Interest (SI & CI)

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Compound Interest

⏱ 4 min read🧩 6 question types🎯 13 practice Q

The idea in one minute

Compound interest adds each year's interest to the principal, so the next year charges more. The amount grows by the same factor every year, not the same rupees.

Use $A = P\left(1 + \dfrac{R}{100}\right)^T$ and $CI = A - P$ . Write the rate as a chip fraction ( $10\% \to \dfrac{11}{10}$ ) and multiply one chip per year.

What compounding means

In compound interest, each year's interest is added to the principal before the next year is charged. Interest starts earning interest, so the amount grows by the same factor every year.

Rs 10,000 at 10%: after one year 11,000, after two years 12,100. The second jump is Rs 1,100, not Rs 1,000 — the extra Rs 100 is interest on interest.

Rule: One chip per year: $A = P \times \left(1 + \dfrac{R}{100}\right)^T$ and $CI = A - P$ .

Chips worth knowing by heart

Rate	Chip	Rate	Chip
5%	21/20	8%	27/25
10%	11/10	12.5%	9/8
20%	6/5	25%	5/4

CI on Rs 12,000 at 10% for 2 years: $12000 \times \dfrac{11}{10} \times \dfrac{11}{10} = 14520$ , so CI $= 2520$ . The same money at simple interest gives only 2,400.

Tip: Keep chips as fractions. They stay exact, and exam numbers cancel cleanly.

Net per cent for two and three years

Square the chip to read the two-year effect: 10% → 21% (not 20%), 20% → 44%, 25% → 56.25%, 5% → 10.25%.

Three years: 10% → 33.1%, 20% → 72.8%. Fraction rates stay clean: $12.5\%$ has chip $\dfrac{9}{8}$ , so two years multiply the money by $\dfrac{81}{64}$ and CI $= \dfrac{17}{64}$ of P.

These small tables answer "the amount is what per cent of P" instantly.

Compounding more often than once a year

Half-yearly: rate halves, periods double — $A = P\left(1 + \dfrac{R}{200}\right)^{2T}$ .
Quarterly: rate quarters, periods quadruple — $A = P\left(1 + \dfrac{R}{400}\right)^{4T}$ .

Rs 10,000 at 20% for $1\dfrac{1}{2}$ years compounded half-yearly is 3 periods at 10%: $10000 \times 1.1^3 = 13310$ .

Careful: $1\dfrac{1}{2}$ years half-yearly is 3 periods, not 2. Count periods before touching any chip.

More frequent compounding always gives more interest than yearly compounding at the same yearly rate.

Broken years

$2\dfrac{1}{2}$ years at 10% compounded yearly: chips for the 2 full years, then simple interest for the half year on the amount.

10000 \times 1.1 \times 1.1 = 12100, \qquad 12100 \times \dfrac{5}{100} = 605, \qquad A = 12705

Watch: The mixed rule (compound full years, simple the leftover fraction) is only for "compounded annually" questions. Half-yearly compounding handles the half year as its own full period.

A different rate each year

Multiply each year's own chip. Rs 25,000 at 10% in year one and 20% in year two: $25000 \times \dfrac{11}{10} \times \dfrac{6}{5} = 33000$ . A loss year simply uses a chip below 1.

Scaling and going backwards

CI is proportional to P: at the same rate and time, CI on Rs 30,000 is 1.5 times CI on Rs 20,000.

From an amount back to P, divide by the chips: Rs 6,655 after 3 years at 10% is $\dfrac{6655}{1.331} = 5000$ .

Note: Know the powers cold: $1.1^2 = 1.21$ , $1.1^3 = 1.331$ , $1.2^2 = 1.44$ , $1.25^2 = 1.5625$ , $1.2^3 = 1.728$ .

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common2 practice Q

Amount or CI by chips

How to spot it:

P, R and whole-year T are given; the amount or CI is asked.

A = P\left(1 + \frac{R}{100}\right)^T

Method

Write the chip 1 + R/100 as a small fraction.
Multiply P by the chip T times.
CI = A − P; check CI is more than SI.

Why it works:

Each year scales the running amount by the same factor.

Try this

Find the compound interest on Rs 12,000 at 10% per annum for 2 years.

Show solution

$12000 \times \dfrac{11}{10} \times \dfrac{11}{10} = 14520$ .
CI $= 14520 - 12000 = 2520$ (SI would be 2400).

Answer

Rs 2,520

Type 2very common2 practice Q

Net per cent or rate from a multiplier

How to spot it:

'Becomes 1.21 times in 2 years' or 'CI as a per cent of P' — the net compounding effect.

\text{net factor} = \left(1 + \frac{R}{100}\right)^T, \qquad R = 100(\sqrt[T]{m} - 1)

Method

Square the chip for 2 years, cube it for 3.
Read off the per cent: 1.21 → 21%, 1.44 → 44%.
From a multiple, root it: m = 1.44 → chip 1.2 → 20%.

Why it works:

Compounding applies the same growth twice, so effects multiply.

Try this

A sum invested at compound interest becomes 1.44 times itself in 2 years. The annual rate is:

Show solution

Chip $= \sqrt{1.44} = 1.2$ .
$R = (1.2 - 1) \times 100 = 20$ .

Answer

20% per annum

Type 3very common2 practice Q

Half-yearly or quarterly compounding

How to spot it:

'Compounded half-yearly / quarterly' appears in the question, often with 1 or 1.5 years.

A = P\left(1 + \frac{R}{200}\right)^{2T} \quad \text{or} \quad P\left(1 + \frac{R}{400}\right)^{4T}

Method

Halve (or quarter) the rate; double (quadruple) the periods.
Run the chips on periods.
1.5 years half-yearly is 3 periods.

Why it works:

Each compounding period uses the rate for its own fraction of a year.

Try this

Find the CI on Rs 10,000 at 20% per annum for 1.5 years, compounded half-yearly.

Show solution

Rate $10\%$ per half-year, 3 periods.
$10000 \times \left(\dfrac{11}{10}\right)^3 = 13310$ .
CI $= 3310$ .

Answer

Rs 3,310

Type 4common2 practice Q

Different rates in different years

How to spot it:

'10% in the first year and 20% in the second, compounded annually' — sometimes a loss year.

A = P\left(1+\frac{r_1}{100}\right)\left(1+\frac{r_2}{100}\right)\cdots

Method

Write each year's own chip.
Multiply them all with P.
CI = A − P, or compare with another scenario.

Why it works:

Each year compounds on the running amount at its own rate.

Try this

Rs 25,000 is invested at 10% for the first year and 20% for the second, compounded annually. The amount is:

Show solution

$25000 \times \dfrac{11}{10} \times \dfrac{6}{5}$ .
$= 33000$ .

Answer

Rs 33,000

Type 5common2 practice Q

Scaling the CI

How to spot it:

The CI for one sum is given; the CI for another sum at the same rate and time is asked.

\frac{CI_2}{CI_1} = \frac{P_2}{P_1}

Method

CI is proportional to P at a fixed rate and time.
Scale by the ratio of the principals.
For a changed time, use the net per cent of the new period instead.

Why it works:

Every term of the CI formula is linear in P.

Try this

The CI on Rs 20,000 for 2 years at 10% is Rs 4,200. The CI on Rs 30,000 for the same period and rate is:

Show solution

$CI_2 = 4200 \times \dfrac{30}{20}$ .
$= 6300$ .

Answer

Rs 6,300

Type 6common

Broken period: full years plus a fraction

How to spot it:

T is something like 2.5 years and the question says compounded annually.

A = P\left(1+\frac{R}{100}\right)^{\lfloor T \rfloor} \times \left(1 + \frac{R \times \text{fraction}}{100}\right)

Method

Split T into whole years and the leftover fraction.
Compound the whole years with chips.
Charge simple interest on the result for the fraction.
Add to finish.

Why it works:

Banks compound full years and pay simple interest for the leftover part year.

Try this

Find the compound interest on Rs 10,000 at 10% per annum for 2.5 years, compounded annually.

Show solution

$10000 \times \dfrac{11}{10} \times \dfrac{11}{10} = 12100$ .
Half-year SI: $12100 \times \dfrac{5}{100} = 605$ .
$A = 12705$ , CI $= 2705$ .

Answer

Rs 2,705

Formula sheet

Compound amount

A = P\left(1 + \frac{R}{100}\right)^T

One chip per year.

Compound interest

CI = A - P = P\left[\left(1 + \frac{R}{100}\right)^T - 1\right]

Year-wise multipliers

A = P \times \frac{100 + r_1}{100} \times \frac{100 + r_2}{100} \times \cdots

For rates that change yearly.

Half-yearly compounding

A = P\left(1 + \frac{R}{200}\right)^{2T}

Rate halves, periods double.

Quarterly compounding

A = P\left(1 + \frac{R}{400}\right)^{4T}

Rate quarters, periods quadruple.

Shortcuts that save time

⚡ Multiply the chips

One chip per year, multiplied. Fraction chips cancel before you multiply.

Example

Find the amount on Rs 10,000 at 10% per annum compound interest for 2 years.

Show solution

$10000 \times \dfrac{11}{10} \times \dfrac{11}{10} = 12100$ .
CI $= 12100 - 10000 = 2100$ .

Answer

Rs 12,100 (CI Rs 2,100)

⚡ Divide the chips to get P

An amount after T years walks back to the principal by dividing by the chip T times.

Example

A sum amounts to Rs 4,840 in 2 years at 10% per annum CI. Find the sum.

Show solution

$P = 4840 \times \dfrac{10}{11} \times \dfrac{10}{11}$ .
$= 4000$ .

Answer

Rs 4,000

⚡ Half-yearly: halve rate, double count

Convert first, then run the chips on the periods.

Example

Find the CI on Rs 10,000 at 20% per annum for 1.5 years, compounded half-yearly.

Show solution

3 half-years at $10\%$ : chip $\dfrac{11}{10}$ cubed.
$10000 \times \dfrac{1331}{1000} = 13310$ ; CI $= 3310$ .

Answer

Rs 3,310

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Raising one chip to a power when rates change year to year.

Multiply each year's own chip separately.

Mistake 02

Halving the time instead of the rate for half-yearly compounding.

Rate halves (R/200) and the number of periods doubles (2T).

Mistake 03

Reporting the amount A when the CI is asked.

CI = A − P; subtract the principal before answering.

Mistake 04

Using R/200 but keeping the exponent T.

The exponent must count periods: 2T for half-yearly, 4T for quarterly.

Mistake 05

Compounding the leftover fraction of a year.

Compound the full years; the leftover fraction earns simple interest.

Quick revision

Read this the night before the exam.

One chip per year; multiply; CI $=$ A $-$ P.
Net two-year effects: 10 $\to$ 21, 20 $\to$ 44, 25 $\to$ 56.25 per cent.
Half-yearly: R/2 and 2T periods; quarterly: R/4 and 4T.
Broken period: full years compound, leftover fraction simple.
Different yearly rates: multiply the year-wise chips.
P from A: divide by the chip power (1.21, 1.331, 1.44).

Practice: 13 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 13 questions

Suggested time 8 min · wrong answers go to your mistake notebook automatically.

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