# Class-8 Compound Interest

Introduction to Compound Interest

Deducing a Formula for Compound Interest

Compound Interest for Different Interest Rate

Compound Interest When The Total Time Is Not a Complete Number of Conversion Periods

## Introduction to Compound Interest

Bank, post office, insurance companies calculate the interest for one year and then yearly interest is added to the principal. The amount we get becomes the principal for second year interest calculation. This process is repeated until the amount is calculated for the whole loan period.

Thus, the difference between the final amount and the original principal is called the compound interest. It is written as C.I.

**C.I = A − P**

**Note: The principal remains constant for the whole loan period in case of simple interest calculation, but the principal keeps on changing every year for compound interest calculation.**

**Example 1.** Calculate the compound interest on 20,000 rupees for 2years at 4% per annum.

**Solution.** Here, rate of interest = 4%, principal for the first year = Rs. 20,000.

Interest for the first year = 20,000 × 4 × ^{1}⁄_{100}

= Rs. 800

Amount at the end of first year = 20,000 + 800 = Rs. 20,800.

Principal for the second year = Rs. 20,800

Interest for the second = 20,800 × 4 × ^{1}⁄_{100} = Rs. 832

Amount at the end of second year = 20,800 + 832 = Rs. 21,632

Compound interest for 2 years = final amount − original

= 21,632 − 20,000 = Rs. 1632.

**Example 2.** Mohan invests 15000 rupees for 3 years at 5% per annum compound interest in bank. Find out the compound interest for the second year and for the 3rd year.

**Solution.** Here, rate of interest = 5%

Principal for the first year = Rs. 15,000.

Interest for the first year = 15,000 × 5 × ^{1}⁄_{100}

= Rs. 750

Amount at the end of first year = 15,000 + 750

= Rs. 15,750

Principal for the second year = Rs. 15,750

Interest for the second = 15,750 × 5 × ^{1}⁄_{100}

= Rs. 787.50

Compound interest for the second year = Rs. 787.50

Amount at the end of second year = 15,750 + 787.50

= Rs. 16,537.50

Principal for the third year = Rs. 16,537.50

Interest for the third year = 16,537.50 × 5 × ^{1}⁄_{100}

= Rs. 826. 87

Compound interest for the third year = Rs. 826.87.

**Example 3.** Calculate the simple interest and compound interest on 20,000 rupees for 2 years at 5% per annum.

**Solution.** For simple interest, Principal for the first year = Rs. 20,000

Rate of interest = 5% per annum

Interest for first year = 20,000 × 5 × ^{1}⁄_{100}

= Rs. 1000

Amount at the end of first year = 20,000 + 1000

= Rs. 21,000

Principal for the second year = Rs. 21,000

Interest for the second = 20,000 × 5 × ^{1}⁄_{100}

= Rs. 1000

Amount end of second year = 21,000 + 1000

= Rs. 22000

Interest earned by simple interest in 2 years = Rs. 22000 − Rs. 20,000

= Rs. 2000

For, compound interest principal for the first year = Rs. 20,000

Rate of interest = 5% per annum

Interest for first year = 20,000 × 5 × ^{1}⁄_{100}

= Rs. 1000

Amount at the end of first year = 20,000 + 1000

= Rs. 21,000

Principal for the second year = Rs. 21,000

Interest for the second = 21,000 × 5 × ^{1}⁄_{100}

= Rs. 1050

Amount end of second year = 21,000 + 1050

= Rs. 22050

Interest earned by compound interest in 2 years = 22050 − 20,000

= Rs. 2050

Now according to question, difference of S.I and C.I is = 2050 − 2000

= Rs. 50

## Deducing a Formula for Compound Interest

To find out compound interest in shorter way, following formulae we should use. Let's discuss how formulae formed.

Suppose a sum of **P** rupees is compounded annually at a rate of **R**% per annum for **n** year. The amount at the end of **n** years must be calculated by using formulae

**A = P(1 + ^{R}⁄_{100})^{n}**

**Compound Interest (C.I) = A − P**

Let's see some solved examples for better understanding

**Example 1.** What principal of money will amount to Rs. 2205 in two years at 7% per annum compound interest?

**Solution.** Here, A = Rs. 2205, R= 5%, n = 2 years

A = P(1 + ^{R}⁄_{100})^{n}

⇒ 2205 = P(1 + ^{5}⁄_{100})^{2}

⇒ 2205 = P(^{21}⁄_{20}) × (^{21}⁄_{20})

⇒ P = 2205(^{21}⁄_{20}) × (^{21}⁄_{20})

= Rs. 2000

Hence the principal is Rs. 2000.

**Example 2.** Find compound interest on 15,500 rupees for 2 years at 10% per annum.

**Solution.** Given P = Rs. 15,500, T = 2 years, R = 10%

Here, we first find out Amount (A) = 15,500 (1 + ^{10}⁄_{100})^{2}

= 15,500 (1 + ^{1}⁄_{10})^{2}

= 15,500 × ^{11}⁄_{10} × ^{11}⁄_{10}

= Rs. 18,755

C.I = A − P = 18,755 − 15,500 = Rs. 3,255

Hence, the compound interest is Rs. 3,255.

## Conversion Period

The interest is added every time after a specified period to form a new principal amount. This period is known as conversion period. If interest is compounded annually, then one conversion period is considered in one year. If interest is compounded semi-annually, then two conversion period is considered in one year. Similarly, if interest is compounded quarterly, then four conversion periods is considered in a year.

**Example 1.** Calculate the amount should be paid at the end of 3 months on Rs. 2800 at 5% per annum compounded quarterly.

**Solution.** As rate of interest is 5% per annum

Then, rate of interest quarterly is = ^{5}⁄_{4}%

As the amount to be paid in 6 month, so, n = 4 quarters

According to question, A = P(1 + ^{R}⁄_{100})^{n}

= 2800{1 + (^{(5⁄4)}⁄_{100})}^{4}

= 2800 (1 + ^{5}⁄_{400})^{4}

= 2800(1 + ^{1}⁄_{80})^{4}

= 2800 × (^{81}⁄_{80}) × (^{81}⁄_{80}) × (^{81}⁄_{80}) × (^{81}⁄_{80})

= Rs. 2942.64

Hence, 2942.64 rupees to be paid at the end of 3 months.

## Compound Interest for Different Interest Rate

When the rates of interest for the successive conversion periods are R_{1}%, R_{2}%, R_{3}%, ...and principal is **P**. Then, amount **A** is calculated by using the formulae.

A = P(1 + ^{R1}⁄_{100}) × (1 + ^{R2}⁄_{100}) × (1 + ^{R3}⁄_{100}) × ... so on

And the calculate compound interest (C.I) = A − P.

**Example 1.** How much will Rs. 20,000 amount in 2 years at compound interest, if the rates for the successive years are 5% and 6% per year?

**Solution.** P = Rs. 20,000, R_{1}= 5% and R_{2}= 6%

A = P(1 + ^{R1}⁄_{100}) × (1 + ^{R2}⁄_{100})

= 20,000(1 + ^{5}⁄_{100}) × (1 + ^{6}⁄_{100})

= Rs. 1,11300

## Compound Interest When The Total Time Is Not a Complete Number of Conversion Periods

If total time period is not a complete number of conversion period, then we should calculate the amount as simple interest for partial time period.

Consider time is given 2 years 4 months and interest is R% per annum compounded annually, then amount A is calculated by using formulae.

A = P(1 + ^{R}⁄_{100})^{2} × [1 + (^{4}⁄_{12}) × ^{R}⁄_{100}]

Then find out compound interest using C.I = A − P

**Example 1.** Calculate the amount and compound interest on Rs. 25,000 compounded annually for 3^{1}⁄_{2} years at 2% per annum.

**Solution.** Here P = Rs. 25000, R = 2%, T = 3^{1}⁄_{2} years

A = 25000(1 + ^{2}⁄_{100})^{3} × {1 + (^{6}⁄_{12}) × ^{2}⁄_{100}}

= 25000 × (^{51}⁄_{100})^{3} × {1 + (^{1}⁄_{100})}

= 25000 × (^{51}⁄_{100}) × (^{51}⁄_{100}) × (^{51}⁄_{100}) × (^{101}⁄_{100})

= Rs. 26795.50

Then calculating Compound interest C.I = A − P

C.I = 26795.50 − 25000 = Rs. 1795.50

## Class-8 Compound Interest Worksheet

Compound Interest Worksheet - 1

Compound Interest Worksheet - 2

## Answer Sheet

**Compound-Interest-Answer**Download the pdf

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