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A) Rs \[10800\]for 3 years \[12\dfrac{1}{2}\% \]per annum compounded annually.

B) Rs. \[18000\]for 2 years \[2\dfrac{1}{2}\% \]year at \[10\% \] per annum compounded annually.

C) Rs \[62500\]for \[1\dfrac{1}{2}\% \] year at \[8\% \] per annum compounded annually.

D) Rs \[8000\] for 1 year at \[9\% \] per annum compounded half annually (you could use the year by year Calculating using Si formula to verify)

E) Rs \[10000\]for 1 year at \[8\% \] per annum compounded half-yearly.

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Compound Interest:- Compound interest is the interest calculated on the principal and the interest accumulated over the previous period. It is different from the simple interest where interest is not added to the principal while calculating the interest during the next period. Compound interest finds its usage in most of the transactions in the banking and finance sectors and also in other areas as well. Some of its applications are:

Increase or decrease in population

The growth of bacteria

Rise or depreciation in the value of an item.

Compound interest formula:-

The compound interest formula is given below:

Compound interest \[ = \]Amount\[ - \] principal

Where the amount is given by:

\[A = P{\left[ {1\dfrac{R}{{100}}} \right]^t}\]

Where

A\[ = \] amount

P\[ = \]Principal

R\[ = \]rate of interest

t\[ = \]interest is compounded per year amount is compound annually.

Compound interest when the rate is compounded half-yearly:

\[{R^1} = \dfrac{R}{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,{T^1} = 2T\]

\[{R^1} = \dfrac{R}{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,{T^1} = 2T\]

When the rate is compounded half-yearly we divided the rate by 2 and multiply the time by 3 before using the general formula for the amount in case of compound interest.

Therefore,

For the first statement:-

Given

P\[ = \] \[C.I = 4577.34\]Rs

P\[ = \]3 years

R\[ = \]\[12\dfrac{1}{2}\% \,\,\,\, = \dfrac{{25}}{2}\% \]

We know the formula to calculate he amount is

\[A = P{\left[ {1\dfrac{R}{{100}}} \right]^t}\]

And calculating the CI is

CI\[ = \]A-P

Now putting the value in the formula-

\[A = 10800{\left[ {1 + \dfrac{{25/2}}{{100}}} \right]^3}\]

\[ = 10800{\left[ {1 + \dfrac{{25}}{{100}}} \right]^3}\]

\[ = 10800{\left[ {\dfrac{{8 + 1}}{8}} \right]^2}\]

\[ = 10800 \times \dfrac{9}{8} \times \dfrac{9}{8} \times \dfrac{9}{8}\]

\[ = \dfrac{{2700 \times 729}}{{64 \times 2}}\]

\[ = \dfrac{{1968300}}{{128}}\]

\[A = 15377.34\]Rs

CI\[ = \]A-P

\[ = 15377.34 - 10800\]

\[C.I = 4577.34\]Rs

b. for the second statement: -

Given

\[P = 18000\]Rs

For calculating the amount we the formula

\[A = P{\left[ {1\dfrac{R}{{100}}} \right]^t}\]

Putting the values in the formula

\[{T^1} = 2\]

\[ = 18000{\left[ {\dfrac{{11}}{{10}}} \right]^2}\]

\[ = 18000 \times \dfrac{{121}}{{100}}\]

\[ = 18000 \times 121\]

\[A = 21780\]Rs

C.I=A-P

\[ = 21780 - 18000\]

\[C.I = 3780\]Rs

Now for \[t = \dfrac{1}{2}\]

\[p = 21780\]Rs \[R = 10\% \]\[T = \dfrac{1}{2}\]

\[S.I = \dfrac{{P \times R \times t}}{{100}}\]

\[ = \dfrac{{21780 \times 10}}{{100}} \times \dfrac{1}{2}\]

\[S.I = 1089\]

Total interest\[ = C.I. + S.I\]

\[ = 3780 + 1089 = 4869\]Rs

\[A = P + I\]

\[ = 18000 + 4869\]

\[A = 22869\]Rs

C. For the third statement we have

\[p = 62500\] Rs

Now we have to found interest in half yearly

So we divide the rate by 2

\[R = 8\% \]

\[R = \dfrac{8}{2} = 4\% \]

And multiply the time by 2

\[T = \dfrac{3}{2}\,\,\,\,{T^1} = \dfrac{3}{2} \times 2 = 3\] years

Now the amount is

\[A = P{\left[ {1\dfrac{{{R^1}}}{{100}}} \right]^{{t^1}}}\]

Putting the values in he formula

\[ = 62500{\left[ {1 + \dfrac{4}{{100}}} \right]^3}\]

\[ = 62500{\left[ {\dfrac{{26}}{{25}}} \right]^3}\]

\[ = 62500 \times \dfrac{{26}}{{25}} \times \dfrac{{26}}{{25}} \times \dfrac{{26}}{{25}}\]

\[A = 70304\]Rs

\[C.I = A - P\]

\[ = 70304 - 62500\]

\[C.I = 7804\]Rs

D. For the fourth statement

Given

\[P = 8000\] Rs

\[R = 9\% \]

For half yearly we divided the rate by 2

\[{R^1} = \dfrac{9}{2} = 4.5\% \]

\[T = 1\]

By multiply the time by 2

\[{T^1} = 2\]

For calculating the amount we have

\[A = P{\left[ {1\dfrac{{{R^1}}}{{100}}} \right]^{{t^1}}}\]

\[ = 8000{\left[ {1\dfrac{{4.5}}{{1000}}} \right]^2}\]

\[ = 8000{\left[ {\dfrac{{1045}}{{1000}}} \right]^2}\]

\[ = 8000 \times \dfrac{{1045}}{{1000}} \times \dfrac{{1045}}{{1000}}\]

\[ = \dfrac{{2 \times 209 \times 209}}{{10}}\]

\[A = 8736.2\]Rs

\[C.I = A - P\]

\[ = 8736.2 - 8000\]

\[C.I = 736.2\]

For the fifth statement

Given,

\[p = 10000\]

For half yearly we multiply the time by 2 and divide the rate by 2 a so we get

\[R = 8\% \Rightarrow {R^1} = \dfrac{8}{2} = 4\% \]

\[T = 1\,year\, \Rightarrow {T^1} = 2 \times 1 = 2\]

For calculating the anoun we have the formula

\[C.I = 816\]

\[ = 10000{\left[ {1\dfrac{4}{{100}}} \right]^2}\]

\[ = 10000{\left[ {\dfrac{{25}}{{25}}} \right]^2}\]

\[ = 10000 \times \dfrac{{26}}{{25}} \times \dfrac{{26}}{{25}}\]

\[A = 10816\]Rs

\[C.I = A - P = 10816 - 10000\]

\[C.I = 816\]Rs

From the data, it is clear that the interest rate for the first year in compound interest is the same as that in case of simple interest i.e

Other than the first year, the interest compound annually is always greater than that in case of simple interest.

Do not confuse between simple interest and compound interest. Simple interest is generally used or applied to the short-term loans, usually, one year or less, that are administered by financial companies on the other hand compound interest is the interest that is calculated on the principal and the interest that is accumulated over the previous tenure.

Thus, the C.I. is also called as “interest on interest”.