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 ${\displaystyle \frac{PRT}{100}}$ 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
$$\begin{gathered} \Rightarrow {\displaystyle \frac{1600t\left(t+1\right)}{2}} \\ \Rightarrow 800t\left(t+1\right) \end{gathered}$$
We know that the interest is calculated using the formula, ${\displaystyle \frac{PRT}{100}}$, where $P$ is the principal amount, $R$ is the rate of interest and $T$ is the time.
Then, interest for the one month is
$$\begin{gathered} \Rightarrow {\displaystyle \frac{800t\left(t+1\right)\times 10\times 1}{100\times 12}} \\ \Rightarrow {\displaystyle \frac{20t\left(t+1\right)}{3}} \end{gathered}$$
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, ${\displaystyle \frac{20t\left(t+1\right)}{3}}=120400-1600t$
On rearranging the above equation, we will get,
$$\begin{gathered} \Rightarrow 20t^2+20t+4800t-361200=0 \\ \Rightarrow 20t^2+4820t-361200=0 \end{gathered}$$
Divide the equation by 20
$$\begin{gathered} \Rightarrow t^2+241t-18060=0 \\ \Rightarrow t^2+301t-60t-18060=0 \\ \Rightarrow t\left(t+301\right)-60\left(t+301\right)=0 \\ \Rightarrow \left(t+301\right)\left(t-60\right)=0 \end{gathered}$$
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 ${\displaystyle \frac{PRT}{100}}$, 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.