Simple & compound interest: Quantitative Aptitude Notes for UPSC Geo-Scientist

Simple & Compound Interest – UPSC Geo‑Scientist Quantitative Aptitude

Target Level – Basic to intermediate (sufficient for most UPSC quantitative aptitude sections).
Goal – Understand concepts, memorise formulas, and solve typical problems quickly.

1. Simple Explanation

Simple Interest (SI)

Interest earned only on the principal amount for the given period.
No compounding – the interest is not added to the principal for further interest calculation.

Compound Interest (CI)

Interest earned on the principal and on interest that has accumulated in prior periods.
Can be compounded annually, semi‑annually, quarterly, monthly, daily (or any specified period).
The longer the compounding frequency, the higher the effective interest.

2. Key Points & Formulas

Concept	Formula	Notes
Simple Interest	( SI = \frac{P \times R \times T}{100} )	(P)=Principal, (R)=annual rate %, (T)=time in years (if not in years, convert).
Total Amount (A)	( A = P + SI ) or ( A = P\left(1 + \frac{R \times T}{100}\right) )	For quick calculation.
Compound Interest (once a year)	( A = P \left(1 + \frac{R}{100}\right)^T )	(T) in years.
Compound Interest (n times a year)	( A = P \left(1 + \frac{R/100}{n}\right)^{nT} )	(n)= number of compounding periods per year.
Effective Annual Rate (EAR)	( EAR = \left(1 + \frac{R/100}{n}\right)^n - 1 )	Useful for comparing investment options.
Interest Rate from Two Amounts	( R = 100 \times \frac{A_2 - A_1}{A_1 \times T} )	For simple interest only.
Time from Two Amounts (CI)	( T = \frac{\log\left( \frac{A_2}{A_1} \right)}{n \log\left(1 + \frac{R/100}{n}\right)} )	For compound interest.

Quick Conversion Tips

Months to Years: Divide by 12.
Days to Years: Divide by 365 (or 360 if approximation).
Percentage to Decimal: Divide by 100.

Common Mistakes

Forgetting to convert rates to decimal form.
Using months as years without dividing by 12.
Mixing up compounding frequency (n) – e.g., quarterly compounding uses n=4.
Assuming simple interest when the problem states compound.

3. Worked Example

Problem:
A geologist invests ₹50,000 in a fixed deposit that offers 8% per annum compounded quarterly.

Find the amount after 3 years.
Find the compound interest earned.
How much would the amount be if the interest were simple (still 8% per annum)?

Step‑by‑Step

Step	Action	Calculation	Result
1	Identify values	(P = 50,000) ₹, (R = 8%), (n = 4) (quarterly), (T = 3) years	—
2	Convert rate to decimal	(R/100 = 0.08)	—
3	Apply CI formula	(A = 50{,}000 \times \left(1 + \frac{0.08}{4}\right)^{4 \times 3})
4	Simplify inside parenthesis	(1 + 0.02 = 1.02)	—
5	Compute exponent	(4 \times 3 = 12)	—
6	Raise to power	(1.02^{12}) ≈ 1.2682418	—
7	Multiply by principal	(A = 50{,}000 \times 1.2682418 ≈ 63{,}412.09)	₹63,412.09
8	Compound interest	(CI = A - P = 63{,}412.09 - 50{,}000 = 13{,}412.09)	₹13,412.09
9	Simple interest (for comparison)	(SI = \frac{50{,}000 \times 8 \times 3}{100} = 12{,}000)	₹12,000
10	Simple final amount	(A_{SI} = 50{,}000 + 12{,}000 = 62{,}000)	₹62,000

Answer

Compound Amount after 3 yrs: ₹63,412.09
Compound Interest earned: ₹13,412.09
Simple Amount after 3 yrs: ₹62,000

4. Quick Revision Summary

Topic	Formula	Typical Question Types
Simple Interest	(SI = \frac{PRT}{100})	“How much interest will be earned?” “What will be the total amount?”
Total / Final Amount	(A = P(1 + \frac{R T}{100}))	“What is the final amount after X years?”
Compound Interest (once/year)	(A = P(1 + \frac{R}{100})^T)	“How much will a deposit grow to after Y years?”
Compound Interest (n times/year)	(A = P(1 + \frac{R/100}{n})^{nT})	“Interest is compounded quarterly/monthly.”
Effective Annual Rate	(EAR = (1 + \frac{R/100}{n})^n - 1)	“Which investment offers better return?”
Solving for unknowns	Inverse formulas (time, rate, principal)	“If you need Z amount in X years, what rate is needed?”