# Simple interest for compound interest and formula

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## Simple Interest vs. Compound Interest: An Overview

Interest is the cost of borrowing money, where the borrower pays a fee to the lender for the loan. Interest, usually expressed as a percentage, can be simple or compound. Simple interest is based on the principal amount of a loan or deposit. In contrast, compound interest is based on the amount of principal and the interest accruing therein at each period. Simple interest is calculated only on the principal amount of a loan or deposit, so it’s easier to determine than compound interest.

Key points to remember

- Interest is the cost of borrowing money, where the borrower pays a fee to the lender for the loan.
- As a rule, simple interest paid or received over a certain period is a fixed percentage of the capital borrowed or loaned.
- Compound interest accumulates and is in addition to the interest accrued from previous periods, so borrowers must pay interest on the interest as well as on the principal.

## Simple interest

Simple interest is calculated using the following formula:

interest rate} &n = text{Term of loan, in years} end{aligned}”>

$$Simple interest

=

P

Ã—

r

Ã—

m

or:

P

=

The principal amount

r

=

Annual interest rate

m

=

Loan term, in years

begin {aligned} & text {Simple interest} = P times r times n & textbf {where:} & P = text {Principal amount} & r = text {Interest rate annual} & n = text {Duration of loan, in years} end {aligned}

Simple interest=PÃ—rÃ—mor:P=The principal amountr=Annual interest ratem=Loan term, in years

As a rule, simple interest paid or received over a certain period is a fixed percentage of the capital borrowed or loaned. For example, suppose a student gets a simple interest loan to pay for a year of college tuition, which costs $ 18,000, and the annual interest rate on the loan is 6%. The student repays the loan over three years. The amount of simple interest paid is:

$$

$

3

,

2

4

0

=

$

1

8

,

0

0

0

Ã—

0

.

0

6

Ã—

3

begin {aligned} & $ 3.240 = $ 18.000 times 0.06 times 3 end {aligned}

$3,240=$18,000Ã—0.06Ã—3

and the total amount paid is:

$$

$

2

1

,

2

4

0

=

$

1

8

,

0

0

0

+

$

3

,

2

4

0

begin {aligned} & $ 21,240 = $ 18,000 + $ 3,240 end {aligned}

$21,240=$18,000+$3,240

## Compound interest

Compound interest accumulates and is in addition to the interest accrued from previous periods; it includes interest on interest, in other words. The compound interest formula is:

$$

Compound interest

=

P

Ã—

(

1

+

r

)

t

–

P

or:

P

=

The principal amount

r

=

Annual interest rate

t

=

Number of years of interest applied

begin {aligned} & text {Compound interest} = P times left (1 + r right) ^ t – P & textbf {where:} & P = text {Principal amount} & r = text {Annual interest rate} & t = text {Number of years of interest applied} end {aligned}

Compound interest=PÃ—(1+r)t–Por:P=The principal amountr=Annual interest ratet=Number of years of interest applied

It is calculated by multiplying the principal amount by one plus the annual interest rate plus the number of compounded periods, then minus the principal reduction for that year. With compound interest, borrowers have to pay interest on the interest as well as on the principal.

## Examples of simple interest and compound interest

Here are some examples of simple and compound interest.

### Example 1

Suppose you put $ 5,000 into a one-year certificate of deposit (CD) that earns simple interest at 3% per annum. The interest you earn after one year would be $ 150:

$$

$

5

,

0

0

0

Ã—

3

%

Ã—

1

begin {aligned} & $ 5,000 times 3 % times 1 end {aligned}

$5,000Ã—3%Ã—1

### Example 2

Continuing with the example above, assume that your certificate of deposit is cashable at any time, with interest payable to you on a pro rata basis. If you cashed in the CD after four months, how much would you earn in interest? You will receive $ 50:

$$$

5

,

0

0

0

Ã—

3

%

Ã—

4

1

2

begin {aligned} & $ 5,000 times 3 % times frac {4} {12} end {aligned}

$5,000Ã—3%Ã—124

### Example 3

Suppose Bob borrows $ 500,000 for three years from his wealthy uncle, who agrees to charge him simple interest of 5% per annum. How much should Bob pay in interest charges each year, and what would his total interest charges be after three years? (Assuming the principal amount stays the same throughout the three years, that is, the full loan amount is paid off after three years.) Bob should pay $ 25,000 in interest charges each. year :

$$

$

5

0

0

,

0

0

0

Ã—

5

%

Ã—

1

begin {aligned} & $ 500,000 times 5 % times 1 end {aligned}

$500,000Ã—5%Ã—1

or $ 75,000 in total interest charges after three years:

$$

$

2

5

,

0

0

0

Ã—

3

begin {aligned} & $ 25,000 times 3 end {aligned}

$25,000Ã—3

### Example 4

Continuing with the example above, Bob needs to borrow an additional $ 500,000 for three years. Unfortunately, her rich uncle is kicked out. Thus, he takes out a loan from the bank at an interest rate of 5% per annum compounded annually, the total amount of the loan and the interest being payable after three years. What would be the total interest paid by Bob?

Since compound interest is calculated on principal and accrued interest, here’s how it adds up:

$$

After the first year, interest payable

=

$

2

5

,

0

0

0

,

Where

$

5

0

0

,

0

0

0

(loan principal)

Ã—

5

%

Ã—

1

After the second year, interest payable

=

$

2

6

,

2

5

0

,

Where

$

5

2

5

,

0

0

0

(Loan principal + first year interest)

Ã—

5

%

Ã—

1

After the third year, interest payable

=

$

2

7

,

5

6

2

.

5

0

,

Where

$

5

5

1

,

2

5

0

Loan principal + interest for the first year

And two)

Ã—

5

%

Ã—

1

Total interest payable after three years

=

$

7

8

,

8

1

2

.

5

0

,

Where

$

2

5

,

0

0

0

+

$

2

6

,

2

5

0

+

$

2

7

,

5

6

2

.

5

0

begin {aligned} & text {After the first year, interest payable} = $ 25,000 text {,} & text {or} $ 500,000 text {(Principal loan)} times 5 % times 1 & text {After the second year, interest payable} = $ 26,250 text {,} & text {or} $ 525,000 text {(loan principal + first year interest) } & times 5 % times 1 & text {After the third year, interest payable} = $ 27,562.50 text {,} & text {or} $ 551250 text {Loan principal + interest for the first year} & text {and Two)} times 5 % times 1 & text {Total interest payable after three years} = $ 78 812.50 text {,} & text {or} $ 25,000 + $ 26,250 + 27,562.50 $ end {aligned}

After the first year, interest payable=$25,000,Where $500,000 (loan principal)Ã—5%Ã—1After the second year, interest payable=$26,250,Where $525,000 (Loan principal + first year interest)Ã—5%Ã—1After the third year, interest payable=$27,562.50,Where $551,250 Loan principal + interest for the first yearAnd two)Ã—5%Ã—1Total interest payable after three years=$78,812.50,Where $25,000+$26,250+$27,562.50

It can also be determined using the compound interest formula above:

$$

Total interest payable after three years

=

$

7

8

,

8

1

2

.

5

0

,

Where

$

5

0

0

,

0

0

0

(loan principal)

Ã—

(

1

+

0

.

0

5

)

3

–

$

5

0

0

,

0

0

0

begin {aligned} & text {Total interest payable after three years} = $ 78 812.50 text {,} & text {or} $ 500,000 text {(Principal loan)} times ( 1 + 0.05) ^ 3 – $ 500,000 end {aligned}

Total interest payable after three years=$78,812.50,Where $500,000 (loan principal)Ã—(1+0.05)3–$500,000

This example shows how the compound interest formula is derived from paying interest on interest as well as on principal.

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