2.3 CI Compounded Half-Yearly and Quarterly
Free study material · concepts, shortcuts & solved questions
Figure: Compounding more often edges the final amount up, but only slightly.
When interest is compounded more often than once a year, you must adjust both the rate and the number of periods before applying the same formula.
The Golden Adjustment Rule:
| Compounding frequency | Rate per period | Number of periods (T years) |
|---|---|---|
| Annually | R | T |
| Half-yearly | R/2 | 2T |
| Quarterly | R/4 | 4T |
| Monthly | R/12 | 12T |
So the general CI formula becomes:
A = P(1 + (R)/(100 × n))^(nT)
where n = number of compounding periods per year (n = 2 for half-yearly, n = 4 for quarterly). This single adjustment — recalculate R and T first, then treat the problem exactly like an ordinary annual-compounding CI question — is the entire skill being tested; examiners rarely test anything more exotic than half-yearly or quarterly compounding at the SSC/RRB level, so mastering this table alone secures every question of this type.
Solved Example 2.3.1: Find the CI on Rs. 10,000 at 10% p.a. compounded half-yearly for 1 year.
Solution: Half-yearly rate = 5%, number of periods = 2 (since 1 year = 2 half-years). A = 10000 × (1.05)² = 10000 × 1.1025 = Rs. 11,025. CI = Rs. 1,025.
(Compare: annual compounding at 10% for 1 year would give CI = Rs. 1,000 exactly — half-yearly compounding always yields slightly more because interest is credited sooner and starts earning further interest.)
Solved Example 2.3.2: Find the CI on Rs. 8,000 at 20% p.a. compounded half-yearly for 1.5 years.
Solution: Half-yearly rate = 10%, number of periods = 1.5 × 2 = 3. A = 8000 × (1.1)³ = 8000 × 1.331 = Rs. 10,648. CI = Rs. 2,648.
Solved Example 2.3.3: Find the CI on Rs. 20,000 at 8% p.a. compounded quarterly for 6 months.
Solution: Quarterly rate = 8/4 = 2%, number of periods = 6 months = 2 quarters. A = 20000 × (1.02)² = 20000 × 1.0404 = Rs. 20,808. CI = Rs. 808.
Solved Example 2.3.4: Find the CI on Rs. 16,000 at 20% p.a. compounded quarterly for 9 months.
Solution: Quarterly rate = 5%, number of periods = 9 months = 3 quarters. A = 16000 × (1.05)³ = 16000 × 1.157625 = Rs. 18,522. CI = Rs. 2,522.
Solved Example 2.3.5: Find the CI on Rs. 12,000 at 12% p.a. compounded monthly for 2 months.
Solution: Monthly rate = 12/12 = 1%, number of periods = 2. A = 12000 × (1.01)² = 12000 × 1.0201 = Rs. 12,241.20. CI = Rs. 241.20. Monthly compounding is rare at this level but follows the exact same “divide rate by n, multiply time by n” rule — here n = 12.
Solved Example 2.3.6: Find the CI on Rs. 40,000 at 12% p.a. compounded quarterly for 1 year.
Solution: Quarterly rate = 12/4 = 3%, number of periods = 4 (a full year of quarterly compounding). A = 40000 × (1.03)⁴ = 40000 × 1.12550881 = Rs. 45,020.35 (approx). CI = Rs. 5,020.35 (approx). This is a good reminder that even “1 full year” changes the period count once compounding is not annual — periods = 4, not 1.