Question

Kavita has a cumulative time deposit account in a bank. She deposits ₹ 800 per month and gets ₹ 16700 as maturity value. If the rate of interest be 5% per annum, find the total time for which the account was held.

[Hint: x² + 481x – 10020 = 0 ⇒ x² + 501x – 20x – 10020 = 0.]

Answer 1

Given:

Monthly deposit P = ₹ 800

Maturity value (amount) = ₹ 16,700

Rate of interest R = 5% p.a.

Let n = number of months the account was held

Step-wise calculation:

1. Use the standard formula for maturity of a monthly recurring (cumulative) deposit:

Maturity = nP + Interest, where Interest = `P xx [(n(n + 1))/(2 xx 12)] xx (R/100)`

2. Substitute P = 800, R = 5 and Maturity = 16700:

`16700 = 800n + 800 xx [(n(n + 1))/(24)] xx (5/100)`

Simplify the interest term:

`800 xx (5/100) = 40` 

So, `16700 = 800n + 40 × [(n(n + 1))/24]` 

= `800n + (40/24) n(n + 1)`

= `40/24`

= `5/3`

So, `16700 = 800n + (5/3) n(n + 1)`.

3. Multiply both sides by 3 to clear denominator:

50100 = 2400n + 5n(n + 1)

= 5n² + 2405n

4. Bring to standard quadratic form and divide by 1:

5n² + 2405n – 50100 = 0

⇒ Divide by 5

⇒ n² + 481n – 10020 = 0

5. Factor the quadratic use the hint: 

n² + 481n – 10020 = n² + 501n – 20n – 10020

= (n + 501)(n – 20)

= 0 

So, n = 20 or n = –501 (reject negative).

Therefore, the account was held for 20 months = 1 year 8 months.