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

Let time be n months.

Given,

M.V. = ₹ 1,20,400

By formula,

M.V. = P × n + P × $\dfrac{n(n + 1)}{2 \times 12} \times \dfrac{r}{100}$

Substituting values we get :

$\Rightarrow 120400 = 1600 × n + 1600 \times \dfrac{n(n + 1)}{2 \times 12} \times \dfrac{10}{100} \\[1em] \Rightarrow 120400 = 1600\Big(n + \dfrac{n(n + 1)}{24} \times \dfrac{10}{100}\Big) \\[1em] \Rightarrow \dfrac{120400}{1600} = n + \dfrac{n(n + 1)}{240} \\[1em] \Rightarrow \dfrac{301}{4} = \dfrac{240n + n^2 + n}{240} \\[1em] \Rightarrow \dfrac{301 \times 240}{4} = 240n + n^2 + n \\[1em] \Rightarrow 301 \times 60 = n^2 + 241n \\[1em] \Rightarrow n^2 + 241n - 18060 = 0 \\[1em] \Rightarrow n^2 + 301n - 60n - 18060 = 0 \\[1em] \Rightarrow n(n + 301) - 60(n + 301) = 0 \\[1em] \Rightarrow (n - 60)(n + 301) = 0 \\[1em] \Rightarrow n = 60 \text{ or } n = -301.$

Since, time cannot be negative.

∴ n = 60 months or $\dfrac{60}{12}$ = 5 years.

Hence, time = 5 years.