CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon


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%
loader
B
10%
loader
C
912%
loader
D
814%
loader

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 $$3\ years=3\div \dfrac { 1 }{ 2 } =6$$ and
the 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

Suggest Corrections
thumbs-up
 
0


similar_icon
Similar questions
View More


similar_icon
People also searched for
View More



footer-image