  Question

Sanju puts equal amount of money, one at $$10\%$$ per annum compound interest payable half-yearly and the second at a certain rate per annum compound interest payable yearly. If he gets equal amounts after $$3$$ years what is the value of the second rate percent?

A
1014%  B
10%  C
912%  D
814%  Solution

The correct option is C $$\displaystyle 10\frac{1}{4}\%$$Given that, Sanju puts the same amount of money, one at $$10\%$$ per annum compound interest, compounded half-yearly, and other at some interest per annum compound interest, compounded annually.He receives the same amount after $$3$$ years.To find out: The rate of interest for the second investment.Let the principal amount be $$Rs.\ x$$. And the amount received after $$3$$ years be $$A$$Also, let the rate of interest in the second case be $$y\%$$ per annum.In the first case, the payment is made half-yearly at $$10\%$$.So the number of time in $$3\ years=3\div \dfrac { 1 }{ 2 } =6$$ andthe rate $$=10\%\div 2=5\%$$In CI, we know that the amount $$(A)=P{ \left( 1+\dfrac { R }{ 100 } \right) }^{ T }$$Here, $$P=x,\ R=5\%\ and \ T=6$$$$\therefore \ A=x{ \left( 1+\dfrac { 5 }{ 100 } \right) }^{ 6 }$$.Now, in the second case, $$P=x,\ R=y\%\ and \ T=3$$$$\therefore \ A=x{ \left( 1+\dfrac { y }{ 100 } \right) }^{ 3 }$$.Given that, the amount received in both cases is the same.Hence, $$x{ \left( 1+\dfrac { y }{ 100 } \right) }^{ 3 }=x{ \left( 1+\dfrac { 5 }{ 100 } \right) }^{ 6 }$$$$\Rightarrow { \left( 1+\dfrac { y }{ 100 } \right) }^{ 3 }={ \left\{ { { \left( 1+\dfrac { 5 }{ 100 } \right) } }^{ 2 } \right\} }^{ 3 }\\$$$$\Rightarrow { \left( 1+\dfrac { y }{ 100 } \right) }={ { \left( 1+\dfrac { 5 }{ 100 } \right) } }^{ 2 }=\dfrac { 441 }{ 400 }$$$$\Rightarrow \dfrac { y }{ 100 } =\dfrac { 441 }{ 400 } -1$$$$\Rightarrow \dfrac{y}{100}=\dfrac { 41 }{ 400 }$$$$\therefore \ y=\dfrac { 41 }{ 4 }=10\dfrac14\%$$Hence, the required rate of interest for the second case is $$10\dfrac14\%$$.Mathematics

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