7.b Compound Interest¶

Introduction¶

Compound interest is a method of calculating interest where the interest earned over time is added to the principal amount, and future interest is calculated on the new total. This means that interest is earned not only on the original principal but also on the accumulated interest. As a result, compound interest grows faster than simple interest.

The formula for calculating compound interest is:

\[ A = P \left(1 + \frac{R}{100}\right)^T \]

Where: - $ A $ = Total amount (Principal + Interest) - $ P $ = Principal amount (the initial sum of money) - $ R $ = Rate of interest per annum (as a percentage) - $ T $ = Time period for which the interest is calculated (in years)

The compound interest itself can be found by subtracting the principal from the total amount:

\[ \text{Compound Interest} (CI) = A - P \]

Example 1: Calculating Compound Interest Annually¶

Problem¶

Emma invests $2,000 in a savings account that offers a 6% annual compound interest rate. Calculate the compound interest and the total amount after 3 years.

Solution¶

Given: - $ P = 2000 $ (Principal amount) - $ R = 6 \% $ (Rate of interest) - $ T = 3 $ years (Time period)

Using the compound interest formula:

\[ A = 2000 \left(1 + \frac{6}{100}\right)^3 \]

\[ A = 2000 \left(1.06\right)^3 \]

Calculating $ 1.06^3 $:

\[ A \approx 2000 \times 1.191016 = 2382.03 \]

So, the total amount after 3 years is approximately $2,382.03.

The compound interest is:

\[ CI = 2382.03 - 2000 = 382.03 \]

Explanation¶

In this example, Emma invested $2,000 at an annual compound interest rate of 6% for 3 years. The interest was calculated on the total amount accumulated each year, resulting in a total interest of approximately $382.03 after 3 years. The power in the formula represents the compounding effect over multiple periods.

Example 2: Calculating Compound Interest Semi-Annually¶

Problem¶

Mark deposits $1,500 in a bank account that offers a 4% annual interest rate compounded semi-annually. Calculate the compound interest and the total amount after 2 years.

Solution¶

Given: - $ P = 1500 $ (Principal amount) - $ R = 4 \% $ (Annual rate of interest) - $ T = 2 $ years (Time period) - Compounding frequency = Semi-annual (2 times a year)

The formula for compound interest with a compounding frequency $ n $ times per year is:

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

Substitute the values:

\[ A = 1500 \left(1 + \frac{4}{100 \times 2}\right)^{2 \times 2} \]

\[ A = 1500 \left(1 + 0.02\right)^4 \]

\[ A = 1500 \times 1.08243216 \approx 1623.65 \]

So, the total amount after 2 years is approximately $1,623.65.

The compound interest is:

\[ CI = 1623.65 - 1500 = 123.65 \]

Explanation¶

In this case, Mark's deposit earned interest at a rate of 4%, compounded semi-annually. The effective interest calculation was done four times over the 2 years, resulting in a slightly higher total amount compared to annual compounding due to the more frequent compounding periods.

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