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Rate Compounded Annually Or Half Yearly (Semi Annually)

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definition

  • Conversion Period: Time period and rate when interest not compounded annually The time period after which the interest is added each time to form a new principal is called the conversion period.

notes

Rate Compounded Annually Or Half Yearly (Semi-Annually):

Time period and rate when interest not compounded annually The time period after which the interest is added each time to form a new principal is called the conversion period.

i) When compound interest is compounded annually,

Amount when interest is compounded annually = `"P"( 1 + "R"/100)^n`; 

P is principal, R is rate of interest, n is time period

P = Rs. 100 at 10% per annum compounded annually
The time period taken is 1 year 

I = ₹ `(100 xx 10 xx 1)/100` = Rs. 10

A = Rs. 100 + Rs. 10

A = Rs. 110.

ii) When compound interest is compounded Semi-annually,

When the interest is compounded half-yearly, there are two conversion periods in a year each after 6 months. In such situations, the half-yearly rate will be half of the annual rate.

Amount when interest is compounded half-yearly

= `"P"( 1 + "R"/200)^(2n)        .......{"R"/2 "is half-yearly rate and"
, 2n = "number of half-years"`

iii) When compound interest is compounded quarterly,

In this case, there are 4 conversion periods in a year and the quarterly rate will be one-fourth of the annual rate.

Amount when interest is compounded quarterly,

= `"P"(1 + "R"/400)^(4n)       ........{ "R"/4 "is half-yearly rate and, 4n = number of half-years"’`

Example

Find CI paid when a sum of Rs. 10,000 is invested for 1 year and 3 months at `8 1/2%` per annum compounded annually.

1 year 3 months = `1 3/12 "year" = 1 1/4 "years"`

A = ₹ `10000(1 + 17/200)^(1 1/4)`
Find the amount for the whole part, i.e., 1 year in this case.
A = ₹ `10000(1 + 17/200)`
A = ₹ `10000 xx 217/200 ` = ₹ 10,850
Now this would act as principal for the next `1/4` year.
SI = ₹ `(10850 xx 1/4 xx 17)/(100 xx 2)`
 
SI = ₹ `(10850 xx 1 xx 17)/(800)` = Rs. 230.56
 
Interest for first year = ₹ 10850 - ₹ 10000 = ₹ 850
And, interest for the next `1/4` year = ₹ 230.56
Therefore, total compound Interest = 850 + 230.56 = ₹ 1080.56.
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