Question

In what time will Rs. \[{\mathbf{8}},{\mathbf{000}}\]amount to Rs. \[{\mathbf{8}},{\mathbf{820}}\]at \[{\mathbf{10}}\% \] p.a. interest compounded half-yearly?
A.2.5 Years
B.4 Years
C.3 Years
D.1 Year

Answer 1

Hint :- Rate of interest compounded half-yearly and whenever interest is compounded half-yearly time becomes double of the time when interest compounded yearly.


Complete step by step solution.
“Here, we have given :-
Principal = Rs. \[{\text{8}},000\]
Amount = Rs. \[{\text{8}},{\text{82}}0\]
and Rate of interest = \[{\text{1}}0\% \] p.a.
Amount is the sum of interest and principal.
By using above hint for half-yearly compound interest, rate of interest is the half of the rate of interest compound yearly so we will use rate of interest = \[\dfrac{{10}}{2}\% = 5\% \](half-yearly)
Now, relation between principal amount, time and rate of interest is –
\[{\text{P}}{\left( {1 + \dfrac{{\text{R}}}{{100}}} \right)^{\text{T}}} = {\text{A}}\]
Put all the values which we have given in the question
\[ \Rightarrow 8000{\left( {1 + \dfrac{5}{{100}}} \right)^{\text{T}}} = 8820\]
\[ \Rightarrow {\left( {1 + \dfrac{5}{{100}}} \right)^{\text{T}}} = \dfrac{{8820}}{{8000}}\]
\[ \Rightarrow {\left( {\dfrac{{105}}{{100}}} \right)^{\text{T}}} = \dfrac{{882}}{{800}}\]
\[ \Rightarrow {\left( {\dfrac{{21}}{{20}}} \right)^{\text{T}}} = \dfrac{{441}}{{400}}................(1)\]
We know that \[{\left( {{\text{2}}0} \right)^{\text{2}}} = {\text{4}}00\]and \[{\left( {{\text{21}}} \right)^{\text{2}}} = {\text{441}}\]
So, equation (1) becomes
\[ \Rightarrow {\left( {\dfrac{{21}}{{20}}} \right)^{\text{T}}} = {\left( {\dfrac{{21}}{{20}}} \right)^2}\]
Comparing both sides
\[{\text{T }} = {\text{ 2}}\](Half Years)
Using hint original time will be two half years = one year
So time in which Rs.\[{\text{8}},000\]amount to Rs. \[{\text{8}},{\text{82}}0\]at \[{\text{1}}0\% \]p.a. interest compounded half-yearly in one year.
So, option D is the right answer


Note- As we all know that fraction value of \[{\text{5}}\% \] is \[\dfrac{5}{{100}}i.e.\dfrac{1}{{20}}\], so now we have to find in how much time we will get one unit interest at \[{\text{2}}0\] unit principal for this we have to compare this fraction i.e. \[\left( {\dfrac{{21}}{{20}}} \right)\] with ratio of amount and principal \[{\left( {\dfrac{{21}}{{20}}} \right)^{\text{T}}} = \left( {\dfrac{{8820}}{{8000}}} \right)\]
And we will get value of T in the way.