Answer
438k+ views
Hint:
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,
Complete step by step answer:
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
Note:
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”.
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,
Complete step by step answer:
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
Note:
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”.
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
When people say No pun intended what does that mea class 8 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Name the states which share their boundary with Indias class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give an account of the Northern Plains of India class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Advantages and disadvantages of science
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Bimbisara was the founder of dynasty A Nanda B Haryanka class 6 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
10 examples of evaporation in daily life with explanations
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
How do you graph the function fx 4x class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)