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# Compound Interest

You have learned about the simple interest and formula for calculating simple interest and amount. Now, we shall discuss the concept of compound interest and the method of calculating the compound interest and the amount at the end of a certain specified period. We shall also study the population growth and depreciation of the value of movable and immovable assets.

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## Compound Interest

If the borrower and the lender agree to fix up an interval of time (say, a year or a half year or a quarter of a year etc) so that the amount (Principal + interest) at the end of an interval becomes the principal for the next interval, then the total interest over all the intervals, calculated in this way is called the compound interest and is abbreviated as C.I.

Compound interest at the end of a certain specified period is equal to the difference between the amount at the end of the period and original principal i.e. C.I. = Amount – Principal. In this section, we shall discuss some examples to explain the meaning and the computation of compound interest. Compound interest when interest is compounded annually.

### Example 1

Find the compound interest on Rs 1000 for two years at 4% per annum.

Solution: Principal for the first year =Rs 1000

$$SI\quad =\frac { P\times R\times T }{ 100 } \\ SI\quad for\quad 1st\quad year\quad =\frac { 1000\times 4\times 1 }{ 100 } \\ SI\quad for\quad 1st\quad year\quad =Rs\quad 40$$

Amount at the end of first year =Rs1000 + Rs 40 = Rs 1040. Principal for the second year = Rs1040

$$SI\quad for\quad 2nd\quad year\quad =\frac { 1040\times 4\times 1 }{ 100 }$$

$$SI\quad for\quad 2nd\quad year\quad =Rs\quad 41.60$$

Amount at the end of second year, $$Amount=Rs1040+Rs41.60=Rs1081.60$$

Therefore, $$Compound\quad interest=Rs(1081.60–1000)=Rs81.60$$

Remark: The compound interest can also be computed by adding the interest for each year.

### Compound Interest when Compounded Half Yearly

Example 2: Find the compound interest on Rs 8000 for 3/2 years at 10% per annum, interest is payable half-yearly.

Solution: Rate of interest = 10% per annum = 5% per half –year. Time = 3/2 years = 3 half-years

Original principal = Rs 8000. $$Interest\quad for\quad the\quad first\quad half\quad year = \frac { 8000\times 5\times 1 }{ 100 } =420$$ . Amount at the end of the first half-year= Rs 8000 +Rs 400 =Rs8400

Principal for the second half-year =Rs 8400

$$Interest\quad for\quad the\quad second\quad half-year=\frac { 8400\times 5\times 1 }{ 100 } =420$$

Amount at the end of the second half year = Rs 8400 +Rs 420 = Rs 8820

$$Interest \; for\; the\; third\; half \; year=\frac { 8820\times 5\times 1 }{ 100 } =Rs441$$

Amount at the end of third half year= Rs 8820+ Rs 441= Rs 9261. Therefore, compound interest= Rs 9261- Rs 8000= Rs 1261. Therefore, $$compound\quad interest=Rs9261-Rs8000=Rs1261$$

## Compound Interest by Using Formula

In this section, we shall obtain some formulae for the compound interest.

### Case 1

Let P be the principal and the rate of interest be R% per annum. If the interest is compounded annually, then the amount A and the compound interest C.I. at the end of n years is given by:

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

Example 3: Find the compound interest on Rs 12000 for 3 years at 10% per annum compounded annually.

Solution: P =Rs 12000, R =10% per annum and n=3. Therefore, amount (A) after 3 years

$$A=P{ \left( 1+\frac { R }{ 100 } \right) }^{ 3 } \\ A=12000{ \left( 1+\frac { 10 }{ 100 } \right) }^{ 3 }\\ A=12000{ \left( \frac { 11 }{ 10 } \right) }^{ 3 }\\ A=12000{ \left( \frac { 11 }{ 10 } \right) }\times { \left( \frac { 11 }{ 10 } \right) }\times { \left( \frac { 11 }{ 10 } \right) }\\ A=15972\\ Compound\quad interest=A-P\\ CI=Rs15972-Rs12000=Rs3972$$

### Case 2

When the interest is compounded half-yearly. $$A=P{ \left( 1+\frac { R }{ 200 } \right) }^{ 2n }\\ CI=P\left[ { \left( 1+\frac { R }{ 200 } \right) }^{ 2n }-1 \right]$$

### Case 3

When the interest is compounded quarterly. $$A=P{ \left( 1+\frac { R }{ 400 } \right) }^{ 4n }\\ CI=P\left[ { \left( 1+\frac { R }{ 400 } \right) }^{ 4n }-1 \right]$$

### Case 4

Let be the principal and the rate of interest be R1% for the first year, R2% for second year, R3% for third year and so on and last Rn% for the nth year . Then the amount (A) and the compound interest C.I. at the end of n years are given by: $$A=P{ \left( 1+\frac { { R }_{ 1 } }{ 100 } \right) }{ \left( 1+\frac { { R }_{ 2 } }{ 100 } \right) }{ \left( 1+\frac { { R }_{ 3 } }{ 100 } \right) }$$

A = P (1+R1/100)(1+R2/100)…(1+Rn/100) and

$$CI=A-P$$respectively.

### Case 5

Let p is the principal and the rate of interest is R% per annum. If the interest is compounded annually but time is the fraction of a year, say 21/4 years, then amount A is given by:

$$A=P{ { \left( 1+\frac { { R } }{ 100 } \right) }^{ 5 } }{ \left( 1+\frac { \frac { R }{ 4 } }{ 100 } \right) }$$ and $$CI = A-P$$

## More Solved Example For You

Example 4: Find the compound interest on Rs 10000 for one year at 20% per annum compounded quarterly.

Solution: Rate of interest = 20% per annum= 20/4%= 5% per quarter. Time = 1 year= 4 quarters

Principal for the first quarter= Rs 10000

$$Interest\quad for\quad the\quad first\quad quarter=\frac { 10000x5x1 }{ 100 } =Rs500$$

Amount at the end of first quarter =Rs (10000+500) = Rs 10500 , Principal for the second quarter = Rs 10500

$$Interest\quad for \quad the \quad second \quad quarter=\frac { 10500x5x1 }{ 100 }=Rs525$$

Amount at the end of second quarter= Rs 10500+ Rs 525= Rs 11025. Principal for the third quarter= Rs 11025

$$Interest\quad for\quad the\quad third\quad quarter=\frac { 11025\times 5\times 1 }{ 100 } =Rs551.25$$

Amount at the end of third quarter= Rs11025 + Rs 551.25= Rs 11576.25 . Principal for the fourth quarter= Rs 11576.25

$$Interest\; for\; the\; fourth\; quarter=\frac { 11576.25\times 5\times 1 }{ 100 } =Rs578.8125$$

Amount at the end of fourth quarter= Rs 11576.25+ Rs 578.8125= Rs 12155.0625. Therefore, compound interest= Rs 12155.0625- Rs 10000= Rs 2155.625

Question- How do you calculate compound interest?

Answer- In order to calculate Compound Interest, you need to multiply the initial principal amount by one plus the annual interest rate which we raise to the number of compound periods minus one.

Question-Where is do we use compound interest?

Answer- Banks usually pay compounded interest on deposits. It acts as an advantage for depositors and credit card holders may make use of compound interest calculations as incentives to pay off their balances swiftly.

Question- What is the compound interest formula?

Answer- The Compound Interest formula is- A = P (1 + r/n) (nt). Over here, the A is the final amount and P is the initial principal balance. Similarly, r is the interest rate and n is the no. of times interest applied per time period. Finally, t is the no. of time periods elapsed.

Question- What is a compound interest rate?

Answer- Compound interest is adding the interest to the principal sum of a loan or deposit or interest. Thus, we see that the simple annual interest rate is the interest amount per period which we multiply by the no. of periods every year.

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