Question

A recurring deposit account of Rs.1,200 per month has a maturity value of Rs.12,440. If the rate of interest is 8% and the interest is calculated at the end of every month, find the time (in months) of this Recurring Deposit Account.

Answer 1

Hint: We have been given the amount deposited every month, the maturity value and the rate of interest. We need to find the number of months which we will assume as n. Since we have been told that it is a recurring deposit account we can calculate the total S.I. by using the formula $$TotalS.I.={\displaystyle \frac{P\times R\times n\left(n+1\right)}{100\times 12\times 2}}$$ in which we will substitute the values of P and R to get the S.I. in terms of n. Similarly as we have the monthly deposit we can calculate the total deposit in terms of n by using the formula: Total amount Deposited= Monthly deposit$$\times No.ofmonths$$$$=1200\times n$$. Finally to get an equation in terms of n, we will use the formula for Maturity Value, which is Maturity Value= Total Deposited Amount + Total Simple Interest. As we have been given the maturity value we will then use this equation to find the value of n.

Complete step by step answer:
Let us first understand the concept of recurring deposit.
In a recurring deposit account, a fixed amount of money is deposited in regular intervals for a certain duration of time. The total amount at the end of this time period is called the maturity value. This maturity value is the sum of the total amount deposited in regular intervals and the total interest earned. As we have been given the monthly deposit, rate of interest and the maturity value, we will use this formula to find the number of months:
Maturity Value= Total Deposited Amount + Total Simple Interest.
Let us assume n is the number of months.
We will first identify the formula for calculating the total interest.
S.I. for n month$$={\displaystyle \frac{P\times R\times n}{100\times 12}}$$
S.I. for 1 month$$={\displaystyle \frac{P\times R\times 1}{100\times 12}}$$
S.I. for n-1 months $$={\displaystyle \frac{P\times R\times \left(n-1\right)}{100\times 12}}$$
Therefore, we can say that total S.I would be the sum of S.I. for all the n months.
It can be represented as follows,
Total S.I. $$={\displaystyle \frac{P\times R}{100\times 12}}\left(1+2+...+\left(n-1\right)+n\right)$$
$$={\displaystyle \frac{P\times R\times n\left(n+1\right)}{100\times 12\times 2}}$$
Now from this formula we can calculate S.I.
We know that,
P=Rs.1200
Total amount Deposited= Monthly deposit$$\times No.ofmonths$$
$$=1200\times n$$
Maturity Value= Rs.12,440
Rate=8%
We have the formula,
Total S.I. $$={\displaystyle \frac{P\times R\times n\left(n+1\right)}{100\times 12\times 2}}$$
Substituting the above values in the formula we get
Total S.I.$$={\displaystyle \frac{1200\times n\left(n+1\right)8}{2\times 12\times 100}}$$
Total S.I. $$=4n\left(n+1\right)$$
Now we will use the maturity value equation which is as follows.
Maturity Value= Total Deposited Amount + Total Simple Interest.
We will substitute values in the above equation.
$$12440=1200n+4n\left(n+1\right)$$
From this we can see that we get an equation with only one unknown, which is n.
We will now open the brackets to simplify it further.
$$12440=1200n+4n^2+4n$$
We will take all the terms on the L.H.S.
$$4n^2+1204n-12440=0$$
We will solve this quadratic equation by factorization to find the roots.
To simplify the equation we can divide it throughout by 4.
$$n^2+301n-3110=0$$
On factorization we will get,
$$n^2+311n-10n-3110=0$$
$$\Rightarrow n\left(n+311\right)-10\left(n+311\right)=0$$
$$\Rightarrow \left(n+311\right)\left(n-10\right)=0$$
$$\Rightarrow n=-311orn=10$$
Now, n is no of months which cannot be negative.
Therefore, the acceptable value of n is 10.

Hence, time for the Recurring Deposit account is 10 months.

Note: It is important to note that the question mentions recurring deposits that are repeated deposits where all the simple interests for individual months are added to give total interest. It is different from Compound interest and thus we should be careful and use simple interest formula only. Also as we know other values, we will aim to form an equation in terms of n, which is no of months.