Anil borrows Rs 2 lakhs at an interest rate of 8% per annum, compounded half-yearly. He repays Rs 10320 at the end of the first year and closes the loan by paying the outstanding amount at the end of the third year. Then, the total interest, in rupees, paid over the three years is nearest to

45311

51311

33130

40991

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Concepts Covered - 1

Compound Interest: Relation Among Principal, Time, Rate Percent of Interest Per Annum and Total Interest

Definition: Compound Interest (CI) refers to the interest that accumulates on both the initial principal and the accumulated interest from previous periods.

Formula:

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

Where,

A = Total amount after T years

P = Principal (initial amount)

R = Rate of interest per annum (in %)

T = Time period (in years)

The Compound Interest (CI) is given by:

$\mathrm{CI=A-P}$

Tip 1: Always be clear about the compounding frequency - annually, semi-annually, quarterly, or monthly. Adjust the rate and time accordingly. If compounding is semi-annually, divide R by 2 and multiply T by 2.

Foundation Building Questions:

Question: A sum of ₹10,000 is invested at 10% per annum compounded annually. What will be the amount after 3 years?

Solution:

Using the formula:

$\mathrm{\begin{gathered} \mathrm{A}=P\left(1+\frac{R}{100}\right)^T \\ \mathrm{A}=10,000\left(1+\frac{10}{100}\right)^3 \\ \mathrm{A}=10,000(1.1)^3 \\ \mathrm{A}=13,310 \end{gathered}}$

So, the amount after 3 years will be ₹13,310.

Tip 2: To quickly calculate CI for two years at rate R%, use the formula:

\CI = P \times R% \times (1 + R%) \

This is particularly handy for quick calculations and objective exams.

Tip 3: When dealing with multiple compounding periods in a year, always adjust the time and rate. For example, for quarterly compounding over 2 years at 8% p.a., R becomes 2% (i.e., 8/4) and T becomes 8 (i.e., 2 X 4).

Application of Previous Concept (Simple Interest)

Remember that for the first year, Simple Interest and Compound Interest are the same. This understanding can help speed up calculations for certain problems.

Example:

Question: An amount of ₹15,000 is lent out at a compound interest rate of 6% per annum compounded semi-annually. What is the interest earned after 2 years?

Solution:

First, adjust for semi-annual compounding:

$\mathrm{R = 6/2 = 3% and T = 2 \times 2 = 4.}$

Using the CI formula:

$\mathrm{\begin{gathered} \mathrm{A}=P\left(1+\frac{R}{100}\right)^T \\ \mathrm{~A}=15,000\left(1+\frac{3}{100}\right)^4 \\ \mathrm{~A}=16,813.81 \end{gathered}}$

CI = A - P

CI = ₹16,813.81 - ₹15,000

= ₹1,813.81

Interest earned is ₹1,813.81.

Tip 4: In questions involving both SI and CI, compute the SI for the entire period first. Then, calculate the CI for the period minus the SI to find the additional interest due to compounding.

By continually practising, not only do you familiarise yourself with the formulas, but you also develop shortcuts and strategies to tackle any kind of problem on this topic in management entrance exams.