2.4 CI on Fractional Time (e.g., 2½ years) — Annual Compounding

When time is not a whole number of years but compounding is still annual (a common trap), SSC uses this rule: apply CI formula for the whole-year part, then apply simple interest for the fractional part, calculated on the amount already accumulated at the end of the whole years.

A = P(1+(R)/(100))^(n)(1 + (R × f)/(100))

where n = whole number of years, f = fractional part of the year.

Solved Example 2.4.1: Find the CI on Rs. 10,000 at 10% p.a. for 2½ years.

Solution: First 2 years (compounded): A₂ = 10000 × (1.1)² = 12,100. Remaining half-year treated as SI on 12,100 at 10% for ½ year: SI = 12100 × 10 × 0.5/100 = 605. Final amount = 12100 + 605 = Rs. 12,705. CI = Rs. 2,705.

Solved Example 2.4.2: Find the CI on Rs. 8,000 at 10% p.a. for 3 years 3 months, compounded annually.

Solution: Whole-year part (3 years): A₃ = 8000 × (1.1)³ = 8000 × 1.331 = Rs. 10,648. Remaining 3 months = ¼ year, treated as SI on 10,648 at 10% for ¼ year: SI = 10648 × 10 × 0.25/100 = Rs. 266.20. Final amount = 10648 + 266.20 = Rs. 10,914.20. CI = Rs. 2,914.20.

Solved Example 2.4.3: Find the CI on Rs. 6,000 at 20% p.a. for 1 year 3 months, compounded annually (edge case: fractional part applied after only 1 whole year).

Solution: Whole-year part (1 year): A₁ = 6000 × 1.2 = Rs. 7,200. Remaining 3 months = ¼ year, treated as SI on 7,200 at 20% for ¼ year: SI = 7200 × 20 × 0.25/100 = Rs. 360. Final amount = 7200 + 360 = Rs. 7,560. CI = Rs. 1,560.