Compound Interest Formula
Compound Interest Formula (CI & Amount Equations)
The Compound Interest Formula is an essential mathematical tool used to calculate the interest earned on both the initial principal amount and the accumulated interest from previous periods. Widely known as "interest on interest", it is a core chapter in CBSE Class 8 Mathematics and highly scoring in competitive exams like SSC CGL, Banking, and Insurance tests.
Topic: Comparing Quantities
Exams: CBSE · SSC · Banking · UPSC CSAT
On this page
What is Compound Interest?
Unlike Simple Interest, where the interest is computed solely on the primary deposit, Compound Interest (CI) reinvests your earnings. At the end of every period, the interest earned is added back to your principal amount. This updated figure then becomes the base for calculating the interest for the next term, helping money grow at an exponential rate.
The Total Compound Interest earned is always found by subtracting the original Principal from the final Total Accumulated Amount:
Compound Interest (CI) = Total Amount (A) − Principal (P)
1. Yearly/Annual Compound Interest Formula
When interest is compounded once every year, the total maturity amount equation is structured as follows:
A = P (1 + R100)n
To isolate the pure interest component, we use:
- A = Total accumulated amount after 'n' years (Principal + Interest)
- P = Principal / Initial capital value
- R = Rate of interest per annum (percentage)
- n = Number of years / total time periods
2. Half-Yearly & Quarterly Compounding Formulas
If interest is processed multiple times throughout a calendar year, we must scale down the interest rate and scale up the time intervals based on a value 't' (number of times compounded annually):
A = P (1 + Rt × 100)t × n
- For Half-Yearly (Semi-Annually): Interest cycles twice a year, so we set t = 2. This divides the rate by 2 and multiplies the periods by 2.
- For Quarterly: Interest cycles four times a year, so we set t = 4. This divides the rate by 4 and multiplies the periods by 4.
Master Reference Chart of All Variations
| Compounding Interval | Amount Formula (A) | Effective Rate & Periods |
|---|---|---|
| Annually (Once a year) | A = P(1 + R/100)n | Rate = R % · Period = n |
| Half-Yearly (Twice a year) | A = P(1 + R/200)2n | Rate = R/2 % · Period = 2n |
| Quarterly (Four times a year) | A = P(1 + R/400)4n | Rate = R/4 % · Period = 4n |
| Monthly (12 times a year) | A = P(1 + R/1200)12n | Rate = R/12 % · Period = 12n |
Differences: Compound Interest vs. Simple Interest
Understanding the functional variance prevents confusion when reading word problems:
- Simple Interest (SI): Calculated strictly on the initial capital across all years. The annual interest yields a constant value.
- Compound Interest (CI): Calculated on the changing terminal balance. The interest amount increases with each passing period.
For a duration of exactly 2 years, the difference between Compound Interest and Simple Interest can be extracted directly using:
CI − SI = P (R100)2
Common Student Errors to Avoid
The primary equation calculates the total maturity **Amount (A)**, not just the interest component. Students frequently forget to subtract the original Principal **(P)** from their total at the end to get the isolated Compound Interest **(CI)**.
In half-yearly questions, if the rate is 10% per annum, you must change it to 5% for your inner brackets ($R = 5$) and double the number of years for the exponent ($2n$).
Solved Numerical Examples
Find the compound interest earned on ₹10,000 for 2 years at an interest rate of 10% compounded annually.
Step 1: Compute Total Amount (A)
A = 10000 × (1 + 10/100)2
A = 10000 × (1.1)2
A = 10000 × 1.21 = ₹12,100
Step 2: Isolate Compound Interest (CI)
CI = A − P = 12100 − 10000 = ₹2,100
Answer: Compound Interest = ₹2,100
Find the total maturity amount on a principal sum of ₹20,000 for 1 year at 12% per annum, compounded half-yearly.
Since it is compounded half-yearly, the rate is divided by 2 and the time is multiplied by 2.
Modified Rate ($R'$) = 12 / 2 = 6% per half-year
Modified Time periods ($n'$) = 1 × 2 = 2 periods
Calculation:
A = P × (1 + R'/100)n'
A = 20000 × (1 + 6/100)2
A = 20000 × (1.06)2
A = 20000 × 1.1236 = ₹22,472
Answer: Maturity Amount = ₹22,472