Question

The maturity value of a cumulative deposit account is Rs. 1,20,400. If each monthly instalment for this account is Rs. 1,600 and the rate of interest is 10% per year, find the time for which the account was held.

Answer 1

Hint: We will calculate the expression for principal amount from the given conditions. Then, find the interest using the formula $\dfrac{{PRT}}{{100}}$ and by subtracting principal amount from total amount, and equate both the values. Solve the quadratic equation to get the required time.

Complete step-by-step answer:
We are given that the maturity value of a cumulative deposit is Rs. 1,20,400 and deposit per month is Rs. 1,600.
Also, the rate of interest per year is 10%.
Let the required time be $t$ months.
Then, the principal amount for one month will be
$
   \Rightarrow \dfrac{{1600t\left( {t + 1} \right)}}{2} \\
   \Rightarrow 800t\left( {t + 1} \right) \\
$
We know that the interest is calculated using the formula, $\dfrac{{PRT}}{{100}}$, where $P$ is the principal amount, $R$ is the rate of interest and $T$ is the time.
Then, interest for the one month is
$
   \Rightarrow \dfrac{{800t\left( {t + 1} \right) \times 10 \times 1}}{{100 \times 12}} \\
   \Rightarrow \dfrac{{20t\left( {t + 1} \right)}}{3} \\
$
Also, interest can be calculated by subtracting principal from the total amount.
Total amount is Rs. 1,20,400 and principal amount for one year will be $1600t$, where 1600 is the monthly instalment for $t$ months.
Hence, $\dfrac{{20t\left( {t + 1} \right)}}{3} = 120400 - 1600t$
On rearranging the above equation, we will get,
$
   \Rightarrow 20{t^2} + 20t + 4800t - 361200 = 0 \\
   \Rightarrow 20{t^2} + 4820t - 361200 = 0 \\
$
Divide the equation by 20
$
   \Rightarrow {t^2} + 241t - 18060 = 0 \\
   \Rightarrow {t^2} + 301t - 60t - 18060 = 0 \\
   \Rightarrow t\left( {t + 301} \right) - 60\left( {t + 301} \right) = 0 \\
   \Rightarrow \left( {t + 301} \right)\left( {t - 60} \right) = 0 \\
$
Equate each factor to 0
$t = - 301,t = 60$
But, time can never be negative.
Hence, the required time is 60 months or 12 years.

Note: The formula for calculating simple interest is $\dfrac{{PRT}}{{100}}$, where $P$ is the principal amount, $R$ is the rate of interest and $T$ is the time. And the total amount is the summation of principal amount and interest on that amount.