Question

Kavita has a cumulative time deposit account in a bank. She deposits ₹ 600 per month and gets ₹ 6165 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].

Answer 1

Given:

Monthly deposit P = ₹ 600

Maturity amount MV = ₹ 6165

Rate of interest R = 6% per annum

Let n = Total number of months the account was held

Step-wise calculation:

1. For a recurring/cumulative time deposit, interest earned is given by `I = P xx [(n(n + 1))/(2 xx 12)] xx (R/100)`.

2. Maturity value MV = Total deposits + Interest

= P × n + I

Substitute P, R and simplify:

`MV = 600n + 600 × [(n(n + 1))/24] × (6/100)` 

= `600n + 600 × [(n(n + 1))/24] xx 0.06`

= `600n + (600 xx 0.06/24) xx n(n + 1)` 

= `600n + (36/24) xx n(n + 1)`

= `600n + (3/2) xx n(n + 1)`

3. Set MV = 6165:

`600n + (3/2)(n^2 + n) = 6165`

Multiply both sides by 2 to remove fraction:

1200n + 3(n² + n) = 12330 

3n² + 1203n – 12330 = 0 

Divide by 3: n² + 401n – 4110 = 0.

4. Factor the quadratic or use the hint:

n² + 411n – 10n – 4110 = 0

⇒ (n + 411)(n – 10) = 0.

Thus, n = 10 or n = –411. Reject negative value.

Total time the account was held = 10 months.