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Maths / Comparing Quantities / Compound Interest Compounded Annually

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Compound Interest Compounded Annually

Definition: Compound interest is calculated as a percentage as a total amount a the end of the previous compounding period.

Formula:

First Type: $F_{v} = P_{v}(1 + i )^{n}$ ,

where $F_{v}$ = Future Value

$P_{v}$ = Present Value or Original Amount

$i$ = Annual Interest Rate as a Decimal

$(i + 1)$ = multiplier

n = number of years of investment

Illustration: What will Rs 5000 invested at 8% p.a compound interest amount to after 2 years ?

Solution : An interest rate of 8% indicates that i = 0.08.

For 2 years , n = 2 and so $F_{v} = P_{v}(1 + i )^{n}$ = Rs 5000 $times$ $(1.08)^{2}$ = Rs 5832

Second Type:

To find interest only, we use

Compound interest = $F_{v}$ - $P_{v}$

Illustration: How much interest is earned according to the above example1 .

Solution : interest earned = $ 5832 - $ 5000 = $ 832

now , To Find Compound Interest (CI) when Interest is Compounded Annually:

In such cases where interest is compounded yearly, the interest accrued during the first year is added to the principal and the amount so obtained becomes the principal for the second year. The amount at the end of the second year becomes the principal for the third year, and so on.

Illustration: Find the compound interest on Rs 25000 for 3 years at 6% per annum, compounded annually.

Solution: Principal for the first year = Rs. 25000.

$Interest ;for; the; first; year = frac{25000;X;6;X;1}{100}= Rs. 1500.$

Amount at the end of the first year = 25000 + 1500 = Rs 26500.

Principal for the second year = Rs. 26500.

$Interest ;for; the; second; year = frac{26500;X;6;X;1}{100}= Rs. 1590.$

Amount at the end of the second year = Rs 26500 + 1590 = Rs 28090.

Principal for the third year = Rs 28090.

$Interest ;for; the; third; year = frac{28090;X;6;X;1}{100}= Rs. 1685.40$

Amount at the end of the third year = 28090 + 1685.40 = Rs 29775.40.

Therefore, compound interest = 29775.40 - 25000 = Rs 4775.40.

Compound Interest By Formula:

$Amount= Principalleft [1+frac{Rate}{100} right ] ^{time}$

$Compound ;Interest= Amount- Principal$

For the above example:

Principal = 25000

Rate= 6%

time= 3 years

$Amount= Principalleft [1+frac{Rate}{100} right ] ^{time}$

$Amount= 25000left [1+frac{6}{100} right ] ^{3}$

$= 25000left [1+frac{3}{50} right ] ^{3}$

$= 25000left [frac{53}{50} right ] ^{3}$

$= 25000 X ; frac{53}{50};X; frac{53}{50};X; frac{53}{50}$ $= frac{53;X;53;X; 53}{5}=frac{148877}{5}=29775.4$

Therefore, compound interest = 29775.40 - 25000 = Rs 4775.40.

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