Understanding Simple Interest (SI) and Compound Interest (CI) is essential for acing aptitude tests and solving real-life financial problems. In this article, we’ll dive into the key formulas, shortcuts, and tricks to help you master both concepts and tackle questions with ease.
Simple Interest is calculated on the principal (the original sum borrowed or invested) for a fixed period at a constant rate of interest. Unlike compound interest, the interest amount remains the same each year.
The formula for calculating Simple Interest is:
SI=P×R×T100SI = \frac{P \times R \times T}{100}
Where:
The Total Amount (A) after adding the simple interest is:
A=P+SI=P+P×R×T100A = P + SI = P + \frac{P \times R \times T}{100}
Compound Interest differs from simple interest in that the interest is calculated on the initial principal plus any interest that has already been added. This results in “interest on interest,” causing the total amount to grow exponentially.
The compound interest for a certain period is given by:
CI=P(1+R100)T−PCI = P \left(1 + \frac{R}{100}\right)^T – P
Where:
The total amount after compound interest is:
A=P(1+R100)TA = P \left(1 + \frac{R}{100}\right)^T
For cases where the interest is compounded more than once a year (e.g., quarterly, half-yearly), adjust the rate and time accordingly.
When the interest is compounded semi-annually (twice a year), the formula adjusts as:
A=P(1+R200)2TA = P \left(1 + \frac{R}{200}\right)^{2T}
This is useful when dealing with investments or loans that compound every six months.
For quarterly compounding, use the formula:
A=P(1+R400)4TA = P \left(1 + \frac{R}{400}\right)^{4T}
When compounding occurs continuously, the formula becomes:
A=P×eR×T100A = P \times e^{\frac{R \times T}{100}}
Where ee is Euler’s constant (approximately 2.718).
The population of a country is 10 crore and will grow to 13.31 crore in 3 years. What is the annual rate of growth?
Solution:
Using the compound interest formula for population growth:
Population after 3 years=P(1+R100)3\text{Population after } 3 \text{ years} = P \left(1 + \frac{R}{100}\right)^3
Substitute the given values:
13.31=10(1+R100)313.31 = 10 \left(1 + \frac{R}{100}\right)^3 R=10%R = 10\%
Anoop borrowed Rs. 800 at 6% p.a. and Rs. 1200 at 7% p.a. for the same duration. He had to pay Rs. 1584 in total as interest. Find the time period.
Solution:
First, calculate the interest for both loans:
The total interest is given by:
1584=800×6100×t+1200×7100×t1584 = 800 \times \frac{6}{100} \times t + 1200 \times \frac{7}{100} \times t
Solving for tt, we find the time period is 12 years.
When interest is compounded more frequently than annually, the effective rate of interest will differ. For instance, if interest is compounded half-yearly, it’s essential to adjust the rate to find the effective annual rate (EAR).
Example:
If the annual interest rate is 8%, compounded half-yearly, the formula becomes:
A=P(1+8200)2A = P \left(1 + \frac{8}{200}\right)^2
This will give you the effective rate for the year.
For varying rates, apply the formula:
A=P(1+R1100)(1+R2100)(1+R3100)A = P \left(1 + \frac{R_1}{100}\right) \left(1 + \frac{R_2}{100}\right) \left(1 + \frac{R_3}{100}\right)This applies the different rates to each year sequentially.
Compound interest grows exponentially, while simple interest grows linearly. Over a long period, compound interest will always yield a higher return than simple interest.
By mastering these formulas and techniques, you can solve any Simple Interest and Compound Interest question with ease. Remember to practice applying different formulas depending on the type of compounding (annually, semi-annually, quarterly, or continuously) and to use shortcuts for quick calculations.