Compound Interest

Suppose you keep ₹10,000 in a bank that pays 10% a year. With simple interest you get ₹1,000 every year. With compound interest after year one the interest of ₹1,000 joins your money, making ₹11,000; year two’s interest is on ₹11,000, i.e. ₹1,100 — the interest itself starts earning. That is “interest on interest”, and it grows faster and faster with time.

A = P (1 + \frac{R}{100})^{n}, C I = A - P

The core formula for annual compounding

1

The formula and its uses

Finding any one of

A

,

P

,

R

,

n

Finding the amount, the compound interest, the principal, the rate and the time.

Annual compounding

One multiplier per year

Multiplying by $1 + \frac{R}{100}$ , $n$ times, and the reverse questions.

Factor	Rate $R$	2-year factor	3-year factor
1.05	5%	1.1025	1.157625
1.10	10%	1.21	1.331
1.20	20%	1.44	1.728
1.25	25%	1.5625	1.953125
1.04	4%	1.0816	1.124864

MemoryLearn these factors:

1. 1^{2} = 1.21

,

1. 1^{3} = 1.331

,

1. 2^{2} = 1.44

,

1. 2^{3} = 1.728

,

1.0 5^{2} = 1.1025

. Most exam questions rest on them.

Example 1 · Basic

Find the amount and the compound interest on ₹10,000 at 10% per annum for 3 years.

Solution, step by step

1 $A = 10000 \times (1.1)^{3} = 10000 \times 1.331 = 13310$ .
2 $C I = 13310 - 10000 = 3310$

Answer ₹13,310 and ₹3,310

Example 2 · Standard

A sum amounts to ₹6,760 in 2 years at 4% compound interest per annum. What was the sum?

Solution, step by step

1 $P = \frac{6760}{( 1.04 ) ^{2}} = \frac{6760}{1.0816}$ .
2 $= 6250$

Answer ₹6,250

Example 3 · Standard

₹2,500 becomes ₹2,809 in 2 years at compound interest. What is the rate per annum?

Solution, step by step

1 $(1 + \frac{R}{100})^{2} = \frac{2809}{2500} = 1.1236$ .
2 $1.1236 = 1.06$ ⟹ $R = 6%$

Answer 6%

BasicTry it yourself first

What is the amount on ₹8,000 after 2 years at 5% compound interest per annum?

BasicTry it yourself first

What is the amount on ₹5,000 after 2 years at 10% compound interest per annum?

StandardTry it yourself first

What is the amount on ₹12,500 after 3 years at 4% compound interest per annum?

BasicTry it yourself first

What is the compound interest on ₹4,000 at 10% per annum for 2 years, compounded annually?

StandardTry it yourself first

What is the compound interest on ₹25,000 at 12% per annum for 2 years, compounded annually?

StandardTry it yourself first

What is the compound interest on ₹16,000 at 5% per annum for 3 years, compounded annually?

AdvancedTry it yourself first

What is the compound interest on ₹50,000 at 20% per annum for 3 years, compounded annually?

BasicTry it yourself first

A sum amounts to ₹6,050 in 2 years at 10% compound interest per annum. What was the sum?

StandardTry it yourself first

A sum amounts to ₹8,820 in 2 years at 5% compound interest per annum. What was the sum?

AdvancedTry it yourself first

A sum amounts to ₹14,060.8 in 3 years at 4% compound interest per annum. What was the sum?

BasicTry it yourself first

₹2,500 becomes ₹2,704 in 2 years at compound interest. What is the rate per annum?

StandardTry it yourself first

₹4,000 becomes ₹4,840 in 2 years at compound interest. What is the rate per annum?

AdvancedTry it yourself first

₹8,000 becomes ₹9,261 in 3 years at compound interest. What is the rate per annum?

StandardTry it yourself first

In how many years will ₹5,000 become ₹6,655 at 10% compound interest per annum?

AdvancedTry it yourself first

In how many years will ₹2,000 become ₹3,456 at 20% compound interest per annum?

Half-yearly, quarterly and varying rates

Half the rate, twice the periods

When compounding happens more than once a year, the time has a fraction, or the rate differs each year.

Compounding	Rate per period	Periods (in $n$ years)
annual	$R$	$n$
half-yearly	$\frac{R}{2}$	$2 n$
quarterly	$\frac{R}{4}$	$4 n$

A fractional year (say 2 years 4 months): compound for the whole years, then charge simple interest on that amount for the leftover months.
A different rate each year: its own factor each year, all multiplied: $A = P (1 + \frac{R _{1}}{100}) (1 + \frac{R _{2}}{100}) \dots$ .

Example 4 · Standard

Find the quarterly compound interest on ₹8,000 at 20% per annum for 9 months.

Solution, step by step

1Rate per quarter $= 5%$ , periods $= 3$ .
2 $A = 8000 \times 1.0 5^{3} = 9261$ ⟹ $C I = 1261$

Answer ₹1,261

Example 5 · Standard

What will ₹6,000 amount to in 3 years at 5%, 10% and 20% in successive years?

Solution, step by step

1Factor $1.05 \times 1.10 \times 1.20 = 1.386$ .
2 $A = 6000 \times 1.386 = 8316$

Answer ₹8,316

BasicTry it yourself first

What is the half-yearly compound interest on ₹10,000 at 20% per annum for 1 year?

StandardTry it yourself first

What is the half-yearly compound interest on ₹16,000 at 20% per annum for 1.5 years?

StandardTry it yourself first

What is the half-yearly compound interest on ₹16,000 at 10% per annum for 2 years?

AdvancedTry it yourself first

What is the quarterly compound interest on ₹32,000 at 20% per annum for 0.75 years?

StandardTry it yourself first

What is the interest on ₹5,000 at 10% per annum compounded annually for 2 years 6 months? (Simple interest on the part year.)

AdvancedTry it yourself first

What is the interest on ₹8,000 at 10% per annum compounded annually for 1 years 9 months? (Simple interest on the part year.)

BasicTry it yourself first

₹5,000 is invested at compound interest with rates of 10%, 20% in successive years. What is the final amount?

StandardTry it yourself first

₹10,000 is invested at compound interest with rates of 5%, 10%, 20% in successive years. What is the final amount?

AdvancedTry it yourself first

₹8,000 is invested at compound interest with rates of 25%, 10%, 20% in successive years. What is the final amount?

2

Compound versus simple

Difference, relation and multiple-time

The difference of the two interests, getting one from the other, and the “how many times and when” questions.

The CI–SI difference

Interest on interest = the gap

In one year CI = SI; the 2- and 3-year difference formulas.

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How many times, and when

One multiplication, repeated

If a sum becomes $x$ times in $T$ years then in $k T$ years it becomes $x^{k}$ times.

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3

Instalments, population and depreciation

The same multiplier, a new story

Repaying a debt in equal annual instalments, and population or value rising and falling.

Equal annual instalments

The present value of each instalment

The loan = each instalment divided by the factor’s power for its year, added up.

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Population and depreciation

The same compound formula

Growing and shrinking quantities — a minus sign for a fall.

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4

Traps and fast tricks

Slips · tricks

Where marks go in compound interest.

The commonest mistakes

Six traps

The slips that cost marks.

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Fast tricks

Remembered factors, difference formulas

Habits that shorten the work.

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5

The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct rules

One-rule questions.

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Level 2 — Standard

Two ideas combined

Exam-level questions.

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Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

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6

Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.

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The formula and its uses

Annual compounding

Half-yearly, quarterly and varying rates

Compound versus simple

The CI–SI difference

How many times, and when

Instalments, population and depreciation

Equal annual instalments

Population and depreciation

Traps and fast tricks

The commonest mistakes

Fast tricks

The practice ladder

Level 1 — Basic

Level 2 — Standard

Level 3 — Advanced

Revision — formula cards

Free All Access Passevery mind map · Hindi + English · 12 months

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