Simple and Compound Interest

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Simple interest (SI) grows linearly: interest depends only on the original principal P, the annual rate r (as a decimal), and time t in years.

SI formula: I = P · r · t
Amount under SI: A = P(1 + r · t)

Compound interest (CI) grows exponentially because each period earns interest on principal and on previously accumulated interest.

Amount under CI: A = P(1 + r/n)^(n·t)
CI earned: CI = P[(1 + r/n)^(n·t) − 1]

Quick pointers for HAT-UG:

Always convert rate percent → decimal (12% → 0.12) before substituting.
Match time units: quarterly compounding with t measured in quarters, not years.
For 2-year annual compounding, CI − SI = P · r² — a one-line trick worth memorising.

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Core Definitions

Principal P is the original sum borrowed or invested. Rate r is the yearly percentage expressed as a decimal (5% = 0.05). Time t is the loan/tenure in years. Under SI, interest is paid only on P, so the amount after t years is A = P(1 + r·t). Under CI, interest is added back to the principal at the end of every compounding period, so the amount grows geometrically as A = P(1 + r/n)^(n·t), where n is the number of compounding periods per year.

Why CI > SI Over Time

The multiplier (1 + r·t) for SI is linear in t, while (1 + r/n)^(n·t) for CI is exponential. For any r > 0, t > 0, n ≥ 1, the CI amount strictly exceeds the SI amount. The gap equals P·r² for t = 2 years at annual compounding, and P·r²(3 + r) for t = 3 years — useful shortcut identities.

Comparing Compounding Frequencies

Compounding	n	Effective annual rate
Annual	1	r
Semi-annual	2	(1 + r/2)² − 1
Quarterly	4	(1 + r/4)⁴ − 1
Monthly	12	(1 + r/12)¹² − 1
Continuous	∞	eʳ − 1

The effective annual rate rises with n and converges to eʳ − 1 as n → ∞.
EMI for equal monthly instalments: EMI = P · r · (1+r)ⁿ / [(1+r)ⁿ − 1], with r = monthly rate and n = total months.

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Equivalent-Rate Conversion

To compare a rate compounded m₁ times yearly with another compounded m₂ times yearly, convert: r₂ = m₂ · [(1 + r₁/m₁)^(m₁/m₂) − 1]. A 10% nominal rate compounded semi-annually (m₁ = 2) gives the same annual yield as 9.65% compounded quarterly (m₂ = 4). HAT-UG items sometimes test this equivalence using a stated “simple-rate equivalent,” computed as k·[(1 + r/k)^k − 1]% for one year.

Worked Micro-Example

A deposit of PKR 50,000 earns 10% compounded quarterly for 2 years. Compute the maturity amount.

n = 4, t = 2, so n·t = 8 quarters.
A = 50,000 · (1 + 0.10/4)^8 = 50,000 · (1.025)^8 ≈ 50,000 · 1.21840 ≈ PKR 60,920.
CI earned ≈ PKR 10,920; under SI for the same 2 years the interest would be only PKR 10,000, confirming CI > SI.

Common Mistakes and Exam Strategy

Mistake	Correction
Substituting r = 5 instead of 0.05	Convert percent to decimal first
Mixing years with monthly compounding	Use t in the same unit as the period
Equating different compounding frequencies directly	Apply the equivalent-rate formula
Treating EMI’s r as annual rate	r in EMI is the periodic (monthly) rate

The HAT-UG Quantitative Reasoning section tests this topic through roughly 1–2 MCQs at about 3% weightage, mostly calculation-based with a 60–90 second time budget per item. Master the difference-of-CI-and-SI shortcut and the effective-rate table to save time.

Tip: For 2-year annual compounding, the gap CI − SI = P·r² is faster than recomputing both amounts separately.

Practice Prompts

Find the SI and CI on PKR 20,000 at 8% p.a. for 3 years compounded annually, and verify CI − SI = P·r²(3 + r).
A sum doubles in 6 years under annual compounding. Using the Rule of 72, estimate r, then compute the exact effective rate.

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Simple and Compound Interest