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 –
$p={\displaystyle \frac{a}{i}}{\displaystyle \frac{{\left(1+i\right)}^n-1}{{\left(1+i\right)}^{m+n}}}$
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%,
$\Rightarrow {\displaystyle \frac{12}{100}}\times {\displaystyle \frac{1}{2}}=0.06$
So we will put the values of p, m + n and i in the formula,
$\Rightarrow 20,000={\displaystyle \frac{a}{0.06}}\times {\displaystyle \frac{{\left(1+0.06\right)}^7-1}{{\left(1+0.06\right)}^{10}}}$
$\Rightarrow 20,000={\displaystyle \frac{a}{0.06}}\times {\displaystyle \frac{{\left(1.06\right)}^7-1}{{\left(1+0.06\right)}^{10}}}$
$\Rightarrow 20,000={\displaystyle \frac{a}{0.06}}\times {\displaystyle \frac{1.503-1}{1.791}}$
$\Rightarrow 20,000={\displaystyle \frac{a}{0.06}}\times {\displaystyle \frac{0.503}{1.791}}$
$\Rightarrow a={\displaystyle \frac{20,000\times 0.06\times 1.791}{0.503}}$
$\Rightarrow a={\displaystyle \frac{2149.2}{0.503}}$
$\therefore a=427.76$
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.