Simple & Compound Interest: Money That Grows

Introduction

You deposit ₹10,000 in a bank. After a year, you get ₹10,800 back. The extra ₹800 is the interest — the bank's payment for using your money.

There are two ways banks can calculate this interest:

1. Simple Interest (SI): Always calculated on the original amount.
2. Compound Interest (CI): Calculated on the growing amount (interest on interest).

This difference seems small at first, but over years, it creates a massive gap — which is exactly why this topic appears in CAT every year.

Why This Topic Matters

SI/CI questions appear 2–4 times in CAT. They also appear in Data Interpretation sets involving financial data. More importantly, understanding this topic gives you real-life financial literacy.

Basic Concepts

Simple Interest Formula

SI = (P × R × T) / 100

Where:

• P = Principal (original amount)
• R = Rate of interest per year (%)
• T = Time (in years)

Amount (A) = P + SI = P(1 + RT/100)

Compound Interest Formula

A = P(1 + R/100)^T
CI = A - P = P[(1 + R/100)^T - 1]

When compounded half-yearly: A = P(1 + R/200)^(2T)
When compounded quarterly: A = P(1 + R/400)^(4T)

Detailed Explanation

Section 1 — SI vs CI Comparison

For the same P, R, T:

• SI grows linearly (same amount each year).
• CI grows exponentially (more each year).

CI is always greater than SI (for T > 1 year and R > 0).

After 1 year: SI = CI. After 2+ years: CI > SI.

₹5000 at 8% for 3 years. Find SI and CI.

SI = 5000 × 8 × 3 / 100 = ₹1200

CI:
A = 5000 × (1.08)³ = 5000 × 1.259712 = ₹6298.56
CI = 6298.56 - 5000 = ₹1298.56

Difference = ₹98.56

Section 2 — The CI-SI Difference Shortcuts

For 2 years:

CI - SI = P × (R/100)²

For 3 years:

CI - SI = P(R/100)² × (R/100 + 3)

P = ₹10,000, R = 10%, T = 2 years.
CI - SI = 10000 × (10/100)² = 10000 × 0.01 = ₹100

This is faster than calculating CI and SI separately!

Section 3 — Finding P, R, or T

If SI for T years at R% is known:
R = (SI × 100) / (P × T)

Rule of 72: For compound interest, doubling time ≈ 72/R years.

At 8%: doubles in ≈ 9 years.
At 12%: doubles in ≈ 6 years.

For simple interest: doubles when RT = 100, so T = 100/R.

At what rate does ₹1500 become ₹2500 in 5 years (SI)?

SI = 2500 - 1500 = ₹1000
R = (1000 × 100) / (1500 × 5) = 100000/7500 = 13.33%

Section 4 — Effective Annual Rate

When interest is compounded more than once a year, the effective rate is higher than the stated rate.

Effective Rate = (1 + R/n)^n - 1

where n = compounding periods per year.

12% compounded monthly:
Effective rate = (1 + 0.12/12)^12 - 1 = (1.01)^12 - 1 ≈ 12.68%

This is why your credit card APR (compounded daily) is much more than the stated rate.

Section 5 — Installment Problems

Equal installments: If x is paid each year for T years at rate R%:

Loan = x/(1+R/100) + x/(1+R/100)² + ... + x/(1+R/100)^T

(This is a geometric series.)

A sum is to be paid in 2 equal annual installments of ₹2,420 at 10% CI. Find the principal.

Let P = loan amount.
1st installment (at end of year 1): PV = 2420/1.1 = ₹2200
2nd installment (at end of year 2): PV = 2420/1.21 = ₹2000
Total principal = ₹4200

Section 6 — Growth and Depreciation

The same formula applies to population growth and depreciation:

Growth: A = P(1 + R/100)^T
Depreciation: A = P(1 - R/100)^T

A car bought for ₹8,00,000 depreciates at 15% annually. Value after 3 years?

A = 800000 × (0.85)³ = 800000 × 0.614 = ₹4,91,300 (approx.)

Important Terms

Quick Revision

• SI = PRT/100
• Amount (SI) = P(1 + RT/100)
• Amount (CI) = P(1 + R/100)^T
• CI - SI (2 years) = P(R/100)²
• CI - SI (3 years) = P(R/100)²(R/100 + 3)
• Half-yearly: replace R with R/2 and T with 2T
• Doubling time (SI) = 100/R years
• Doubling time (CI) ≈ 72/R years (Rule of 72)

Remember This

Think of SI as stairs (same step height each time) and CI as escalator (speeds up as you go higher). Both start the same but CI always ends up higher for longer durations.

Exam Tips

1. Difference shortcut: CI - SI = P(R/100)² for 2 years. Saves 30 seconds.
2. Half-yearly compounding: Always substitute R→R/2 and T→2T in the CI formula.
3. Installments: Work backwards from year of repayment using present value.
4. If answer choices differ by ₹1-2: You're rounding correctly; pick the closest.
5. Ratio approach: If SI₁:SI₂ = x:y with same R and T, then P₁:P₂ = x:y.

Common Mistakes

❌ Applying CI formula but forgetting to adjust for half-yearly/quarterly compounding.
❌ Confusing Amount with Interest: CI = A - P, not A.
❌ Using CI formula for depreciation without changing sign.
❌ Not reading "per annum" vs "per half-year" rates.

Fun Facts

• Albert Einstein reportedly called compound interest "the eighth wonder of the world."
• The concept of interest appears in ancient Mesopotamian texts from 2000 BCE — charged at 33% per year!
• A ₹1 investment at 1% interest, compounded annually for 100 years, becomes ₹2.70. At 10% it becomes ₹13,780!

Practice Questions

Beginner Level

1. Find SI on ₹3000 at 8% per annum for 3 years.
2. Find CI on ₹5000 at 10% per annum for 2 years.
3. In how many years will ₹4000 double at 5% SI?
4. Amount after CI for ₹2000 at 15% for 2 years?
5. Find CI-SI difference for ₹10,000 at 10% for 2 years.

Intermediate Level

1. ₹8000 gives ₹2000 as SI in 5 years. Find rate.
2. CI on ₹6250 for 2 years at 8% compounded half-yearly.
3. A sum triples itself in 12 years under SI. Find rate.
4. CI for ₹4000 at 10% for 3 years vs SI for same. Find difference.
5. Machine bought for ₹1,50,000. Depreciates 20% per year. Value after 2 years?

Advanced Level

1. A sum at CI grows by 5% in year 1 and 8% in year 2. Find effective 2-year rate.
2. Two equal installments of ₹3,025 repay a loan at 10% CI compounded annually. Find loan.
3. If CI for 3 years is ₹1,655 and for 2 years is ₹1,100, find rate and principal.
4. Population of city = 50,000. Grows 4% in year 1, declines 2% in year 2. New population?
5. A borrowed ₹10,000 at 10% CI. Paid ₹3,100 at end of year 1. Remaining debt after year 2?

Summary

Simple Interest uses only the original principal — grows linearly. Compound Interest uses growing principal — grows exponentially. Key shortcuts: CI-SI = P(R/100)² for 2 years. Half-yearly: halve the rate, double the time. Depreciation uses 1-R/100 instead of 1+R/100. These formulas model everything from bank deposits to population growth.

Related Topics

• [[Percentages: The Multiplier Method]](/notes/percentages-multiplier-method)
• [[Ratio & Proportion]](/notes/ratio-proportion-cat)
• [[Sequence & Series]](/notes/sequence-series-cat)