Kavita has a cumulative time deposit account in a bank. She deposits ₹600 per month and gets ₹6,165 at the time of maturity. If the rate of interest be 6% per annum, find the total time for which the account was held. (Hint: x² + 411x − 10x − 4110 = 0)

Given,

P = ₹600

Maturity Value = ₹6,165

r = 6% per annum

Let the number of months be 'x'.

Sum deposited = P × x = 600x

Interest (I) = $P \times \dfrac{n(n+1)}{2 \times 12} \times \dfrac{r}{100}$

$I = 600 \times \dfrac{x(x+1)}{24} \times \dfrac{6}{100}\\[1em] I = 600 \times \dfrac{6x(x+1)}{2400} \\[1em] = \dfrac{3600x(x+1)}{2400} \\[1em] = \dfrac{3}{2} x(x+1)$

Maturity value = Sum deposited + Interest

$\Rightarrow 600x + \dfrac{3x(x + 1)}{2} = 6165 \\[1em] \Rightarrow \dfrac{1200x + 3x^2 + 3x}{2} = 6165 \\[1em] \Rightarrow 3x^2 + 1203x = 12330 \\[1em] \Rightarrow 3x^2 + 1203x - 12330 = 0 \\[1em] \Rightarrow 3(x^2 + 401x - 4110) = 0 \\[1em] \Rightarrow x^2 + 401x - 4110 = 0 \\[1em] \Rightarrow x^2 + 411x - 10x - 4110 = 0 \\[1em] \Rightarrow x(x + 411) - 10(x + 4110) = 0 \\[1em] \Rightarrow (x - 10)(x + 411) = 0 \\[1em] \Rightarrow x = 10 \text{ or } x = -411.$

Since the number of months cannot be negative

∴ x = 10 months.

Hence,total time for which the account was held = 10 months.