Mr Rashid invested rupees 60000 in a company but found that is investment was losing 6% of its value per annum after 2 years he decided to pull out what was left of the investment and place at 4% interest compound at twice the year he would recover his original investment in the ______ year after investing at 4%
To find the year after which Mr. Rashid would recover his original investment, we need to calculate the future value of his remaining investment after 2 years at a 4% compound interest rate.
After 2 years, Mr. Rashid's investment would have lost 6% of its value per annum. We can calculate the remaining value of his investment after 2 years as follows:
Remaining Value = 60000 * (1 - (6/100))^2 = 60000 * (1 - 0.06)^2 = 60000 * (0.94)^2 = 60000 * 0.8836 = 53016.
Now, we need to find the number of years it takes for this remaining value of 53016 to grow to 60000 at a 4% compound interest rate.
Let's assume that it takes n years to reach 60000. The future value of the remaining investment after n years at a 4% compound interest rate can be calculated as follows:
Future Value = 53016 * (1 + (4/100))^n = 53016 * (1.04)^n.
We need to solve for n:
53016 * (1.04)^n = 60000.
Dividing both sides by 53016:
(1.04)^n = 60000 / 53016.
Taking the logarithm (base 1.04) of both sides:
n = log_base1.04(60000 / 53016).
Using a calculator, we find that n is approximately 2. According to the rounding rule, the value of n should be 2.
Therefore, Mr. Rashid would recover his original investment in the 2 + 2 = 4th year after investing at a 4% compound interest rate.
To find the year when Mr. Rashid would recover his original investment after investing at 4% interest compounded annually, we need to calculate the final value of his initial investment and then determine the number of years it would take for the final value to equal or exceed the original investment.
Step 1: Calculate the final value of the initial investment after 2 years with a 6% annual loss.
Original investment = Rs 60,000
Annual loss rate = 6%
Time = 2 years
Final value after 2 years = Original investment * (1 - Annual loss rate)^Time
Final value after 2 years = 60,000 * (1 - 0.06)^2
Final value after 2 years = 60,000 * (0.94)^2
Final value after 2 years ≈ Rs 52,986.40
Step 2: Calculate the number of years it would take for the final value to equal or exceed the original investment at 4% compound interest.
Original investment = Rs 60,000
Interest rate = 4%
Final value = Rs 60,000
Final value = Original investment * (1 + Interest rate)^Time
60,000 = 60,000 * (1 + 0.04)^Time
1 = (1.04)^Time
Taking the logarithm of both sides:
log(1) = log((1.04)^Time)
0 = Time * log(1.04)
Solving for Time:
Time = 0 / log(1.04)
Since the logarithm of 1 is 0, the equation becomes:
Time = 0 / 0
The expression 0/0 is undefined, so we need to use a limit:
lim Time as x approaches 0 of Time / log(1.04)
Using L'Hopital's rule (differentiating numerator and denominator):
= lim Time as x approaches 0 of 1 / (1.04)^Time * log(1.04)
= 1 / log(1.04)
Using a calculator, the approximate value of log(1.04) is 0.01703.
Time ≈ 1 / 0.01703
Time ≈ 58.78 years
Therefore, Mr. Rashid would recover his original investment in approximately 59 years after investing at 4% compound interest annually.