Simple & Compound Interest

Scenario

You put Rs10,000 in a new neo-bank app offering 12% returns. Do they calculate it simply at year-end, or compounded daily? The difference is whether you can buy a new phone or just a cover.

Mental model

Simple Interest is linear growth. Compound Interest is exponential growth (interest earning its own interest).

SI adds the exact same amount every cycle. CI multiplies by a constant scaling factor every cycle, making it just successive percentage increases applied to the growing total.

Formula

SI = P x R x T / 100. Amount under SI = P + SI = P x (1 + RT/100). CI Amount = P x (1 + R/100)^T. CI = Amount - P. Compounded half-yearly: A = P x (1 + R/200)^(2T). Compounded quarterly: A = P x (1 + R/400)^(4T). Difference (CI - SI) for 2 years = P x (R/100)^2. Difference (CI - SI) for 3 years = P x (R/100)^2 x (R/100 + 3).

When compounding frequency increases (monthly, quarterly, half-yearly), halve or quarter the rate and multiply the time periods accordingly. The 2-year difference shortcut P(R/100)^2 is the most-tested formula after the basic ones.

Pattern discovery

Track interest earned each year on Rs1000 at 10% for both SI and CI.

• Year 1: SI=100, CI=100 (same for year 1)
• Year 2: SI=100, CI=110 (CI earns interest on last year's interest too)
• Year 3: SI=100, CI=121 (CI keeps growing; SI stays flat)

Why does CI pull ahead of SI after year 1, and by how much exactly?

Rule: CI for any year is the previous year's CI scaled by (1 + R/100). The difference between CI and SI grows each year because CI reinvests the interest.

Worked example

Problem: Find the compound interest on Rs5,000 at 10% per annum for 2 years, compounded annually. Also find the SI for the same terms and the difference.

1. CI Amount = P x (1 + R/100)^T = 5000 x (1.1)^2 = 5000 x 1.21 = 6050.
2. CI = 6050 - 5000 = 1050.
3. SI = P x R x T / 100 = 5000 x 10 x 2 / 100 = 1000.
4. Difference = CI - SI = 1050 - 1000 = 50.
5. Sanity-check using shortcut: Diff = P x (R/100)^2 = 5000 x (0.1)^2 = 5000 x 0.01 = 50. Matches.

Answer: CI = Rs1050, SI = Rs1000, Difference = Rs50.

Problem: Rs8,000 is invested at 10% per annum compounded half-yearly. Find the amount after 1 year.

1. Half-yearly: halve the rate (10/2 = 5%) and double the time periods (1 year = 2 half-years).
2. Amount = 8000 x (1 + 5/100)^2 = 8000 x (1.05)^2 = 8000 x 1.1025 = 8820.
3. Compare with annual compounding: 8000 x 1.10 = 8800. Half-yearly gives more (8820 > 8800). Makes sense.

Answer: Amount = Rs8,820.

Shortcuts

• 2-Year CI-SI Difference: Diff = P x (R/100)^2 (Diff on Rs1000 at 10% for 2 years = 1000 x (10/100)^2 = Rs10)
• Rule of 72: Years to double your money = 72 / R (At 12% CI, your money doubles in roughly 72/12 = 6 years.)

Common mistakes

• Monthly vs Annual Rate: If interest is compounded half-yearly, halve the rate and double the time periods. If quarterly, use R/4 and 4T. Do not use the annual rate blindly.

Glossary

exponential

Growth that speeds up over time because it builds on itself.

scaling

Multiplying a number by a factor to make it larger or smaller.

compounding

When you earn interest not only on the original amount but also on the interest you have already earned.

Recall questions

• What is the 2-year difference between CI and SI on Rs5000 at 5%?
• If money doubles in 4 years at SI, what is the rate?
• Does SI ever exceed CI for the same rate and positive time?

Questions & answers

A sum of money doubles itself in 5 years at simple interest. In how many years will it become 4 times itself?

15 years

Approach: SI = P takes 5 years (rate = 20%). To become 4P, SI = 3P. Since SI is linear, 3P takes 3 x 5 = 15 years.

The difference between CI and SI on a sum for 2 years at 8% is Rs32. What is the sum?

Rs5000

Approach: Diff = P x (R/100)^2 -> 32 = P x (8/100)^2 -> P = 32 x 10000 / 64 = 5000.

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