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In what time will Rs. 64,000 amount to Rs.68921 at 5% per annum interest being compounded half yearly?
- a)3 years
- b)2 years
- c)2(1/2) years
- d)1(1/2) years
- e)None of these
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
In what time will Rs. 64,000 amount to Rs.68921 at 5% per annum intere...
R = 2.5%, A = 68921 , P = 64000 and t= 2n
A/P = [1+(R/100)]2n
68921/64000 = [1+(2.5/100)]2n
(41/40)3 = (41/40)2n
2n = 3
n=3/2 =1(1/2) years
A/P = [1+(R/100)]2n
68921/64000 = [1+(2.5/100)]2n
(41/40)3 = (41/40)2n
2n = 3
n=3/2 =1(1/2) years
Most Upvoted Answer
In what time will Rs. 64,000 amount to Rs.68921 at 5% per annum intere...
Understanding the Problem
To find the time required for an investment of Rs. 64,000 to grow to Rs. 68,921 at a compound interest rate of 5% per annum, compounded half-yearly, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
- A = the amount of money accumulated after n years, including interest.
- P = the principal amount (initial investment).
- r = annual interest rate (decimal).
- n = number of times that interest is compounded per year.
- t = time the money is invested for in years.
Identifying Values
- P = Rs. 64,000
- A = Rs. 68,921
- r = 5% = 0.05
- n = 2 (since it's compounded half-yearly)
Plugging in the Values
Using the formula:
68,921 = 64,000(1 + 0.05/2)^(2t)
Now simplifying:
68,921 = 64,000(1 + 0.025)^(2t)
68,921 = 64,000(1.025)^(2t)
Calculating the Time
1. Divide both sides by 64,000:
(68,921 / 64,000) = (1.025)^(2t)
1.076 = (1.025)^(2t)
2. To find t, we use logarithms:
Taking log on both sides:
log(1.076) = 2t * log(1.025)
3. Solve for t:
t = log(1.076) / (2 * log(1.025))
After calculating, t approximately equals 1.5 years.
Conclusion
Thus, the time required for Rs. 64,000 to amount to Rs. 68,921 at a 5% interest rate compounded half-yearly is 1.5 years. Hence, the correct answer is option 'D'.
To find the time required for an investment of Rs. 64,000 to grow to Rs. 68,921 at a compound interest rate of 5% per annum, compounded half-yearly, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
- A = the amount of money accumulated after n years, including interest.
- P = the principal amount (initial investment).
- r = annual interest rate (decimal).
- n = number of times that interest is compounded per year.
- t = time the money is invested for in years.
Identifying Values
- P = Rs. 64,000
- A = Rs. 68,921
- r = 5% = 0.05
- n = 2 (since it's compounded half-yearly)
Plugging in the Values
Using the formula:
68,921 = 64,000(1 + 0.05/2)^(2t)
Now simplifying:
68,921 = 64,000(1 + 0.025)^(2t)
68,921 = 64,000(1.025)^(2t)
Calculating the Time
1. Divide both sides by 64,000:
(68,921 / 64,000) = (1.025)^(2t)
1.076 = (1.025)^(2t)
2. To find t, we use logarithms:
Taking log on both sides:
log(1.076) = 2t * log(1.025)
3. Solve for t:
t = log(1.076) / (2 * log(1.025))
After calculating, t approximately equals 1.5 years.
Conclusion
Thus, the time required for Rs. 64,000 to amount to Rs. 68,921 at a 5% interest rate compounded half-yearly is 1.5 years. Hence, the correct answer is option 'D'.
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