examples.

Compound interest, specifically monthly compound interest, is a way of calculating interest where not only the initial amount borrowed or invested is considered, but also the accumulated interest over time. The formula for monthly compound interest takes into account the principal amount (the initial sum), the interest rate (expressed as a decimal), and the time period over which the interest is applied. Essentially, it's interest on interest, with interest being added back to the principal each month. This results in the total compound interest, which is the final amount minus the principal.

The formula for monthly compound interest can be expressed as:

`CI = P \left(1 + \frac{r}{12}\right)^{12t} - P`

Where,

`P : \text{Principal Amount}`

`r : \text{Interest Rate (in decimal form)}`

`t : \text{Time (in years)}`

The formula to calculate compound interest looks like this:

\(CI = P \left(1 + r\right)^n - P\)

Here, `P` represents the principal amount,

`r` is the rate of interest (in decimal)

`n` signifies the frequency of compounding

`t` stands for the overall tenure

If you're calculating interest monthly, you'll compound it `12` times a year because there are `12` months in a year. So, for monthly compounding, the formula becomes:

`CI = P\left(1 + \frac{r}{12}\right)^{12t} - P`.

Here, the interest is calculated for each month, meaning `r` is divided by `12`, and the time period is `12` times. So, whenever you see "monthly compounding," remember to use this adjusted formula.

**Example `1`: Lisa invested `$2000` at an annual interest rate of `5%`. Calculate the compounded monthly interest for `2` years.**

**Solution: **

Principal `(P)` \( = $2000 \)

Rate `(R)` \( = 5\, \text{%} = 0.05\)

Time `(T)` \( = 2\, \text{years} \)

Using monthly compound interest formula:

`CI = P\left(1 + \frac{r}{12}\right)^{12t} - P`

`= 2000\left(1 + \frac{0.05}{12 }\right)^{12 \times 2} - 2000`

`= 2000\left(\frac{12.05}{12}\right)^{24} - 2000`

`= 2000 (1.004167) - 2000`

`= 2008.33 - 2000`

`= 8.33`

Hence, the monthly compound interest \( = $8.33 \)

**Example `2`: Sam invested `$3000` and after `2` years, it grew to `$3500`, compounded monthly. What was the monthly interest rate?**

**Solution: **

Given:

Principal `(P)` \( = $3000 \)

Time `(T)` \( = 2\, \text{years} \)

Amount \( = $3500 \)

Amount is given by:

`A = P\left(1 + \frac{r}{12}\right)^{12 \times t}`

`3500 = 3000\left(1 + \frac{r}{12}\right)^{12 \times 2}`

`\frac{3500}{3000} = \left(1 + \frac{r}{12}\right)^{24}`

`(7/6)^(1/24) = 1 + r/12`

`1.006444 = 1 + \frac{r}{12}`

`\frac{r}{12} = 1.006444 - 1`

`\frac{r}{12} = 0.006444`

`r = 0.006444 \times 12`

`r = 0.077328`

Hence, monthly rate of interest \(= 0.0773\) or \(7.73\, \text{%} \).

**Example `3`: Ron invested `$7000` at an annual interest rate of `9%`. Calculate the compounded monthly interest for `7` years?**

**Solution: **

Principal `(P)` \( = $7000 \)

Rate `(R)` \( = 9\, \text{%} = 0.09 \)

Time `(T)` \( = 7\, \text{years} \)

Using monthly compound interest formula:

`CI = P\left(1 + \frac{r}{12}\right)^{12t} - P`

`= 7000\left(1 + \frac{0.09}{12 }\right)^{12 \times 7} - 7000`

`= 7000\left(\frac{12.09}{12}\right)^{84} - 7000`

`= 7000 (1.0075) - 7000`

`= 52.5`

Hence, the monthly compound interest \( = $52.50 \)

**Example `4`: Find the annual compound interest for principal `=` Rs `6000`, Rate \( = 4\, \text{%} \), Time \( = 6\, \text{years} \). **

**Solution: **

Principal `(P) =` Rs `6000`

Rate `(R)` \( = 4\, \text{%} \)

Time `(T)` \( = 6\, \text{years} \)

Compound Interest Formula `=` Amount `-` Principal

Here,

`A = P \left(1 + \frac{R}{100}\right)^{n}`

`= 6000 \left(1 + \frac{4}{100}\right)^{6}`

`= 6000 \left(\frac{26}{25}\right)^{6}`

`= 6000 (1.2653)`

`= 7591.8`

So, Amount `=` Rs `7591.8`

`\text{Compound Interest} = 7591.8 - 6000`

`=` Rs `1591.8`

**Q`1`. A sum of `$15000` is borrowed and the rate is `8%`. Calculate the monthly compound interest for `3` years?**

- \( $3821.882 \)
- \( $ 3192.87 \)
- \( $4053.555 \)
- \( $2095.745 \)

**Answer:** c

**Q`2`. A sum of `$5000` is borrowed from the bank as a home loan where the interest rate is `8%` per annum, and the amount is borrowed for a period of `2` years. How much will be monthly compounded interest charged by the bank on loan provided.**

- \( $864.44 \)
- \( $879.62 \)
- \( $912.87 \)
- \( $982.1 \)

**Answer:** a

**Q`3`. Sophie invested Rs `1000` during the year `2010`. After `10` years, she sold the investment for Rs `1500` in the year `2020`. Calculate the rate of the investment if compounded monthly.**

- \(3.99\, \text{%} \)
- \(4.06\, \text{%} \)
- \(5.15\, \text{%} \)
- \(6.63\, \text{%} \)

**Answer:** b

**Q`4`. If Adam lends `$1,500` to his friend at an annual interest rate of `4.3%`, compounded per month. Calculate the interest after the end of the year by using the compound interest formula.**

- \( $65.786 \)
- \( $71.63 \)
- \( $87.6 \)
- \( $91.54 \)

**Answer: **a

**Q`1`. What is the formula for monthly compound interest?**

**Answer:** The formula for monthly compound interest is:

`CI = P \left(1 + \frac{r}{12}\right)^{12t} - P`

where:

- \(P\) is the principal amount (the initial amount of money).
- \(r\) is the annual interest rate (in decimal).
- \(t\) is the time the money is invested for in years.

**Q`2`. How often is the interest compounded in the monthly compound interest formula?**

**Answer:** In the monthly compound interest formula, the interest is compounded monthly.

**Q`3`. What does \(r\) represent in the monthly compound interest formula?**

**Answer:** In the formula \(r\) represents the annual interest rate, expressed as a decimal.

**Q`4`. Can the monthly compound interest formula be used to calculate interest for irregular intervals?**

**Answer:** No, the formula assumes that the interest is compounded monthly at regular intervals.

**Q`5`. What is the significance of \(12\) in the formula for monthly compound interest?**

**Answer:** The \(12\) represents the number of compounding periods per year, which aligns with the monthly compounding frequency.