Question

Sam invested R6800,00 at 12,0% interest per year, compounded monthly. The number of months he has to wait for this amount to grow to R9165,37, rounded to two decimal places, is

Answer 1

Given:

Sam invested a sum of Rs. 6,80,000

The rate of interest per year = 12%

Amount = Rs. 9,16,537

To find:

The no. of months Sam has to wait for the sum amount to grow

Formula to be used where the interest is compounded monthly:

$\boxed{\bold{A = P [ 1\:+\:\frac{\frac{R}{12}}{100}]^(12n)}}$

where

A = amount

P = Principal

R = rate of interest

n = time period in terms of years

Solution:

Now, by substituting the given values in the formula, we will find the value of "12n" i.e., the no. of months Sam has to wait to receive an amount of Rs. 916537 rounde to two decimal places.

916537 = 680000 [1 + $\frac{\frac{12}{12}}{100}$ ]¹²ⁿ

⇒ $\frac{916537}{680000}$ = [1 + $\frac{1}{100}$ ]¹²ⁿ

⇒ 1.3478 = [ $\frac{101}{100}$ ]¹²ⁿ

⇒ 1.3478 = [ 1.01 ]¹²ⁿ

taking log on both sides of the equation

⇒ log 1.3478 = 12n log [ 1.01 ] ........ [Formula: log aᵇ = b log a]

⇒ 0.1296 = 12n × 4.321 × 10⁻³

⇒ 12n = $\frac{0.1296}{4.321\:*\:10^-3}$

⇒ 12n = 29.993 ≈ 30.00

Thus, the no. of months Sam has to wait for this amount of Rs. 680000 to grow to Rs. 9165,37, rounded to two decimal places, is 30.00.

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Answer 2

The number of months he has to wait for amount to grow to ₹ 9,16,537 is 30

Step-by-step explanation:

The compound interest is calculated with formula given below:

A = P (1 + r/n)ⁿᵃ

Where,

A = Compounded amount = ₹ 9,16,537

P = Initial principal amount = ₹ 6,80,000

r = Interest rate = 12%

n = Number of times interest applied in a period of time = 12

a = Number of time periods elapsed = ?

On substituting the values, we get,

9,16,537 = 6,80,000 (1 + (12/100)/12)¹²ᵃ

9,16,537/6,80,000 = (1 + (0.12)/12)¹²ᵃ

1.347 = ((12 + 0.12)/12)¹²ᵃ

1.347 = ((12.12)/12)¹²ᵃ

1.347 = (1.01)¹²ᵃ

On taking log on both sides, we get,

log (1.347) = log (1.01)¹²ᵃ

0.129 = 12a (0.0043)

0.129/0.0043 = 12a

30 = 12a

∴ 12a = 30