Question

A man borrows Rs. 20,000 at 12% per annum, compounded semi – annually and agrees to pay it in 10 equal semi-annual instalments. Find the value of each instalment, if the first payment is due at the end of two years.

Answer 1

Hint: Here we will proceed by assuming each instalment, m and n as variables. Then we will use the concept of deferred annuity to find each instalment using the given rate of interest i.e. 12% and principal amount be Rs. 20,000.

Complete step-by-step answer:

Deferred annuity is an annuity which commences only after a lapse of some specified time after the final purchase premium has been paid.
Formula of deferred annuity –
$p = \dfrac{a}{i}\dfrac{{{{\left( {1 + i} \right)}^n} - 1}}{{{{\left( {1 + i} \right)}^{m + n}}}}$
Here we will assume that each instalment will be a, m be the semi-annual instalment and n be the remaining instalments.
$ \Rightarrow $ m=7, n=3 and m + n=10,
 Also given that p is Rs. 20,000
Now we will calculate i @ 12%,
$ \Rightarrow \dfrac{{12}}{{100}} \times \dfrac{1}{2} = 0.06$
So we will put the values of p, m + n and i in the formula,
$ \Rightarrow 20,000 = \dfrac{a}{{0.06}} \times \dfrac{{{{\left( {1 + 0.06} \right)}^7} - 1}}{{{{\left( {1 + 0.06} \right)}^{10}}}}$
$\Rightarrow 20,000 = \dfrac{a}{{0.06}} \times \dfrac{{{{\left( {1.06} \right)}^7} - 1}}{{{{\left( {1 + 0.06} \right)}^{10}}}}$
$\Rightarrow 20,000 = \dfrac{a}{{0.06}} \times \dfrac{{1.503 - 1}}{{1.791}}$
$\Rightarrow 20,000 = \dfrac{a}{{0.06}} \times \dfrac{{0.503}}{{1.791}}$
$\Rightarrow a = \dfrac{{20,000 \times 0.06 \times 1.791}}{{0.503}}$
$\Rightarrow a = \dfrac{{2149.2}}{{0.503}}$
$\therefore a = 427.76$
Hence each instalment is of Rs. 427.67.

Note: In order to solve this question, one mistake that many of us can do is we do not convert the given rate into i i.e. instalment. Also we must be careful about the semi-annual instalment in which m is semi-annual instalment and n is remaining seven instalments as one can get confused in this statement. Hence we will get the desired result.