Question

Rachna borrows \[Rs.12000\] at \[10%\] annum interest compounded half – yearly. She repays \[Rs.4000\] at the end of every 6 months. Calculate the third payment she has to make at the end of 18 months in order to clear the entire loan.
(a) \[Rs.5281.5\]
(b) \[Rs.2678.9\]
(c) \[Rs.2345.6\]
(d) \[Rs.2098.7\]

Answer 1

Hint: Use the formula \[A=P{{\left( 1+\dfrac{R}{200} \right)}^{2T}}\] to calculate the amount Rachna paid after 1 year. Subtract this amount from the total money she borrowed. Use this money to calculate the amount again that she paid in the second instalment. Repeat the process to calculate the amount she paid in the third instalment.

Complete step-by-step answer:
We know that Rachna borrows \[Rs.12000\] which is compounded at a rate of \[10%\] per annum compounded half yearly. She repays \[Rs.4000\] after every six months. We have to calculate the amount she pays after the third instalment.
We will use the formula \[A=P{{\left( 1+\dfrac{R}{200} \right)}^{2T}}\], where A is the amount paid after interest is added, P is the money on which interest is added, T is the time for which loan is taken and R is the rate of interest, to calculate the amount she pays after each instalment.
We will firstly calculate the amount she will pay in the first instalment. Substituting \[P=Rs.12000,R=10%,T=\dfrac{1}{2}\] in the above formula, we have \[A=12000{{\left( 1+\dfrac{10}{200} \right)}^{2\times \dfrac{1}{2}}}=12000\left( 1+\dfrac{1}{20} \right)=12000\left( \dfrac{21}{20} \right)=Rs.12600\].
We know that she paid \[Rs.4000\] in the first instalment. So, the amount left to be paid \[=Rs.12600-Rs.4000=Rs.8600\]. We will now calculate the interest on this amount now.
Thus, we have \[A=8600{{\left( 1+\dfrac{10}{200} \right)}^{2\times \dfrac{1}{2}}}=8600\left( 1+\dfrac{1}{20} \right)=8600\left( \dfrac{21}{20} \right)=Rs.9030\]. We know that she paid \[Rs.4000\] in the second instalment. So, the amount left to be paid \[=Rs.9030-Rs.4000=Rs.5030\].
We will now calculate the interest on this amount now. Thus, we have \[A=5030{{\left( 1+\dfrac{10}{200} \right)}^{2\times \dfrac{1}{2}}}=5030\left( 1+\dfrac{1}{20} \right)=5030\left( \dfrac{21}{20} \right)=Rs.5281.5\].
Hence, the amount she paid in the third instalment is \[Rs.5281.5\], which is option (a).

Note: One must be careful while using the principal amount on which interest is calculated. After each instalment, the principal amount will be deducted. One must also keep in mind that the time to be used for calculating interest is 6 months as the instalment is paid after 6 months.