When you deposit money in a bank, the bank usually adds interest once a year. But some banks add interest twice a year (half-yearly) or four times a year (quarterly). Each time interest is added, it becomes part of the new principal, and the next interest is calculated on this larger amount. This is why more frequent compounding gives you more interest
💡Key Rules
For half-yearly CI: The rate is divided by 2, and time is multiplied by 2.
For quarterly CI: The rate is divided by 4, and time is multiplied by 4
🔑Formula
A = P × (1 + R/(100×n))^(t×n)
Where:
- A = Amount after interest
- P = Principal (original money)
- R = Annual rate of interest (%)
- n = Number of times compounded per year (2 for half-yearly, 4 for quarterly)
- t = Time in years
- CI = A − P
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Exam Patterns
What examiners ask — read before attempting PYQs
SSC CGL typically asks: Compare CI for different compounding periods, find CI amount, or calculate effective rate.
Shortcut/Trick:
For half-yearly: Use R/2 and 2t. For quarterly: Use R/4 and 4t. Always remember the rate gets divided and time gets multiplied by the same number.
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Worked Example
Solve this step-by-step before moving on
1
Step 1
Identify n = 4 (quarterly)
2
Step 2
Apply formula: A = 8000 × (1 + 20/(100×4))^(1×4)
3
Step 3
A = 8000 × (1 + 5/100)^4
4
Step 4
A = 8000 × (1.05)^4
5
Step 5
A = 8000 × 1.2155 = 9724
6
Step 6
CI = 9724 − 8000 = Rs 1724
Common Mistake:
Students forget to divide the rate by the compounding frequency. They use the full annual rate instead of R/2 or R/4, leading to wrong answers. Always reduce the rate first.
🔑 Key Points
Half-yearly CI: Divide rate by 2, multiply time by 2
Quarterly CI: Divide rate by 4, multiply time by 4
Formula: A = P(1 + R/(100n))^(tn) where n = compounding frequency
More frequent compounding = higher final amount
CI = Amount − Principal (always calculate both separately)
In 1 year, quarterly compounding gives more interest than half-yearly
📌 Exam Facts
For half-yearly compounding, the effective rate formula is: (1 + R/200)^2 − 1
For quarterly compounding in 1 year, total compounding periods = 4
Half-yearly means n = 2, so rate becomes R/2 for each period
Quarterly means n = 4, so rate becomes R/4 for each period
If time is 2 years with quarterly compounding, total periods = 8
Compound Interest formula with frequency: A = P(1 + r/100)^n where r is periodic rate and n is total periods
For half-yearly: 1 year = 2 periods, 2 years = 4 periods, 3 years = 6 periods
💪 Practice Questions (15) · Showing 3
Q 1easy
A sum of ₹8,000 is invested at 12% per annum compound interest, compounded half-yearly. What is the amount after 1 year?
Q 2easy
₹5,000 is invested at 8% per annum compound interest, compounded quarterly. Find the compound interest earned in 6 months.
Q 3easy
A principal amount becomes ₹10,404 in 1 year at 8% per annum compound interest, compounded half-yearly. What is the principal?
🚀 60-Second Revision
Remember: Divide rate by compounding frequency (2 for half-yearly, 4 for quarterly), multiply time by the same number
Formula: A = P × (1 + R/(100×n))^(t×n) — this works for ALL compounding frequencies
Trap: Don't forget CI = Amount − Principal; calculate both separately
Quick Check: In 1 year with quarterly CI at 20% p.a., effective rate ≈ 21.55% (not 20%)
Pattern: More frequent compounding always gives MORE interest for same P, R, and t
Always verify: After substitution, ensure exponent = compounding periods per year × time in years
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