Question

True discount on a bill is \[\text{Rs}.1800\] and banker's discount is \[{\rm{Rs}}.1887\]. If the bill is due 10 months hence, find the rate of interest.

Answer 1

Hint: Here we need to find the rate of interest. We have been given the true discount value, banker’s discount and the duration of time. We will use the formula of the banker’s discount which will be in terms of true discount, rate of interest and time period. We will substitute the value of true discount, banker’s discount and the time period and after simplifying the terms, we will get the value of the required rate of interest.

Formula used:
\[B.D = T.D\left( {1 + \dfrac{{R \times T}}{{100}}} \right)\] , where, \[B.D\] refers to the banker’s discount, \[T.D\] refers to the true discount, \[T\] refers to the time period and \[R\] refers to the rate of interest.

Complete step-by-step answer:
Here we need to find the rate of interest.
It is given that:
True discount \[=\text{Rs}.1800\]
Banker's discount \[ = {\rm{Rs}}.1887\]
Time period \[ = 10{\rm{months}}\]
We know that \[1{\rm{month}} = \dfrac{1}{{12}}{\rm{year}}\]
Therefore,
Time period \[ = 10 \times \dfrac{1}{{12}} = \dfrac{5}{6}{\rm{years}}\]
Now, we will substitute the value of true discount, banker’s discount and the time period in the formula \[B.D = T.D\left( {1 + \dfrac{{R \times T}}{{100}}} \right)\]. Therefore, we get
\[1887 = 1800\left( {1 + \dfrac{{{\rm{R}} \times \dfrac{5}{6}}}{{100}}} \right)\]
On further simplifying the terms, we get
\[ \Rightarrow 1887 = 1800\left( {1 + \dfrac{{\rm{R}}}{{120}}} \right)\]
 Now, we will divide both sides by 1800. So, we get
\[ \Rightarrow \dfrac{{1887}}{{1800}} = 1 + \dfrac{{\rm{R}}}{{120}}\]
Now, we will subtract 1 from both sides of the equation.
\[\begin{array}{l} \Rightarrow \dfrac{{1887}}{{1800}} - 1 = 1 + \dfrac{{\rm{R}}}{{120}} - 1\\ \Rightarrow \dfrac{{1887 - 1800}}{{1800}} = \dfrac{{\rm{R}}}{{120}}\end{array}\]
On subtracting the terms, we get
\[ \Rightarrow \dfrac{{87}}{{1800}} = \dfrac{{\rm{R}}}{{120}}\]
Now, multiplying 120 on both sides, we get
\[\begin{array}{l} \Rightarrow \dfrac{{87}}{{1800}} \times 120 = \dfrac{{\rm{R}}}{{120}} \times 120\\ \Rightarrow \dfrac{{87}}{{15}} = {\rm{R}}\end{array}\]
On dividing the numerator by denominator, we get
\[ \Rightarrow {\rm{R}} = 5.8\% \]
Hence, the required rate of interest is equal to \[5.8\% \].

Note: Here we have used the formula to calculate the banker’s discount. True Discount is defined as the method of calculation and collection of Interest upfront from the borrower while disbursing the loan at a specific rate of interest and at the same time, the same rate of interest is considered to calculate the interest on interest and transfer the same to the customer. The rate of interest is defined as the proportion of a loan that is charged as interest.