Question

The difference between CI and SI for $3$ years is $992$ . if the rate of interest is $10\%$ find the principal?

Answer 1

Hint: From the question given that we have to find the principal, given that the difference between CI and SI for $3$ years is $992$, and the rate of interest is $10\%$. As we know that the formula of simple interest is $S.I=\dfrac{principal\times rate\times time}{100}$ and the formula for compound interest is $C.I=principal\times {{\left( 1+rate \right)}^{time}}-principal$, from these we will get the principal.

Complete step by step solution:
From the question given the difference between CI and SI is
$\Rightarrow C.I-S.I=992$
For a time period is,
$\Rightarrow T=3$
And also, the rate of interest is,
$\Rightarrow R=10\%$
Let the principal is P
As we know that the formula for the simple interest is
$\Rightarrow S.I=\dfrac{principal\times rate\times time}{100}$
now by substituting the values in their respective positions we will get,
$\Rightarrow S.I=\dfrac{P\times 10\times 3}{100}=\dfrac{3P}{10}$
As we know that the formula for the compound interest is
$\Rightarrow C.I=principal\times {{\left( 1+rate \right)}^{time}}-principal=principal\left( {{\left( 1+rate \right)}^{time}}-1 \right)$
now by substituting the values in their respective positions we will get,
$\Rightarrow C.I=P\left( {{\left( 1+\dfrac{1}{10} \right)}^{3}}-1 \right)$
As we know that the difference between the simple interest and compound interest is
$\Rightarrow C.I-S.I=992$
now by substituting the values in their respective positions we will get,
$\Rightarrow C.I-S.I=P\left( {{\left( 1+\dfrac{1}{10} \right)}^{3}}-1 \right)-\dfrac{3P}{10}=992$
Now we will take P common from the all the terms, then we will get,
$\Rightarrow P\left( {{\left( 1+\dfrac{1}{10} \right)}^{3}}-1-\dfrac{3}{10} \right)=992$
Now by further simplification we will get,
$\Rightarrow P\left( {{\left( \dfrac{11}{10} \right)}^{3}}-1-\dfrac{3}{10} \right)=992$
 Now by further simplification we will get,
$\Rightarrow P\left( \dfrac{1331-1000-300}{1000} \right)=992$
Now by further simplification we will get,
$\Rightarrow P\left( \dfrac{31}{1000} \right)=992$
Now by further simplification we will get,
$\Rightarrow P=32000$
Therefore, the principal is $32000$

Note: Students should know the formulas of simple interest and compound interest, students should not write rate in simple interest as $R=\dfrac{1}{10}$ because already the percentage is included in the formula, so students should write rate as $R=10$.