Compound Interest • Topic 4 of 4

SI vs CI

Simple interest is computed only on the original principal (SI = P·r·n/100), so it grows in a straight line; compound interest is computed on the growing amount, so it pulls ahead every period. For the first year they are equal; the gap opens from year two onward, and CAT loves to test that gap with two clean shortcuts. For 2 years, CI − SI = P(r/100)² — purely the interest earned on the first year’s interest. For 3 years, CI − SI = P(r/100)²·(3 + r/100). These let you back out P or r from a single difference in seconds. A useful mental check: the year-on-year CI amounts form a geometric progression while SI amounts form an arithmetic one, so any "the difference is x" question is really asking about that one extra layer of interest-on-interest. Always confirm whether the rate or time given is annual before applying these.

✅ Solved examples

1. Find the difference between CI and SI on ₹10,000 at 10% per annum for 2 years.
CI − SI = P(r/100)² = 10000 × (0.1)² = 10000 × 0.01 = ₹100.
2. The difference between CI and SI on a sum at 5% per annum for 2 years is ₹20. Find the sum.
P(r/100)² = 20 ⇒ P × (0.05)² = 20 ⇒ P × 0.0025 = 20 ⇒ P = ₹8,000.
3. Find the difference between CI and SI on ₹15,000 at 10% per annum for 3 years.
CI − SI = P(r/100)²(3 + r/100) = 15000 × 0.01 × 3.1 = 150 × 3.1 = ₹465.
4. The difference between CI and SI for 2 years on a sum at some rate is ₹15, and the SI for 2 years is ₹600. Find the rate.
SI for 1 year = 300, so P(r/100) = 300. CI−SI = P(r/100)² = 300 × (r/100) = 15 ⇒ r/100 = 0.05 ⇒ r = 5%.

✏️ Practice — try these, take hints as needed

1. CI − SI on ₹20,000 at 10% p.a. for 2 years?
Use P(r/100)².
20000 × (0.1)².
20000 × 0.01.
₹200
2. The difference between CI and SI at 4% p.a. for 2 years is ₹32. Find the sum.
P(r/100)² = 32.
P × (0.04)² = 32.
P × 0.0016 = 32.
₹20,000
3. CI − SI on ₹8,000 at 5% p.a. for 3 years?
Use P(r/100)²(3 + r/100).
8000 × 0.0025 × 3.05.
20 × 3.05.
₹61
4. For a sum at 10% p.a., the SI for 2 years is ₹400. Find the CI for 2 years.
SI per year = 200, so P = 2000.
CI − SI = P(r/100)² = 20.
CI = SI + 20.
₹420
5. The difference between CI and SI for 2 years on ₹12,500 is ₹128. Find the rate.
P(r/100)² = 128.
12500 × (r/100)² = 128.
(r/100)² = 0.01024.
≈ 10.12% (r/100 = 0.1012)

📝 Topic test — 8 questions

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