Question

A man has a recurring deposit account in a bank for \[3\dfrac{1}{2}\] years. If the rate of interest is 12% per annum and the man gets \[Rs.10,206\] on maturity, find the value of monthly installments.

Answer 1

Hint: The rate of interest, time for deposit of money, and maturity money are 12%, \[3\dfrac{1}{2}\] years, and \[Rs.10,206\] respectively. Now, convert \[3\dfrac{1}{2}\] years into months using the relation \[1year=12\,months\] . Assume that the installments per month is Rs. y. We know the formula for the interest, \[\left( Installment\,per\,month \right)\times \dfrac{Time\left( Time+1 \right)}{2\times 12}\times \dfrac{Rate}{100}\] . Use this formula and get the value of the Interest. Now, use the formula, Maturity money = \[Time\left( installment\,per\,month \right)+Interest\] to get the value of the maturity money. It is given that the maturity money is \[Rs.10,206\] . Now, solve it further and get the value of y.

Complete step-by-step solution:
According to the question, it is given that
The rate of interest = 12% per annum …………………………………….(1)
The maturity money = \[Rs.10,206\] ……………………………………(2)
The time for which the amount is deposited = \[3\dfrac{1}{2}\] years ………………………(3)
We know that \[1year=12\,months\] ………………………………….(4)
Now, from equation (3) and equation (4), we get
The time for which the amount is deposited = \[3\dfrac{1}{2}\] years = \[\dfrac{7}{2}\times 12\] months = \[7\times 6\] months = 42 months ……………………………………..(5)
Let us assume that the installment per month is Rs. y ……………………………………………(6)
We know the formula for the interest, \[\left( Installment\,per\,month \right)\times \dfrac{Time\left( Time+1 \right)}{2\times 12}\times \dfrac{Rate}{100}\] ………………….(7)
From equation (1), we have the rate of interest.
From equation (5), we have the time for the deposition of money.
From equation (6), we have the installment per month.
Now, from equation (1), equation (5), equation (6), and equation (7), we get
Interest = Rs. \[y\times \dfrac{42\left( 42+1 \right)}{2\times 12}\times \dfrac{12}{100}\] = Rs. \[y\times \dfrac{1806}{24}\times \dfrac{12}{100}\] = Rs. \[9.03y\] ………………………..(8)
We know that maturity money is the summation of interest, and product of time and installment per month.
We know the formula, Maturity money = \[Time\left( installment\,per\,month \right)+Interest\] ………………………………………………(9)
Now, from equation (5), equation (6), and equation (8), we get
\[Maturity\,money=Rs.\,42\times y+9.03y=Rs.\,42y+9.03y=Rs.\,51.03y\] ………………………………….(10)
From equation (2), we have the maturity money.
Now, from equation (2) and equation (10), we get
\[\begin{align}
  & \Rightarrow Rs.\,10206=Rs.\,51.06y \\
 & \Rightarrow \dfrac{10206}{51.06}=y \\
 & \Rightarrow 200=y \\
\end{align}\]
Therefore, the installment per month is Rs. 200.

Note: In this question, one might use the formula \[\dfrac{principal\times rate\times time}{100}\] for finding out the interest. This is wrong because, in the question, it is given that we have a recurring deposit amount so, this formula won’t work here because a recurring deposit is a special kind of term deposit offered by banks which help people with regular incomes to deposit a fixed amount every month into their recurring deposit account and earn interest at the rate applicable to fixed deposits. The formula for finding out the interest is \[\left( Installment\,per\,month \right)\times \dfrac{Time\left( Time+1 \right)}{2\times 12}\times \dfrac{Rate}{100}\] . Therefore, use this formula for finding out the interest.