Compound Interest
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Important Formulas & Concepts
Study MaterialCompound Interest
Compound Interest is one of the most important chapters in Quantitative Aptitude and Banking Aptitude. Questions from this topic are frequently asked in SSC, Banking, Railway, CDS, NDA, UPSC, CAT, Defence, and placement examinations.
This chapter mainly deals with:
- Compound Interest calculation
- Amount calculation
- Annual, half-yearly and quarterly compounding
- Difference between SI and CI
- Variable rate problems
- Growth and depreciation concepts
Compound Interest is slightly more advanced than Simple Interest because interest is calculated not only on the principal but also on previously earned interest.
What is Compound Interest?
When interest is calculated on both the original principal and the accumulated interest of previous periods, it is called Compound Interest.
Compound Interest = Interest on Principal + Previous Interest
Example:
A person deposits ₹10000 at compound interest.
After first year, interest is added to principal.
Next year's interest is calculated on the increased amount.
Important Terminologies
1. Principal (P)
The original amount borrowed or invested is called principal.
2. Amount (A)
The total money after adding compound interest is called amount.
Amount = Principal + Compound Interest
3. Compound Interest (CI)
The extra money earned on both principal and previous interest is called compound interest.
CI = Amount − Principal
4. Rate of Interest (R)
The percentage charged per year is called rate of interest.
5. Time (T)
The duration for which money is invested or borrowed is called time.
Basic Formula of Compound Interest
Let:
- Principal = P
- Rate = R% per annum
- Time = n years
If interest is compounded annually:
Amount = P(1 + R/100)n
Formula for Compound Interest
CI = Amount − Principal
or
CI = P[(1 + R/100)n − 1]
Compound Interest Compounded Half-Yearly
If interest is compounded half-yearly:
- Rate becomes R/2
- Time becomes 2n
Amount = P(1 + R/200)2n
Compound Interest Compounded Quarterly
If interest is compounded quarterly:
- Rate becomes R/4
- Time becomes 4n
Amount = P(1 + R/400)4n
Compound Interest for Fractional Time
If time contains fractions like 2½ years:
- Apply normal compounding for complete years.
- Apply Simple Interest for remaining fraction.
Different Rates for Different Years
If rates are different every year:
- First year = R₁%
- Second year = R₂%
- Third year = R₃%
Then:
Amount = P(1 + R₁/100)(1 + R₂/100)(1 + R₃/100)
Important Formula Summary
| Concept | Formula |
|---|---|
| Amount (Annual) | P(1 + R/100)n |
| Compound Interest | Amount − Principal |
| CI Direct Formula | P[(1 + R/100)n − 1] |
| Half-Yearly Compounding | P(1 + R/200)2n |
| Quarterly Compounding | P(1 + R/400)4n |
Difference Between Simple Interest and Compound Interest
| Simple Interest | Compound Interest |
|---|---|
| Interest only on principal | Interest on principal + previous interest |
| Interest remains constant | Interest increases every year |
| Linear growth | Exponential growth |
| Easy calculations | Comparatively advanced calculations |
Growth Concept in Compound Interest
Compound Interest follows exponential growth.
Each year:
- Interest increases.
- Total amount grows faster.
Depreciation Formula
Compound Interest concepts are also used in depreciation problems.
If value decreases by R% every year:
Value = P(1 − R/100)n
Special Formula for Difference Between CI and SI
1. For 2 Years
CI − SI = P(R/100)2
2. For 3 Years
CI − SI = P(R/100)2(300 + R)/100
Important Observations
1. CI is Always Greater Than SI
✔ For more than one year, Compound Interest is always greater than Simple Interest.
2. Faster Growth in CI
✔ Compound Interest grows faster because interest is added to principal repeatedly.
3. Compounding Frequency Matters
More frequent compounding gives higher amount.
| Compounding Type | Amount |
|---|---|
| Annual | Lower |
| Half-Yearly | Higher |
| Quarterly | Even Higher |
Common Mistakes in Compound Interest
- Using SI formula instead of CI formula.
- Incorrect power calculations.
- Ignoring compounding frequency.
- Wrong conversion in half-yearly problems.
- Calculation mistakes in percentage operations.
Important Exam Tips
- Memorize all CI formulas.
- Learn square and cube values for fast calculations.
- Use direct amount formula.
- Carefully identify compounding frequency.
- Practice SI vs CI comparison problems.
- Simplify percentage calculations early.
- Verify powers and exponents carefully.
Quick Revision Table
| Required Quantity | Formula |
|---|---|
| Amount | P(1 + R/100)n |
| Compound Interest | Amount − Principal |
| Half-Yearly | P(1 + R/200)2n |
| Quarterly | P(1 + R/400)4n |
| Depreciation | P(1 − R/100)n |
Compound Interest is one of the most important arithmetic chapters for competitive examinations. Strong understanding of powers, percentage calculations, and compounding concepts helps candidates solve questions quickly and accurately.