Question

The simple interest and the compound interest on a sum of money at a certain rate for 2 years is Rs. 1,260 and Rs. 1,323 respectively.
Find the sum and the rate.

Answer 1

Hint: First, we will use the formula of simple interest, \[{\text{S.I.}} = \dfrac{{{\text{P}} \times {\text{T}} \times {\text{R}}}}{{100}}\], where \[{\text{P}}\] is principal starting amount of money, \[{\text{R}}\] is the interest rate per year and \[{\text{T}}\] is the time the money is invested in years and the amount when interest is compounded is \[A = {\text{P}}{\left( {1 + \dfrac{{\text{R}}}{{100}}} \right)^n}\], where \[{\text{R}}\] is the interest rate per year. Apply these formulae, and then use the given conditions to find the required value.

Complete step by step answer:
Let us assume that \[{\text{P}}\] represents the principle and R represents the rate of interest.
We are given that the simple interest and the compound interest on a sum of money at a certain rate for 2 years are Rs. 1,260 and Rs. 1,323 respectively.
We know that the formula of simple interest, \[{\text{S.I.}} = \dfrac{{{\text{P}} \times {\text{T}} \times {\text{R}}}}{{100}}\], where \[{\text{P}}\] is principal starting amount of money, \[{\text{R}}\] is the interest rate per year and \[{\text{T}}\] is the time the money is invested in years.
First, we have to find the values of \[{\text{T}}\] for the above formula of simple interest.
\[{\text{T}} = 2\]
We will now substitute the above value and simple interest to compute the simple interest using the above formula.

$   \Rightarrow 1260 = \dfrac{{{\text{P}} \times {\text{R}} \times 2}}{{100}} $
$   \Rightarrow 1260 = \dfrac{{{\text{PR}}}}{{50}}{\text{ ......eq(1)}} $
We know that the formula to calculate the amount when interest is compounded is \[A = {\text{P}}{\left( {1 + \dfrac{{\text{R}}}{{100}}} \right)^n}\], where \[{\text{R}}\] is the interest rate per year.
Substituting the value of T and A in the formula of compound interest, \[{\text{C.I.}} = {\text{A}} - {\text{P}}\], where A is the amount when interest is compounded, we get
$   \Rightarrow {\text{C.I.}} = {\text{P}}{\left( {1 + \dfrac{{\text{R}}}{{100}}} \right)^2} - {\text{P}} $
$   \Rightarrow 1323 = {\text{P}}\left( {1 + {{\left( {\dfrac{{\text{R}}}{{100}}} \right)}^2} + \dfrac{{\text{R}}}{{50}}} \right) - {\text{P}} $
$   \Rightarrow 1323 = {\text{P}} + {\text{P}}{\left( {\dfrac{{\text{R}}}{{100}}} \right)^2} + \dfrac{{{\text{PR}}}}{{50}} - {\text{P}} $
$   \Rightarrow 1323 = {\text{P}}{\left( {\dfrac{{\text{R}}}{{100}}} \right)^2} + \dfrac{{{\text{PR}}}}{{50}} $
$  \Rightarrow 1323 = \dfrac{{{\text{PR}}}}{{50}}\left( {\dfrac{{\text{R}}}{{200}} + 1} \right){\text{ ......eq.(2)}} $
Dividing the equation (2) by the equation (1), we get
\[ \Rightarrow \dfrac{{1323}}{{1260}} = 1 + \dfrac{{\text{R}}}{{200}}\]
Subtracting the above equation by 1 on both sides, we get
$   \Rightarrow \dfrac{{1323}}{{1260}} - 1 = 1 + \dfrac{{\text{R}}}{{200}} - 1 $
$   \Rightarrow \dfrac{{1323 - 1260}}{{1260}} = \dfrac{{\text{R}}}{{200}} $
$   \Rightarrow \dfrac{{63}}{{1260}} = \dfrac{{\text{R}}}{{200}} $
Multiplying the above equation by 200 on both sides, we get
$   \Rightarrow \dfrac{{63 \times 200}}{{1260}} = 200\left( {\dfrac{{\text{R}}}{{200}}} \right) $
$   \Rightarrow \dfrac{{12600}}{{1260}} = {\text{R}} $
$   \Rightarrow {\text{R = 10% }} $
Substituting the above value of R in equation (1) to find the value of principal or sum, we get
$ \Rightarrow 1260 = \dfrac{{2{\text{P}}}}{{50}} $
$   \Rightarrow 1260 = \dfrac{{\text{P}}}{{25}} $
Multiplying the above equation by 25 on both sides, we get
$ \Rightarrow 25 \times 1260 = {\text{P}} $
$   \Rightarrow {\text{P = Rs. 6,300}} $

Thus, the required sum is Rs. 6,300.

Note: In solving these types of questions, you should be familiar with the formulae of simple interest and compound interest. Students should note here that the sum of compound interest and simple interest is the principal amount. It is also important to understand in applying both the simple interest and compound interest formula accordingly. One should remember that simple interest is computed only on the principle, but the compound interest is calculated on both the accumulated interest and the principal or else the answer will be wrong.