Question

A bank advertises that you can double the money if you invest it with them for \[8\] years, what is the rate of interest offered by them?

Answer 1

Hint:In this question we are going to find the rate of interest offered by a bank thus our investment will double in \[8\] years. This question is based on the concept of interest. Additional payment made by the borrower to the lender for using the money is called Interest ( \[I\] ). The interest may be simple or compound. Let us use simple interest to solve our problem. Simple interest is the interest on a sum borrowed for a certain period is the fixed percentage of the principal amount.

Formula used:
\[S.I = \dfrac{{P \times R \times T}}{{100}}\]
where \[P\] is the principal amount, \[R\] is the rate and \[T\] is the time period.

Complete step by step answer:
In the question it is given that the time period is \[8\] years, that is, \[T = 8\].Let us assume that the amount we are going to invest is Rs. \[500\], that is, \[P = 500\].Also given that \[8\] years will take to double the investment money. Thus as per our assumption, to earn it double, interest has to be Rs. \[500\](double the money \[ = \] principal amount \[ + \] interest), that is, \[I = 500\] or\[S.I = 500\].

Let substitute these values in the formula, \[S.I. = \dfrac{{P \times R \times T}}{{100}}\] to find the rate of interest.
\[S.I. = \dfrac{{P \times R \times T}}{{100}}\]
\[500 = \dfrac{{500 \times R \times 8}}{{100}}\]
Zeroes in \[500\] and \[100\] will get cancel thus,
\[500 = \dfrac{{5 \times R \times 8}}{1}\]
Any number divided by \[1\] is the number itself therefore,
\[500 = 5 \times R \times 8\]

Now multiplying \[5\] and \[8\] we will get \[40,(5 \times 8 = 40)\], and multiplying \[40\] and \[R\] will result \[40R\].
\[500 = 40R\]
Therefore, \[R = \dfrac{{500}}{{40}}\],
A zero in \[500\] and \[40\] will get cancel,
\[R = \dfrac{{50}}{4}\]
Now by dividing \[50\] by \[4\], we will get,
\[\therefore R = 12.5\]
Rate of interest is in the form percentage, \[R = 12.5\% \].

Hence, if the bank follows simple interest to double the money if we invest it with them for \[8\] years, then the rate of interest offered by them is \[12.5\% \].

Note: We already saw that the interest may be simple or compound interest.Compound interest is the interest calculated on the principal and interest accumulated over a previous period.
\[C.I. = P\left[ {{{\left( {1 + \dfrac{R}{{100}}} \right)}^n} - 1} \right]\].
In a bank if compound interest is used, then the interest is compounded either annually or half yearly or quarterly.We can find the rate of interest using the rule of \[70\], is a way to calculate the time period taken to double the money.
\[\dfrac{{70}}{R} = T\].
Here we need to find \[R\], so let's change this formula according to \[R\] by interchanging \[T\] and \[R\].
\[R = \dfrac{{70}}{T}\], in our question the time taken to double the money is \[8\] years,
\[\therefore R = \dfrac{{70}}{8} = 8.75\% \].
Thus, if the bank follows compound interest to double the money if we invest it with them for \[8\] years, then the rate of interest offered by them is \[8.75\% \].