Aptitude · Compound Interest

The base moves, and that changes everything

Simple interest charges the same principal every year. Compound interest adds each year’s interest to the principal first, so the next year is charged on a larger sum. One sentence of difference, and it turns a straight line into a curve.

Set the terms and walk the balance year by year →

New principal = old principal + the interest just earned. That single line is the whole chapter; the formula is only a shortcut for repeating it.

01 The idea

Interest on interest

Put ₹1000 at 10% a year. After one year you have ₹1100 — and simple and compound interest agree exactly, because there is no earlier interest for anything to compound on yet.

Year two is where they part. Simple interest charges 10% of the original ₹1000 again, another ₹100. Compound interest charges 10% of the ₹1100 now standing, which is ₹110. So the balance goes to ₹1210 rather than ₹1200.

Year three widens it again: 10% of ₹1210 is ₹121, taking the balance to ₹1331 against simple interest’s ₹1300. The yearly interests run 100, 110, 121 — each one bigger than the last, because the sum being charged keeps growing.

Repeating the same multiplication is a power, which is where the formula comes from. Each year multiplies the balance by 1.1, so after T years it is 1000 × 1.1^T. Write the multiplier as (1 + R/100) and you have the standard form.

Each year multiplies the balance by (1 + R/100). Do that T times and you have the formula: A = P(1 + R/100)^T.

Compound interestInterest charged on the principal and on the interest already added. Written CI = A − P — it is always found by taking the principal off the amount.

AmountThe total standing at the end, A = P(1 + R/100)^T. Unlike simple interest there is no separate formula for the interest; you get the amount first and subtract.

Compounding periodHow often interest is added. Annually is the default; half-yearly, quarterly and monthly all work the same way with the rate and period count adjusted.

02 Worked example

₹1000 at 10% for 2 years, worked both ways

This sum runs through the whole module. Find the compound interest on ₹1000 at 10% per annum for 2 years, compounded annually.

Year one, exactly as simple interestTen per cent of the original ₹1000. Nothing has compounded yet, so this is identical to the simple case.10% of 1000 = 100 ⇒ balance = ₹1100

Year two, on the new balanceTen per cent of ₹1100, not of ₹1000. This is the only real difference in the entire chapter.10% of 1100 = 110 ⇒ balance = ₹1210

Take the principal offThere is no direct formula for compound interest — you always find the amount and subtract.CI = 1210 − 1000 = ₹210

Now the formula, which must agreeTwo years of multiplying by 1.1 is 1.1 squared.A = 1000(1 + 10/100)² = 1000 × (11/10)² = 1000 × 121/100 = 1210

Compare against simple interestSimple interest on the same terms would have added ₹100 twice.SI = 1000×10×2/100 = 200, so CI beats it by ₹10

The ₹10 is exactly 10% of the ₹100 of first-year interest — the interest earned by the interest. That is what “compound” names, and it is the quantity the whole chapter is about. Extend to three years and the gap grows to ₹31, because now there is interest on interest on interest.

03 The method

The formula, and the four compounding periods

One formula, and one adjustment that covers every compounding frequency you will be asked about.

A = P(1 + R/100)^T and CI = A − P. Note there is no shortcut straight to CI — the amount comes first, always.

For any compounding frequency, divide the rate and multiply the time. Half-yearly: A = P(1 + R/200)^2T. Quarterly: P(1 + R/400)^4T. Monthly: P(1 + R/1200)^12T. Same idea each time — the rate per period is R/k and the number of periods is kT.

Compounded	Rate per period	Periods	Formula
Annually	`R`	`T`	`P(1+R/100)^T`
Half-yearly	`R/2`	`2T`	`P(1+R/200)^2T`
Quarterly	`R/4`	`4T`	`P(1+R/400)^4T`
Monthly	`R/12`	`12T`	`P(1+R/1200)^12T`
Any frequency k	`R/k`	`kT`	`P(1+R/100k)^kT`
CI directly	`no such formula`	—	`A − P`

05 Cheat sheet

Compound interest on one page

The formulas, and the two facts about them that get asked as questions in their own right.

Case	Formula	On ₹1,000 at 10%
Amount, annual	`P(1+R/100)^T`	2 yr → 1,210
Compound interest	`A − P`	1,210 − 1,000 = 210
Half-yearly	`P(1+R/200)^2T`	2 yr → 1,215.51
Quarterly	`P(1+R/400)^4T`	2 yr → 1,218.40
First period	`same as simple interest`	both give 100
CI vs SI	`CI > SI for T > 1`	210 vs 200
Averaging the balances	`never`	growth is a power

The first period is always a tieWith no earlier interest to compound, simple and compound interest agree for one period. Any answer claiming they differ in year one is wrong.

There is no direct CI formulaFind the amount and subtract the principal. Students who look for a CI equivalent of PRT/100 waste time on something that does not exist.

More frequent compounding earns moreThe same 10% a year gives 1,210 annually, 1,215.51 half-yearly and 1,218.40 quarterly, because interest gets more chances to earn interest.

06 Where & why

Where this shows up

Compound interest is the model behind almost every real financial product, which is why it carries more marks than simple interest.

Bank PO · SSC CGL

Two- and three-year direct calculations

Usually with rates chosen so the powers stay tidy — 10% gives 1.21 and 1.331, which is why those numbers recur.

TCS NQT · Infosys

Compounding frequency comparisons

“Half-yearly instead of annually — find the difference.” One formula with the rate halved and the periods doubled.

Real finance

Deposits, loans, EMIs, inflation

Every one of these compounds. Simple interest is largely a teaching model; this is the one that describes what banks actually do.

Population and depreciation

The same formula with the sign flipped

Growth at R% a year is the identical power; depreciation is P(1 − R/100)^T.

The one sentence to carry into the rest of the module: the base moves every period. Every model that follows is a consequence of that, including the ones where the rate changes or you are asked for a single year’s interest.

07 Interview questions

What gets asked

Ten, from the definition through to the comparison interviewers use to check you actually understand compounding.

Define compound interest.

Interest calculated on the principal and on the interest already added to it. After each period the interest is added to the principal, so the next period is charged on a larger sum. The amount is P(1 + R/100)^T and the interest is that minus P.

Why is there no direct formula for CI the way there is for SI?

Because the interest is not proportional to anything simple — it is the difference between a power and the principal. So you compute the amount first and subtract. Looking for a CI analogue of PRT/100 is a common waste of time.

Find the compound interest on ₹1000 at 10% for 2 years.

₹210. The amount is 1000 × 1.1² = ₹1210, so the interest is ₹210. Year by year: ₹100 in the first year and ₹110 in the second, because the second year is charged on ₹1100.

Do simple and compound interest ever agree?

Yes, for exactly one period. With no earlier interest to compound on, the first year is identical under both rules — ₹100 on ₹1000 at 10% either way. They separate from the second period onward.

Why does compound interest grow faster and faster?

Because the base grows. On ₹1000 at 10% the yearly interests are 100, 110, 121, 133.1 and so on — each is 10% of a larger balance than the last. Simple interest adds a constant ₹100 every year, so it is a straight line while compound interest curves upward.

How do you handle half-yearly compounding?

Halve the rate and double the number of periods: A = P(1 + R/200)^(2T). So ₹1000 at 10% for 2 years half-yearly is 1000 × 1.05⁴ = ₹1215.51, against ₹1210 compounded annually.

Why does more frequent compounding give a bigger amount?

Because interest is added to the principal sooner, so it starts earning interest itself sooner. The same nominal 10% a year gives ₹1210 annually, ₹1215.51 half-yearly and ₹1218.40 quarterly on a ₹1000 principal over two years.

What is the quarterly formula?

A = P(1 + R/400)^(4T) — the rate divided by four and the periods multiplied by four. The general pattern for k periods a year is P(1 + R/100k)^(kT), which is worth remembering instead of four separate formulas.

₹2000 at 20% for 4 years, compounded annually. Find the amount and the CI.

Hard

₹1,000 is invested at 10% per annum for 3 years. (a) Find the simple interest and the compound interest, and hence the difference. (b) Show that the difference of ₹31 is made up of exactly two identifiable pieces of “interest on interest”, and name each. (c) Predict the difference for a fourth year without computing the full four-year amount, then check your prediction.

Follow-up

Part (b) is the point of the whole lesson: the ₹31 is not an arbitrary number but 10% of the first year’s ₹100, plus 10% of the second year’s ₹210 of accumulated interest. Decomposing it turns compounding from a formula into a mechanism you can reason about.

Show the hint

For (b), write out the yearly interests under both rules and line them up: 100/100, 100/110, 100/121.