Question

The maturity value of a R.D account is Rs.16,176. If the monthly installments are Rs.400 and the rate of interest is 8%. Find the time period of this R.D account.
(a)2 years
(b)3 years
(c)4 years
(d)None of the above

Answer 1

Hint – In this question use the concept that the maturity value is the sum of the principal value and the interest value, so first let the time duration of the R.D account is some x months, then find the principal value using the concept that principal value is the product of the monthly installment and the time period. This will help approaching the problem.

Complete step-by-step answer:
Given data:
Monthly installment = Rs. 400
Rate of interest is 8% per annum.
Maturity value of an R.D account is Rs 16,176.
Now the maturity value is the sum of the principal value and the interest value.
Let the time period for this R.D account is x months.
So the principal value is the product of the monthly installment and the time period.
So the principal value = $x \times 400$ Rs.
Now as we know that the total simple interest (calculated on monthly basis so we have to divide by 12) on any principal amount say (P) having r% interest and n months is given as
S.I = $P \times \dfrac{r}{{100}} \times \dfrac{{\left( {1 + 2 + 3 + ...... + n} \right)}}{{12}}$
Where, the sum of first natural numbers is given as $\dfrac{{n\left( {n + 1} \right)}}{2}$
Total S.I in n months = $P \times \dfrac{r}{{100}} \times \dfrac{{n\left( {n + 1} \right)}}{{12\left( 2 \right)}}$
Here we assume n = x and r = 8 (given)
So the total simple interest in x months is = $400 \times \dfrac{8}{{100}} \times \dfrac{{x\left( {x + 1} \right)}}{{24}}$
So the maturity value is
16176 = $400x + 400 \times \dfrac{8}{{100}} \times \dfrac{{x\left( {x + 1} \right)}}{{24}}$
Now simplify this equation we have,
$16176 = 400\left( {x + \dfrac{1}{{300}} \times x\left( {x + 1} \right)} \right)$
$ \Rightarrow \dfrac{{16176 \times 300}}{{400}} = \left( {300x + x\left( {x + 1} \right)} \right)$
\[ \Rightarrow 12132 = \left( {300x + x\left( {x + 1} \right)} \right)\]
\[ \Rightarrow {x^2} + 301x - 12132 = 0\]
Now factorize this equation we have,
\[ \Rightarrow {x^2} + 337x - 36x - 12132 = 0\]
\[ \Rightarrow x\left( {x + 337} \right) - 36\left( {x + 337} \right) = 0\]
\[ \Rightarrow \left( {x + 337} \right)\left( {x - 36} \right) = 0\]
\[ \Rightarrow x = - 337,36\]
As negative months are not possible so the time period is 36 months.
As we know that in 1 year = 12 months.
Therefore, 36 months = 12(3) months = 1(3) = 3 years.
So this is the required time period of this R.D amount.
Hence option (B) is the correct answer.

Note – When we talk about bonds then generally the maturity value concept comes into play. Maturity value in general is simply the principal value of that specific bond that is to be paid. There are in general two types of interest that are the simple interest and the compound interest. Simple interest is simply calculated at the principal amount value whereas compound interest is calculated over both the principal value as well as the accumulated interest.