Compound Interest: Relation Among Principal, Time, Rate Percent of Interest Per Annum and Total Interest (interest is reckoned half yearly or quarterly or monthly)

When interest is compounded more frequently than annually, it means the interest earned in one period is added to the principal for the calculation of interest in the next period.

Formula:

$\mathrm{A=P\left(1+\frac{R}{n}\right)^{n T}}$

Where,

A = Total amount after T years

P = Principal (initial amount)

R = Rate of interest per annum (in %)

T = Time period (in years)

n = Number of compounding periods in a year (e.g., n=2 for half yearly, n=4 for quarterly)

Tip 1: A key thing to remember is the role of ‘n’. It affects both the rate of interest and the time period. Always adjust these according to the frequency of compounding.

Foundation Building Questions:

Question: A principal amount of ₹5,000 is invested at 8% per annum compounded quarterly. What will be the compound interest earned after 2 years?

Solution:

Given, R = 8% and T = 2 years, and since it's compounded quarterly, n=4.

Adjusting for quarterly compounding:

$\mathrm{Rate : R^{\prime}=\frac{R}{n}=\frac{8}{4}=2 \%\\\ \\ }$

$\mathrm{Time: T^{\prime}=T \times n=2 \times 4=8}$

Using the formula:

$\mathrm{\begin{gathered} \mathrm{A}=P\left(1+\frac{R^{\prime}}{100}\right)^{T^{\prime}} \\ \mathrm{A}=5,000\left(1+\frac{2}{100}\right)^8 \\ \mathrm{~A}=5,831.41 \end{gathered}}$

Compound Interest, CI = A - P

CI = ₹5,831.41 - ₹5,000 = ₹831.41

Tip 2: When transitioning from yearly to half-yearly or quarterly compounding, interest grows slightly faster due to more frequent addition to the principal. It's a tiny effect, but it accumulates over large sums and long durations.

Application of Previous Concepts (Simple & Compound Interest)

For the first year, both SI and CI are the same, regardless of the compounding frequency. From the second year onwards, CI starts surpassing SI due to its compound nature.

Solved Example:

Question: ₹10,000 is invested at 6% per annum compounded half-yearly. How much more will be the interest earned in two years when compared to simple interest?

Solution:

For SI:

$\mathrm{\mathrm{S} I=\frac{P \times R \times T}{100}=\frac{10,000 \times 6 \times 2}{100}=1,200}$

For CI with half-yearly compounding, n=2:

$\mathrm{Rate: R^{\prime}=\frac{R}{n}=\frac{6}{2}=3 \% }$

$\mathrm{Time: T^{\prime}=T \times n=2 \times 2=4 }$

Using the CI formula:

$\mathrm{\begin{gathered} \mathrm{A}=P\left(1+\frac{R^{\prime}}{100}\right)^{T^{\prime}} \\ \mathrm{A}=10,000\left(1+\frac{3}{100}\right)^4 \end{gathered}}$

A = ₹11,268.13

Compound Interest, ( CI = A - P = ₹11,268.13 - ₹10,000 = ₹1,268.13 )

Difference between CI and SI = ₹1,268.13 - ₹1,200 = ₹68.13

Tip 3: When tackling problems where both SI and CI are involved, always calculate SI first. Since it's straightforward, it often helps anchor the more complicated CI calculations.

Tip 4: For quick estimations, remember that the difference between CI and SI for two years is. This difference gives the additional interest gained by compounding compared to simple interest.

Regular practice and understanding the intricacies of compound interest are crucial. The nuances of different compounding frequencies can often lead to errors, so being cautious and practising regularly will help you master these problems.