2.6 Difference Between CI and SI for 3 Years
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Derivation: Let r = R/100. Over 3 years: SI = 3Pr, CI = P[(1+r)^3 - 1] = P(3r + 3r^2 + r^3)
Subtracting SI from CI, the 3Pr terms cancel:
CI - SI = 3Pr^2 + Pr^3 = Pr^2(3+r)
Substituting r = R/100 back and simplifying (multiply out the fraction fully):
[CI - SI (3 years) = (PR2(300+R))/(10(6))]
Notice that setting R = 0 inside the (300+R) term and simplifying would reduce this to something resembling the 2-year formula multiplied by a factor close to 3 — a useful sanity check that this 3-year formula is a natural extension of the 2-year one, not an unrelated new relation to memorise separately.
Solved Example 2.6.1: Find the difference between CI and SI on Rs. 10,000 for 3 years at 10% p.a.
Solution: Diff = (10000 × 100 × 310)/10⁶ = 310,000,000/1,000,000 = Rs. 310.
Check by direct computation: SI = 3 × 10000 × 10/100 = 3,000. CI = 10000 × [(1.1)³ − 1] = 10000 × 0.331 = 3,310. Difference = 3310 − 3000 = 310. ✓ Matches.
Solved Example 2.6.2: The difference between CI and SI for 3 years on a sum at 5% p.a. is Rs. 122. Find the sum.
Solution: 122 = P × 25 × 305/10⁶ = P × 7625/1,000,000. So P = 122 × 1,000,000/7625 = Rs. 16,000.
Solved Example 2.6.3: The difference between CI and SI for 3 years on a sum of Rs. 20,000 is Rs. 620. Find the rate of interest (a new variant: finding R, not P, from the 3-year difference).
Solution: Diff = PR²(300+R)/10⁶. Substituting P = 20,000: 620 = 20000 × R²(300+R)/10⁶ = R²(300+R)/50. Trying the “nice” value R = 10: R²(300+R) = 100 × 310 = 31,000, and 31000/50 = 620 — matches exactly. R = 10% p.a. Check by direct computation: SI = 3×20000×10/100 = 6,000; CI = 20000×[(1.1)³−1] = 20000×0.331 = 6,620; difference = 620. ✓ (At this level, such cubic-in-R equations are always solved by testing a clean rate value rather than by algebraically solving the cubic — the options in an MCQ make this fast.)
Solved Example 2.6.4: The difference between CI and SI for 3 years on a sum at 4% p.a. is Rs. 121.60. Find the sum.
Solution: Diff = PR²(300+R)/10⁶ = P × 16 × 304/10⁶ = P × 4864/1,000,000. So P = 121.60 × 1,000,000/4864 = Rs. 25,000. Check: SI = 3×25000×4/100 = 3,000; CI = 25000×[(1.04)³−1] = 25000×0.124864 = 3,121.60; difference = 121.60. ✓