Simple and Compound Interest

Overview

Simple and Compound Interest appears about 2.3 times per paper across the last four AFCAT solved papers, placing it in the highest weight band of Numerical Ability.

Simple and Compound Interest carry an average of 2.25 marks per AFCAT paper. The formulas are short, the arithmetic is bounded, and the trap library is small — which is exactly why this topic is one of the most reliable mark-banks in Numerical Ability. Once you internalise the four shortcuts in this module — the SI-vs-CI 2-year difference, the rate-derivation by inspection, the compounding-frequency adjustment and the doubling-time identity — almost every AFCAT interest item collapses to a one-line answer.

The topic rewards memorisation of a small table of squares and a handful of formulas. Unlike percentages, you do not need to be quick with mental arithmetic across long chains; you need to recognise which of six standard sub-types you are looking at and apply the matching identity. This module walks through each sub-type with a derivation, a trap pattern and at least one worked example.

Why interest carries reliable marks in AFCAT

Across the last four solved AFCAT papers (2022 II to 2024 I), interest items appear at a steady rate of 2 to 3 per paper. The question-writers favour this topic because it sits at the intersection of three skills the Air Force values for officer cadres: arithmetic discipline, formula recall under time pressure, and reading-the-question-twice. The third matters because almost every CI trap turns on a single word — annually, half-yearly, quarterly, or in two equal installments — that flips the formula entirely.

The good news for a candidate is that the variety of sub-types is narrow:

• Direct SI calculation — plug into PRT/100.
• Direct CI calculation — usually for 2 or 3 years.
• Half-yearly or quarterly compounding.
• CI − SI difference, asking for the principal or the rate.
• Rate-derivation when P and A are both given.
• Doubling-time, tripling-time, or population-growth framing.

Six sub-types, six identities. If you have them in muscle memory by the night before the paper, expect to bank both interest marks in under three minutes total.

Simple Interest — derivation and core formula

Simple Interest is interest charged only on the original principal, regardless of how long the loan has run. If you borrow ₹100 at 10% per annum simple interest, you owe ₹10 of interest each year — year five looks exactly like year one.

Derivation. Interest per year on a sum P at rate R% per annum is (R/100) × P. Over T years that interest is repeated T times, giving:

SI = (P × R × T) / 100

The formula is symmetric in R and T, which is what makes the doubling-time and tripling-time shortcuts work.

Time-to-double identity. A sum doubles when the accumulated interest equals the principal, so SI = P. Substituting:

P = (P × R × T) / 100 ⇒ R × T = 100

Time-to-triple identity. For tripling, accumulated interest equals twice the principal, so SI = 2P, giving R × T = 200. More generally, to grow a sum k times under SI, R × T = 100(k − 1).

Amount under Simple Interest

The amount A is principal plus interest. Under SI:

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

The factor (1 + RT/100) is the SI growth factor. It is linear in T — that is the entire difference between SI and CI.

This linearity is what lets you back-solve. If ₹4,000 grows to ₹5,000 in 5 years, the growth factor is 1.25, so RT/100 = 0.25 and RT = 25. With T = 5 the rate is 5% per annum. No formula manipulation needed.

Compound Interest — derivation and compounding frequency

Compound Interest is interest charged on the running balance — the previous period's interest itself earns interest. The classic derivation walks through period by period.

Start with principal P. After one year at R% per annum, the amount is P(1 + R/100). Treat that as the new principal; after the second year the amount is P(1 + R/100)(1 + R/100) = P(1 + R/100)². Extending to n years:

A = P (1 + R/100)ⁿ     CI = A − P = P [(1 + R/100)ⁿ − 1]

Half-yearly compounding. If the annual rate is R% but interest is added every six months, each half-year applies a rate of R/2 and the total number of compounding periods in n years is 2n. The amount becomes:

A = P (1 + R/200)²ⁿ

Quarterly compounding. Each quarter applies R/4 and there are 4n periods in n years:

A = P (1 + R/400)⁴ⁿ

The general rule: if the annual rate is R% and interest is compounded k times per year, the per-period rate is R/k and the number of periods in n years is kn.

Quick CI shortcuts and the squares table

For 2-year and 3-year horizons the CI formula collapses to short identities you can apply by inspection.

2-year CI. Expand (1 + R/100)² = 1 + 2R/100 + (R/100)². So:

CI₂ = P [2R/100 + (R/100)²] = 2 × SI₁ + P (R/100)²

Since 2-year SI is 2 × SI₁ = 2PR/100, the difference is:

CI₂ − SI₂ = P (R/100)²

3-year CI. Expand (1 + R/100)³ = 1 + 3R/100 + 3(R/100)² + (R/100)³. After subtracting the 3-year SI you get:

CI₃ − SI₃ = P (R/100)² × (3 + R/100)

For most AFCAT rate values (5%, 6%, 8%, 10%, 12%) you should memorise the small squares table below — it lets you compute (1 + R/100)² by inspection, which is the same calculation you need for the 2-year amount.

Lock this table by the second week of preparation. Almost every rate-derivation item in the last four AFCAT papers turns on one of these squares.

SI vs CI side-by-side for 2 and 3 years

The most cited identity in AFCAT interest items is the 2-year SI-vs-CI difference. Memorise the entire comparison so you can move between them without re-deriving.

The 1-year row is worth memorising for one reason: SI and CI are identical for one year when compounding is annual. So if the question says ‘1 year at 8% per annum compound interest’ and stays silent on frequency, the answer is the SI value. The only thing that breaks this equality at 1 year is a half-yearly or quarterly compounding directive.

Reverse usage. The CI − SI = P(R/100)² identity is also a back-solve weapon. If you are told the difference is ₹50 at 10% for 2 years, then P × (0.1)² = 50, so P = ₹5,000. Two seconds of arithmetic.

Rate derivation when P and A are given

A frequent AFCAT format gives you principal and amount with no rate, asking you to derive R. The technique varies with the time horizon.

• 1 year (CI or SI — they agree at 1 year annual): Growth factor A/P = 1 + R/100. So R = 100 × (A/P − 1).
• 2 years (CI): A/P = (1 + R/100)². Take the square root and read off R from the squares table.
• 3 years (CI): A/P = (1 + R/100)³. Match against the cube column of the squares table; the rate is whichever row equals A/P.

For example, if ₹1,000 becomes ₹1,331 in 3 years at CI, then A/P = 1.331, which is the 10% row in the squares table — so R = 10%. You do not need to extract a cube root by hand; you read the rate by inspection.

When the time horizon is not a friendly integer or the ratio does not match a memorised square, the question is signalling that you should use SI instead. AFCAT does not push candidates into hand-rooting non-standard numbers.

Population growth and depreciation as compound interest

Compound interest is the same mathematics that drives population growth and the depreciation of capital assets. The formula is identical; only the language changes.

Population growth. If a population P grows by r% per year, the population after n years is P(1 + r/100)ⁿ. AFCAT typically asks for the population after 2 or 3 years, which puts you right in the squares table.

Depreciation. If the value of a machine V depreciates at d% per year, its value after n years is V(1 − d/100)ⁿ. Same compounding logic, but the factor is less than 1.

Mixed framing — growth in one period, depreciation in the next. Multiply the factors: if a town grows by 10% one year and loses 5% the next, the net factor is 1.10 × 0.95 = 1.045, so the population is 4.5% higher than the start of the two-year window. This is the same as the successive-percentages identity from the percentages module.

Equal installments — the present-value identity

An installment problem gives you a sum borrowed at a stated rate and asks for the equal annual installment that retires the debt in n years. AFCAT uses this format rarely — perhaps once in four papers — but when it appears, it is worth knowing the structure.

Present-value identity. Each installment paid k years from now is worth less in today's money than its face value, because today's money would have earned interest in the interim. The sum of all installments' present values must equal the principal borrowed.

Under simple interest, an installment x paid at the end of year k retires a present-value debt of x / (1 + kR/100). For an n-year SI loan, the identity becomes:

P = x × [1/(1 + R/100) + 1/(1 + 2R/100) + … + 1/(1 + nR/100)]

In practice AFCAT will set up the problem so that the sum collapses to a small fraction. Recognise the structure, write the present-value sum, and substitute — the arithmetic is usually two-digit by design.

For compound interest installments, the identity is the standard annuity formula, but AFCAT does not push candidates into that territory. If you see installments under CI, expect the question to ask for the first installment or the unpaid balance after one year, not the full annuity computation.

Trap patterns AFCAT uses

Across the last four papers the same three traps recur. Lock these in muscle memory.

• The CI − SI difference shortcut. The question gives the difference and one of (P, R). You must apply P(R/100)² for 2 years. The trap is reading ‘3 years’ instead of ‘2 years’ — for which the identity has an extra (3 + R/100) factor. Re-read the time before substituting.
• Hidden half-yearly compounding. The question gives an annual rate and a number of years but slips in the words ‘compounded half-yearly’ or ‘interest credited every six months’. Always halve the rate and double the time. Easy candidates miss this because they plugged into the annual formula on autopilot.
• Doubling-time conversion under SI vs CI. A sum doubles in 8 years under SI does not mean the same sum doubles in 8 years under CI. Under SI, RT = 100 so R = 12.5%. Under CI, doubling at 12.5% takes only about 5.9 years. If the question switches the regime mid-stem, walk through both formulas.
• Quarterly compounding sleight of hand. The phrasing ‘compounded every three months’ is the same as quarterly. Per-period rate is R/4 and number of periods in 1 year is 4. Candidates who do not parse this slowly often divide by 12 instead of 4.
• One-year CI when frequency is annual. If the compounding is annual and the time is exactly 1 year, CI equals SI. The question may bury this inside a more complex stem; recognise that the CI calculation is unnecessary and use PR/100 directly.

Time budget and speed shortcuts

AFCAT gives 72 seconds per Numerical Ability item on average if you take 90 of the 120 minutes for the section. Interest items should run faster than that — they reward formula recall, not creative algebra.

If any single interest item takes more than 90 seconds, mark it for review and move on — the marginal value of the third minute on a CI item is lower than the value of the first 60 seconds on any unattempted item elsewhere.

Strategy summary

• Memorise the squares table for R = 4, 5, 6, 8, 10, 12, 15, 20, 25 by week 2 of preparation.
• Treat the CI − SI 2-year identity P(R/100)² as a reflex — the back-solve direction (given difference, find P) is worth one mark almost every paper.
• For every CI item, the first thing you read is the compounding frequency. If it is not annual, halve or quarter the rate before any other step.
• Use the RT = 100 doubling identity and RT = 200 tripling identity under SI without writing a formula. These appear in roughly one paper in three.
• Skip rate-derivation items where A/P does not match a memorised square. AFCAT is rarely that hostile, but when it is, the time cost is not worth one mark.

Worked AFCAT-style examples

Exam-day strategy

1. Drill the squares table — (1.05)², (1.06)², (1.08)², (1.10)², (1.12)², (1.20)² — until conversion to the rate by inspection is automatic.
2. Treat the CI − SI = P(R/100)² 2-year identity as a reflex; both directions (given P find difference, given difference find P) appear in roughly one paper in two.
3. On every CI item, read the compounding frequency before anything else. Half-yearly halves the rate and doubles the time; quarterly does the same with 4.
4. For SI doubling and tripling problems, use RT = 100 and RT = 200 directly — never plug into the long formula.
5. Skip rate-derivation items where the ratio A/P does not match a memorised square. The probability of a clean integer rate is high in AFCAT; the probability of needing to hand-extract a root is near zero.
6. Aim for 30 to 60 seconds per item across the topic. If a single item runs past 90 seconds, mark it for review and move to the next.

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