Question

The simple interest and compound interest of a certain sum of money for 2 years are Rs 869.40 respectively. Let us calculate that sum of money and the rate of interest.​

Answer 1

$\\\;\underbrace{\underline{\sf{Understanding\;the\;Concept\;:-}}}$

Here the concept of Simple Interest and Compound Interest has been used. We see that we are given the time period We are give that SI and CI are equal. So firstly we can make equations from both and then apply values and find the answer.

Let's do it !!

____________________________________________

★ Formula Used :-

$\\\;\boxed{\sf{\pink{S.I.\;=\;\bf{\dfrac{P\;\times\;R\;\times\;T}{100}}}}}$

$\\\;\boxed{\sf{\pink{CI\;=\;\bf{Amount\;-\;Principal}}}}$

$\\\;\boxed{\sf{\pink{Amount\;=\;\bf{P\:\times\:\bigg(1\;+\;\dfrac{R}{100}\bigg)^{T}}}}}$

____________________________________________

★ Correct Question :-

The simple interest and compound interest of a certain sum of money for 2 years are Rs. 840 and Rs 869.40 respectively. Let us calculate that sum of money and the rate of interest.

____________________________________________

★ Solution :-

Given,

» Simple Interest = S.I. = Rs. 840

» Compound Interest = C.I. = Rs. 869.40

» Time Period = T = 2 year

• Let the Principal be P

• Let the Rate be R

____________________________________________

~ For the relation of S.I. ::

We know that,

$\\\;\sf{:\rightarrow\;\;S.I.\;=\;\bf{\dfrac{P\;\times\;R\;\times\;T}{100}}}$

By applying the values, we get

$\\\;\sf{:\Longrightarrow\;\;840\;=\;\bf{\dfrac{P\;\times\;R\;\times\;2}{100}}}$

$\\\;\bf{:\Longrightarrow\;\;840\;=\;\bf{\green{\dfrac{PR}{50}}}}$

____________________________________________

~ For the relationship of C.I. ::

We know that,

$\\\;\sf{:\rightarrow\;\;CI\;=\;\bf{Amount\;-\;Principal}}$

Also,

$\\\;\sf{:\rightarrow\;\;Amount\;=\;\bf{P\:\times\:\bigg(1\;+\;\dfrac{R}{100}\bigg)^{T}}}$

Combining both the equations, we get,

$\\\;\sf{:\Longrightarrow\;\;C.I.\;=\;\bf{P\:\times\:\bigg(1\;+\;\dfrac{R}{100}\bigg)^{T}\;-\;P}}$

Taking P in common, we get

$\\\;\sf{:\Longrightarrow\;\;869.40\;=\;\bf{P\:\times\:\bigg[\bigg(1\;+\;\dfrac{R}{100}\bigg)^{2}\;-\;1\bigg]}}$

$\\\;\sf{:\Longrightarrow\;\;869.40\;=\;\bf{P\:\times\:\bigg[\bigg(\dfrac{(100\;+\;R)}{100}\bigg)^{2}\;-\;1\bigg]}}$

By using the identity of : (a + b)² = a² + b² + 2ab

$\\\;\sf{:\Longrightarrow\;\;869.40\;=\;\bf{P\:\times\:\bigg[\bigg(\dfrac{(R^{2}\;+\;10000\;+\;200R)}{10000}\bigg)\;-\;1\bigg]}}$

$\\\;\sf{:\Longrightarrow\;\;869.40\;=\;\bf{P\:\times\:\bigg[\bigg(\dfrac{R^{2}}{10000}\;+\;\dfrac{10000}{10000}\;+\;\dfrac{200R}{10000}\bigg)\;-\;1\bigg]}}$

$\\\;\sf{:\Longrightarrow\;\;869.40\;=\;\bf{P\:\times\:\bigg[\bigg(\dfrac{R^{2}}{10000}\;+\;1\;+\;\dfrac{R}{50}\bigg)\;-\;1\bigg]}}$

Now taking R, in common we get,

$\\\;\sf{:\Longrightarrow\;\;869.40\;=\;\bf{P\:\times\:\bigg[R\bigg(\dfrac{R}{10000}\;+\;\dfrac{1}{50}\bigg)\;+\;1\;-\;1\bigg]}}$

$\\\;\sf{:\Longrightarrow\;\;869.40\;=\;\bf{P\:\times\:\bigg[R\bigg(\dfrac{R}{10000}\;+\;\dfrac{1}{50}\bigg)\;+\;1\;-\;1\bigg]}}$

Cancelling 1, we get

$\\\;\sf{:\Longrightarrow\;\;869.40\;=\;\bf{PR\:\times\:\bigg[\dfrac{R}{10000}\;+\;\dfrac{1}{50}\bigg]}}$

$\\\;\bf{:\Longrightarrow\;\;869.40\;=\;\bf{\green{\dfrac{PR}{50}\:\times\:\bigg[\dfrac{R}{200}\;+\;1\bigg]}}}$

____________________________________________

~ For the value of R ::

Dividing both C.I. by S.I., we get

$\\\;\sf{\orange{\mapsto\;\;\dfrac{C.I.}{S.I.}\;=\;\bf{\dfrac{869.40}{840}}}}$

$\\\;\sf{\mapsto\;\;\dfrac{\dfrac{PR}{50}\:\times\:\bigg[\dfrac{R}{200}\;+\;1\bigg]}{\dfrac{PR}{50}}\;=\;\bf{\dfrac{869.40}{840}}}$

Cancelling the like terms, we get

$\\\;\sf{\mapsto\;\;\dfrac{\bigg[\dfrac{R}{200}\;+\;1\bigg]}{1}\;=\;\bf{\dfrac{869.40}{840}}}$

$\\\;\sf{\mapsto\;\;\dfrac{R}{200}\;+\;1\;=\;\bf{1.035}}$

$\\\;\sf{\mapsto\;\;\dfrac{R}{200}\;=\;\bf{1.035\;-\;1}}$

$\\\;\sf{\mapsto\;\;\dfrac{R}{200}\;=\;\bf{0.035}}$

$\\\;\sf{\mapsto\;\;R\;=\;\bf{200\;\times\;0.035}}$

$\\\;\sf{\mapsto\;\;R\;=\;\bf{2\;\times\;3.5}}$

$\\\;\sf{\mapsto\;\;R\;=\;\bf{\purple{7\;\%\;p.a.}}}$

$\\\;\underline{\boxed{\tt{Hence,\;\;Rate\;\;=\;\bf{\blue{7\;\%\;p.a.}}}}}$

____________________________________________

~ For the Principal Sum ::

From the formula of S.I. and value of R, we get

$\\\;\sf{:\rightarrow\;\;S.I.\;=\;\bf{\dfrac{P\;\times\;R\;\times\;T}{100}}}$

$\\\;\sf{:\rightarrow\;\;840\;=\;\bf{\dfrac{P\;\times\;7\;\times\;2}{100}}}$

$\\\;\sf{:\rightarrow\;\;840\;=\;\bf{\dfrac{P\;\times\;14}{100}}}$

$\\\;\sf{:\rightarrow\;\;P\;=\;\bf{\dfrac{840\;\times\;100}{14}}}$

$\\\;\sf{:\rightarrow\;\;P\;=\;\bf{60\;\times\;100}}$

$\\\;\sf{:\rightarrow\;\;P\;=\;\bf{\red{Rs.\;\;6000}}}$

$\\\;\underline{\boxed{\tt{Hence,\;\;Principal\;\;sum\;\;=\;\bf{\blue{Rs.\;\;6000}}}}}$

Answer 2

Step-by-step explanation:

$\\\;\underbrace{\underline{\sf{Understanding\;the\;Concept\;:-}}}$

Here the concept of Simple Interest and Compound Interest has been used. We see that we are given the time period We are give that SI and CI are equal. So firstly we can make equations from both and then apply values and find the answer.

Let's do it !!

____________________________________________

★ Formula Used :-

$\\\;\boxed{\sf{\pink{S.I.\;=\;\bf{\dfrac{P\;\times\;R\;\times\;T}{100}}}}}$

$\\\;\boxed{\sf{\pink{CI\;=\;\bf{Amount\;-\;Principal}}}}$

$\\\;\boxed{\sf{\pink{Amount\;=\;\bf{P\:\times\:\bigg(1\;+\;\dfrac{R}{100}\bigg)^{T}}}}}$

____________________________________________

★ Correct Question :-

The simple interest and compound interest of a certain sum of money for 2 years are Rs. 840 and Rs 869.40 respectively. Let us calculate that sum of money and the rate of interest.

____________________________________________

★ Solution :-

Given,

» Simple Interest = S.I. = Rs. 840

» Compound Interest = C.I. = Rs. 869.40

» Time Period = T = 2 year

Let the Principal be P

Let the Rate be R

____________________________________________

~ For the relation of S.I. ::

We know that,

$\\\;\sf{:\rightarrow\;\;S.I.\;=\;\bf{\dfrac{P\;\times\;R\;\times\;T}{100}}}$

By applying the values, we get

$\\\;\sf{:\Longrightarrow\;\;840\;=\;\bf{\dfrac{P\;\times\;R\;\times\;2}{100}}}$

$\\\;\bf{:\Longrightarrow\;\;840\;=\;\bf{\green{\dfrac{PR}{50}}}}$

____________________________________________

~ For the relationship of C.I. ::

We know that,

$\\\;\sf{:\rightarrow\;\;CI\;=\;\bf{Amount\;-\;Principal}}$

Also,

$\\\;\sf{:\rightarrow\;\;Amount\;=\;\bf{P\:\times\:\bigg(1\;+\;\dfrac{R}{100}\bigg)^{T}}}$

Combining both the equations, we get,

$\\\;\sf{:\Longrightarrow\;\;C.I.\;=\;\bf{P\:\times\:\bigg(1\;+\;\dfrac{R}{100}\bigg)^{T}\;-\;P}}$

Taking P in common, we get

$\\\;\sf{:\Longrightarrow\;\;869.40\;=\;\bf{P\:\times\:\bigg[\bigg(1\;+\;\dfrac{R}{100}\bigg)^{2}\;-\;1\bigg]}}$

$\\\;\sf{:\Longrightarrow\;\;869.40\;=\;\bf{P\:\times\:\bigg[\bigg(\dfrac{(100\;+\;R)}{100}\bigg)^{2}\;-\;1\bigg]}}$

By using the identity of : (a + b)² = a² + b² + 2ab

$\\\;\sf{:\Longrightarrow\;\;869.40\;=\;\bf{P\:\times\:\bigg[\bigg(\dfrac{(R^{2}\;+\;10000\;+\;200R)}{10000}\bigg)\;-\;1\bigg]}}$

$\\\;\sf{:\Longrightarrow\;\;869.40\;=\;\bf{P\:\times\:\bigg[\bigg(\dfrac{R^{2}}{10000}\;+\;\dfrac{10000}{10000}\;+\;\dfrac{200R}{10000}\bigg)\;-\;1\bigg]}}$

$\\\;\sf{:\Longrightarrow\;\;869.40\;=\;\bf{P\:\times\:\bigg[\bigg(\dfrac{R^{2}}{10000}\;+\;1\;+\;\dfrac{R}{50}\bigg)\;-\;1\bigg]}}$

Now taking R, in common we get,

$\\\;\sf{:\Longrightarrow\;\;869.40\;=\;\bf{P\:\times\:\bigg[R\bigg(\dfrac{R}{10000}\;+\;\dfrac{1}{50}\bigg)\;+\;1\;-\;1\bigg]}}$

$\\\;\sf{:\Longrightarrow\;\;869.40\;=\;\bf{P\:\times\:\bigg[R\bigg(\dfrac{R}{10000}\;+\;\dfrac{1}{50}\bigg)\;+\;1\;-\;1\bigg]}}$

Cancelling 1, we get

$\\\;\sf{:\Longrightarrow\;\;869.40\;=\;\bf{PR\:\times\:\bigg[\dfrac{R}{10000}\;+\;\dfrac{1}{50}\bigg]}}$

$\\\;\bf{:\Longrightarrow\;\;869.40\;=\;\bf{\pink{\dfrac{PR}{50}\:\times\:\bigg[\dfrac{R}{200}\;+\;1\bigg]}}}$

____________________________________________

~ For the value of R ::

Dividing both C.I. by S.I., we get

$\\\;\sf{\orange{\mapsto\;\;\dfrac{C.I.}{S.I.}\;=\;\bf{\dfrac{869.40}{840}}}}$

$\\\;\sf{\mapsto\;\;\dfrac{\dfrac{PR}{50}\:\times\:\bigg[\dfrac{R}{200}\;+\;1\bigg]}{\dfrac{PR}{50}}\;=\;\bf{\dfrac{869.40}{840}}}$

Cancelling the like terms, we get

$\\\;\sf{\mapsto\;\;\dfrac{\bigg[\dfrac{R}{200}\;+\;1\bigg]}{1}\;=\;\bf{\dfrac{869.40}{840}}}$

$\\\;\sf{\mapsto\;\;\dfrac{R}{200}\;+\;1\;=\;\bf{1.035}}$

$\\\;\sf{\mapsto\;\;\dfrac{R}{200}\;=\;\bf{1.035\;-\;1}}$

$\\\;\sf{\mapsto\;\;\dfrac{R}{200}\;=\;\bf{0.035}}$

$\\\;\sf{\mapsto\;\;R\;=\;\bf{200\;\times\;0.035}}$

$\\\;\sf{\mapsto\;\;R\;=\;\bf{2\;\times\;3.5}}$

$\\\;\sf{\mapsto\;\;R\;=\;\bf{\purple{7\;\%\;p.a.}}}$

$\\\;\underline{\boxed{\tt{Hence,\;\;Rate\;\;=\;\bf{\blue{7\;\%\;p.a.}}}}}$

____________________________________________

~ For the Principal Sum ::

From the formula of S.I. and value of R, we get

$\\\;\sf{:\rightarrow\;\;S.I.\;=\;\bf{\dfrac{P\;\times\;R\;\times\;T}{100}}}$

$\\\;\sf{:\rightarrow\;\;840\;=\;\bf{\dfrac{P\;\times\;7\;\times\;2}{100}}}$

$\\\;\sf{:\rightarrow\;\;840\;=\;\bf{\dfrac{P\;\times\;14}{100}}}$

$\\\;\sf{:\rightarrow\;\;P\;=\;\bf{\dfrac{840\;\times\;100}{14}}}$

$\\\;\sf{:\rightarrow\;\;P\;=\;\bf{60\;\times\;100}}$

$\\\;\sf{:\rightarrow\;\;P\;=\;\bf{\red{Rs.\;\;6000}}}$

$\\\;\underline{\boxed{\tt{Hence,\;\;Principal\;\;sum\;\;=\;\bf{\blue{Rs.\;\;6000}}}}}$