Question

Mr Arora borrowed \[Rs\,40,960\] from a bank to start a play school. If the bank charges \[12\dfrac{1}{2}\% \] per annum compounded half-yearly, what amount will he have to pay after \[1\dfrac{1}{2}\] years?

Answer 1

Hint: We are asked what amount Mr Arora has to pay if the interest is made on the basis of half yearly. For this first convert the parameters in half year basis. Recall the formula for compound interest and using the given values find out the required amount.

Complete step-by-step answer:
Given, Principal amount \[P = Rs\,40,960\]
Rate of interest \[ = 12\dfrac{1}{2}\% = \left( {12 + \dfrac{1}{2}} \right)\% = \dfrac{{25}}{2}\% \]
Time \[ = 1\dfrac{1}{2} = 1 + \dfrac{1}{2} = \dfrac{3}{2}\] years
It is given that the amount is compounded half yearly, so we need to convert the terms in the form of per half yearly.
Rate of interest will be now, \[R = \dfrac{{25}}{2} \times \dfrac{1}{2}\% = \dfrac{{25}}{4}\% \]
Time is \[\dfrac{3}{2}{\text{years}}\] , which we can say as \[3\] half years, in terms of half year \[T = 3\] .
We have the formula for compound interest as,
 \[{\text{amount}} = P\left[ {{{\left( {1 + \dfrac{R}{{100}}} \right)}^T}} \right] \]
Putting the values of \[P\] , \[R\] and \[T\] , we have
 \[{\text{amount}} = 40960\left[ {{{\left( {1 + \dfrac{{\dfrac{{25}}{4}}}{{100}}} \right)}^3}} \right] \\
   = 40960\left[ {{{\left( {1 + \dfrac{{25}}{{400}}} \right)}^3}} \right] \\
   = 40960\left[ {{{\left( {1 + \dfrac{1}{{16}}} \right)}^3}} \right] \\
   = 40960\left[ {{{\left( {\dfrac{{17}}{{16}}} \right)}^3}} \right] \]
 \[= 40960 \times \dfrac{{4913}}{{4096}} \\
   = Rs\,49130 \]
Therefore, Mr Arora will have to pay after one and half year is \[Rs\,49130\] .
So, the correct answer is “ \[Rs\,49130\]”.

Note: Remember the formula for simple and compound interest always. Simple interest depends on the principal amount or loan for a fixed period whereas compound interest depends on the principal amount and also the interest that is accumulated on the amount every year. Simple interest is written as, \[S.I = \dfrac{{P \times R \times T}}{{100}}\] and compound interest is written as \[C.I = P\left[ {{{\left( {1 + \dfrac{R}{{100}}} \right)}^T}} \right] \] . Students usually get confused between these two formulas. So, don’t get confused and always remember these formulas and while answering check which type of interest is mentioned in the question and accordingly proceed for the calculations.