Compound Interest Formula and SI–CI Gap

📑 On this page

⚡ The short answer

Compound interest charges interest on the principal and on interest already added. A = P × (1 + R/100)ⁿ, and CI = A − P. Half-yearly: halve the rate, double the periods. Over two years, CI − SI = P × (R/100)². The trap: 10% for 3 years is 33.1%, not 30%.

🎬 The 30-second story

The quiz prize that grew legs

“Kabir bhai pays 10% a year on my ₹40,000. Simple. Done.”

🏆

Aarav

Aarav and Zoya each win ₹40,000 at a quiz.

“The bank pays 10% too. But it adds the interest back every year.”

🏦

Zoya

Zoya picks a 3-year deposit, compounded annually.

“I've got ₹52,000. You've got ₹53,240?!”

📊

Aarav

Three years later. Same money, rate and time.

“My interest got a job too. It earned interest.”

🪙

Zoya

₹1,240 extra, for doing nothing.

Tara's maths: SI pays 10% of the original ₹40,000 each year: ₹12,000. CI pays 10% of the growing balance: ₹4,000, ₹4,400, ₹4,840 = ₹13,240. The ₹1,240 gap is interest on interest.

What is compound interest, really?

Simple interest is a fixed salary: the principal earns the same cheque every year. Compound interest is a salary with an automatic raise. Last year's interest joins the balance, so this year's 10% is 10% of a bigger number. Watch ₹5,000 at 20%: tied in Year 1, more than ₹2,400 apart by Year 5.

📊 The data

₹5,000 at 20% a year: what you hold at the end of each year

Simple interestCompound interest

Year 1 · identical: no interest yet to earn on

₹6,000

Year 2

₹7,000

₹7,200

Year 3

₹8,000

₹8,640

Year 4

₹9,000

₹10,368

Year 5 · gap = ₹2,441.60

₹10,000

₹12,441.6

SI adds a flat ₹1,000 a year: a straight line. CI multiplies by 1.2 a year: a steepening curve.

What is the compound interest formula, and why does it work?

Adding R% to anything is the same as multiplying it by (1 + R/100). At 10% that multiplier is 1.1. Each year the balance is multiplied once more, so after n years you have multiplied n times. Repeated multiplication is a power. That is the formula.

🧮 The formula

Three years at 10%, one year at a time

💰

start

1️⃣

1.1

Year 1: +10%

2️⃣

1.1

Year 2: +10% of the new total

3️⃣

1.1

Year 3: same again

📈

A = P × 1.331

A = P(1 + R/100)ⁿ

So compound interest is successive percentage change with the same rate each time. If ₹100 → ₹110 → ₹121 makes sense, you can do CI. Then take the last step: CI = A − P. The amount is always sitting in the options for the student who stops one line early.

⚔️ Side by side

Simple vs compound: same P, R and T

📏Simple interest

interest on the original only

•SI = P × R × T ÷ 100
•Same interest every year

📈Compound interest

interest on principal + past interest

•A = P(1 + R/100)ⁿ; CI = A − P
•Interest rises every year

Test	Simple interest	Compound interest
First period	Equal to CI	Equal to SI
After that	Smaller	Always bigger
10% for 3 years	30%	33.1%

How do you calculate compound interest step by step?

✍️ Worked example Warm-up ⏱ 40 sec

Two years, annual compounding

Meher deposits her ₹6,250 internship stipend at 8% a year, compounded annually. What compound interest does she earn in 2 years?

1
Year 1: 8% of 6,250 = 500. Balance ₹6,750.
2
Year 2: 8% of 6,750 = 540. Balance ₹7,290.
3
CI = 7,290 − 6,250 = ₹1,040. Year 2 paid ₹40 more: that is 8% of Year 1's ₹500.

Answer: ₹1,040

⚡ Successive % route

Two years at 8% = 8 + 8 + (8 × 8)/100 = 16.64%. 16.64% of 6,250 = ₹1,040.

✍️ Worked example CLAT level ⏱ 60 sec

Half-yearly compounding on a phone loan

Aarav borrows ₹16,000 for a phone at 10% a year, compounded half-yearly. What interest does he owe after 18 months, and how much more is that than simple interest?

1
Rate per half-year = 10 ÷ 2 = 5%.
2
18 months = 3 half-years: three periods, not 1.5.
3
16,000 → 16,800 → 17,640 → 18,522.
4
CI = 18,522 − 16,000 = ₹2,522.
5
SI = 16,000 × 10 × 1.5 ÷ 100 = ₹2,400. Extra = ₹122.

Answer: ₹2,522, which is ₹122 more than simple interest

⚡ The exam-hall shortcut

1.05³ = 1.157625, so three periods at 5% = 15.7625%. 16,000 × 0.157625 = ₹2,522.

✍️ Worked example Tricky ⏱ 30 sec

Work backwards from the CI–SI gap

On a sum lent for 2 years at 12% a year, compound interest beats simple interest by ₹216. Find the sum.

1
Two-year gap = P × (R/100)² = P × 0.0144.
2
P = 216 ÷ 0.0144 = ₹15,000.
3
Check: SI = 15,000 × 24% = ₹3,600; CI = 15,000 × 25.44% = ₹3,816. Gap = ₹216. ✓

Answer: ₹15,000

⚡ The exam-hall shortcut

216 is 1.44% of P, so P = 21,600 ÷ 1.44 = 15,000.

What changes when interest is compounded half-yearly or quarterly?

Two things change together, never one. Divide the rate by the number of periods in a year and multiply the time by the same number. More frequent compounding means interest starts earning sooner, so the amount creeps up, but only slightly.

Compounding	Rate per period	Periods in 1 year	₹10,000 at 12% becomes
Annually	12%	1	₹11,200
Half-yearly	6%	2	₹11,236
Quarterly	3%	4	₹11,255.09
Monthly	1%	12	₹11,268.25

→ scroll to see more

⚠️ The half-yearly trap cuts both ways

Halve the rate but keep one period and you get ₹10,600. Double the periods but keep 12% and you get ₹12,544. Both get planted as options. And within a single year, half-yearly compounding already opens a CI–SI gap, because there are two periods. CLAT 2019 was built on exactly that.

🎛️ Try it yourself — change any number

Play with Aarav and Zoya's ₹40,000

Prize money (₹) Rate per year (%) Years Times compounded a year

Simple interest

12,000

Compound interest

13,240

CI − SI

1,240

SI = P×R×T/100; CI = P(1 + R/100)³ − P

As set: SI ₹12,000, CI ₹13,240. Set compounding to 2: CI rises to about ₹13,604. Set years and compounding to 1: the gap vanishes.

What is the difference between SI and CI for 2 and 3 years?

Don't memorise this one; see it. Take the story's ₹40,000 at 10%. Simple interest pays a flat ₹4,000 a year. Compound interest pays that same ₹4,000 plus 10% of every rupee of interest already in the account.

🕒 Timeline

Where the extra ₹1,240 comes from

1
Year 1

CI ₹4,000 = SI ₹4,000

No past interest yet. Extra = ₹0.
2
Year 2

CI ₹4,400 = ₹4,000 + ₹400

₹400 is 10% of Year 1's ₹4,000. Extra = P(R/100)².
3
Year 3

CI ₹4,840 = ₹4,000 + ₹400 + ₹400 + ₹40

10% of each earlier ₹4,000, plus 10% of Year 2's ₹400. Extra = ₹840.
Σ
Total

Gap = ₹400 + ₹840 = ₹1,240

3 × P(R/100)² + P(R/100)³ = P(R/100)²(3 + R/100).

📌 Two shortcuts that do real work

2 years: CI − SI = P × (R/100)². 3 years: CI − SI = P × (R/100)² × (3 + R/100). Check: 40,000 × 0.01 × 3.1 = ₹1,240. Given the 2-year gap, flip it: P = gap × (100/R)².

How do you estimate compound interest fast in the exam?

You rarely need the exact power. Learn this small grid of multipliers and most CI questions become one multiplication.

Rate	2 years: total %	3 years: total %	Doubles in about
5%	10.25%	15.76%	14.4 years
8%	16.64%	25.97%	9 years
10%	21%	33.1%	7.2 years
12%	25.44%	40.49%	6 years
20%	44%	72.8%	3.6 years

→ scroll to see more

✓Two years at R%: 2R + R²/100, the a + b + ab/100 rule with a = b = R.
✓Rule of 72: money doubles in roughly 72 ÷ R years (at 10%, exactly about 7.27). Good for options, not final answers.
✓Doubling chains: if it doubles in t years, it is 4 times in 2t and 8 times in 3t. Times multiply, years add.
✓Different rates each year: multiply the factors. 10% then 12% on ₹25,000 = 25,000 × 1.1 × 1.12 = ₹30,800.

How does compound interest hide in a CLAT caselet?

Since 2020, CLAT rarely asks "find the CI" in one line. It writes a paragraph: a family budget, a few savings schemes. Spot the interest, pull the numbers into a table, then calculate. Say Rohan has ₹50,000 to park for 3 years and four offers.

📊 The data

₹50,000 for 3 years: what each offer pays back

10% simple

₹65,000

11% simple

₹66,500

10% compound, yearly

₹66,550

10% compound, half-yearly · best

₹67,004.78

10% compound beats 11% simple over 3 years by ₹50, but loses over 2 (₹60,500 vs ₹61,000). Read the time.

🧭 Decision flow

Before you calculate: four checks

❓ Does the passage say compound, or describe interest being added to the balance?

No →

Use simple interest: P × R × T ÷ 100.

Yes ↓

❓ Is the compounding frequency stated (annual by default)?

No →

"Compounded periodically" with no frequency: cannot be determined.

Yes ↓

❓ Is the principal given, or buildable from the data?

No →

You can find the % gain, not the rupee interest.

Yes ↓

❓ Does it ask for interest, not the amount?

Yes →

Compute A, then subtract P.

No →

A is your answer. Don't subtract.

⚠️ Where the marks leak

Giving the amount when interest is asked. Adding rates over years (10% for 3 years is 33.1%, not 30%). Reading 18 months as 1.8 years. And in savings plans, forgetting that each year's deposit earns only for the years it actually sits in the bank. Estimate first: if zero-interest savings fall just short of a target, check whether one year's interest covers the gap.

How CLAT asked it

📝 Asked in CLAT 2019 · Q60 SI vs CI, half-yearly

The difference between Simple Interest and Compound Interest on Rs. 500 for 1 year at 10% per annum, reckoned half yearly is

ARs. 1

BRs. 1.25

CRs. 1.5

DRs. 2

Reveal answer + reasoning

Answer: B

There is no official key for 2019; this is our solution. SI = 500 × 10 × 1 ÷ 100 = ₹50. Half-yearly means 5% for 2 periods: 500 → 525 → 551.25, so CI = ₹51.25. Difference = ₹1.25.
Faster: two periods behave like the 2-year rule with R = 5: 500 × (5/100)² = ₹1.25. About 30 seconds.

Why the others fail

A — A round-number guess. Any real calculation gives ₹1.25.
C — Matches no compounding route; usually a slip in the 5% steps.
D — Close to quarterly compounding (about ₹1.91). The question says half-yearly: 2 periods, not 4.

📝 Asked in CLAT 2025 · Q116 Savings target with 12% interest

Mr. Das is working in a construction company. … His total monthly income includes a salary of Rs. 9228/- and a 10% house rent allowance. … Out of his total monthly income, he spends 25% on food expenses, 18% on paying the house-rent, 9% on entertainment, 23% on the education of his child, 13% on medical expenses, and he saves 12% of his total monthly income.

If Mr. Das gets 12% annual interest on the savings and he wishes to save Rs. 30,000/- in two years period, how much extra should he save in the next year?

ARs. 1200/-

BRs. 1300/-

CRs. 1400/-

DThere is no need for saving

Reveal answer + reasoning

Answer: D

Income: 9,228 × 1.1 = ₹10,150.80 a month. Savings: 12% = ₹1,218.10 a month, or ₹14,617.15 a year.
No interest: two years = ₹29,234.30, only ₹765.70 short of ₹30,000.
With interest: Year 1's savings earn 12% through Year 2 = ₹1,754.06, more than the shortfall. Total ≈ 16,371.21 + 14,617.15 = ₹30,988.36. Counting interest monthly gives even more (about ₹32,856). He needs nothing extra. About 75 seconds.

Why the others fail

A — Assumes a shortfall; the interest more than closes the ₹765.70 gap.
B — Bait for students who skip the interest.
C — Same mistake: treating savings as idle cash.

💥 Myth vs fact

Myth vs fact

🙅 Myth

10% a year for 3 years means 30% interest.

Tap to bust it →

✅ Fact

That is simple interest. Compounded, it is 33.1%.

🙅 Myth

For half-yearly compounding, just halve the rate.

Tap to bust it →

✅ Fact

Halve the rate and double the periods. 2 years half-yearly = 4 periods.

🙅 Myth

CI and SI can't differ within one year.

Tap to bust it →

✅ Fact

They tie for one period only. Half-yearly gives two in a year (CLAT 2019).

🙅 Myth

A higher simple rate always beats a lower compound rate.

Tap to bust it →

✅ Fact

Depends on time. 11% simple loses to 10% compound over 3 years, but wins over 2.

Can you do these five?

🎯 Quick quiz

Five compound interest checks

0 / 5

Rohan puts ₹12,500 in a fixed deposit at 4% a year, compounded annually, for 2 years.

Q1. What compound interest does he earn?

Kabir borrows ₹24,000 for shop stock at 20% a year, compounded half-yearly, for 1 year.

Q2. How much interest does he pay?

On a sum at 5% a year for 2 years, compound interest exceeds simple interest by ₹45.

Q3. What is the sum?

Meher's savings triple in 6 years at compound interest.

Q4. In how many years will they become 9 times?

Zoya's deposit earned ₹1,210 compound interest in its second year and ₹1,331 in its third, compounded annually.

Q5. What is the rate of interest?

🎯 Compound interest in a nutshell

Interest on principal plus past interest: A = P(1 + R/100)ⁿ, CI = A − P.
It is successive percentage change at one fixed rate.
SI and CI tie for one period; after that, CI always wins.
Half-yearly: rate ÷ 2, periods × 2. Quarterly: ÷ 4, × 4.
CI − SI: P(R/100)² for 2 years; × (3 + R/100) for 3.

📎 Formula card

Amount: A = P × (1 + R/100)ⁿ; CI = A − P
k times a year: A = P × (1 + R/(100k))^nk
CI − SI: 2 years P(R/100)²; 3 years P(R/100)²(3 + R/100)
Sum from the 2-year gap: P = gap × (100/R)²
Multipliers: 1.1² = 1.21, 1.1³ = 1.331, 1.05² = 1.1025, 1.08² = 1.1664, 1.12² = 1.2544
Doubling: about 72 ÷ R years
Rate from consecutive years: R = (later CI ÷ earlier CI − 1) × 100

Frequently asked questions

What is the formula for compound interest?

Amount A = P × (1 + R/100) raised to the power n, where P is the principal, R the yearly rate and n the number of years. Compound interest is A minus P. So ₹40,000 at 10% for 3 years becomes ₹53,240: CI = ₹13,240.

How is compound interest different from simple interest?

Simple interest is charged only on the original principal, so it is the same every year. Compound interest is charged on the principal plus all interest added so far, so it grows every year. They tie for the first period; after that CI is always larger.

How do you calculate compound interest half-yearly?

Divide the yearly rate by 2 and multiply the number of years by 2. So 10% a year for 18 months, compounded half-yearly, is 5% for 3 periods: ₹16,000 grows to 16,000 × 1.05 × 1.05 × 1.05 = ₹18,522, giving ₹2,522 interest.

What is the difference between CI and SI for 2 years and 3 years?

For 2 years the difference is P × (R/100) squared. For 3 years it is P × (R/100) squared × (3 + R/100). On ₹40,000 at 10%, the 2-year gap is ₹400 and the 3-year gap is ₹1,240. Reverse the 2-year rule to find the principal: P = gap × (100/R) squared.

Is compound interest asked in CLAT?

Yes, now mostly inside caselets. CLAT 2019 asked the SI–CI gap on ₹500 at 10% compounded half-yearly for one year. CLAT 2025 put a 12% interest question inside a family budget passage about saving ₹30,000 in two years.