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Given annuity of Rs. 100 amounts to Rs. 3137.12 at 4.5% p.a C. I. The number of years will be
- a)25yrs. (appx.)
- b)20 yrs. (appx.)
- c)22 yrs.
- d)none of these
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
Given annuity of Rs. 100 amounts to Rs. 3137.12 at 4.5% p.a C. I. The ...
We use the formula for the Future Value (FV) of an annuity with compound interest:
Formula:
FV = a × [(1 + i)n - 1] / i
FV = a × [(1 + i)n - 1] / i
Where:
a = Rs. 100 (annuity payment)
i = 4.5% = 0.045 (annual interest rate)
n = number of years
FV = Rs. 3137.12 (future value)
a = Rs. 100 (annuity payment)
i = 4.5% = 0.045 (annual interest rate)
n = number of years
FV = Rs. 3137.12 (future value)
Substitute the given values into the formula:
3137.12 = 100 × [(1 + 0.045)n - 1] / 0.045 3137.12
= 100 × [(1.045)n - 1] / 0.045
31.3712 = (1.045)n - 1 / 0.045
1.4137 = (1.045)n - 1
2.4137 = (1.045)n
= 100 × [(1.045)n - 1] / 0.045
31.3712 = (1.045)n - 1 / 0.045
1.4137 = (1.045)n - 1
2.4137 = (1.045)n
Taking logarithms to solve for n:
log(2.4137) = n × log(1.045)
0.3835 = n × 0.0189 n ≈ 20.3 years
0.3835 = n × 0.0189 n ≈ 20.3 years
Conclusion:
The number of years is approximately 20 years.
The number of years is approximately 20 years.
Answer: B: 20 yrs. (appx.)
Most Upvoted Answer
Given annuity of Rs. 100 amounts to Rs. 3137.12 at 4.5% p.a C. I. The ...
Future value of annuity 3137.25 = 100x{(1.045)n-1}/.045 (1.045)n = 2.4117625. Now multiply 1.045 as many times till you get 2.4117625.
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Community Answer
Given annuity of Rs. 100 amounts to Rs. 3137.12 at 4.5% p.a C. I. The ...
Given:
Annuity = Rs. 100
Amount = Rs. 3137.12
Rate of interest = 4.5% p.a. (C.I.)
To find:
Number of years
Solution:
Step 1: Find the annual rate of interest (R)
Let the annual rate of interest be R. Then,
Amount = Annuity × [(1 + R/100)^n - 1]/(R/100)
Where n is the number of years.
Substituting the given values, we get:
3137.12 = 100 × [(1 + R/100)^n - 1]/(R/100)
Multiplying both sides by (R/100) and simplifying, we get:
31.3712R = (1 + R/100)^n - 1
Step 2: Apply logarithms
Taking logarithm on both sides, we get:
log (31.3712R + 1) = n log (1 + R/100)
Dividing both sides by log (1 + R/100), we get:
n = log (31.3712R + 1)/log (1 + R/100)
Step 3: Substitute values and solve
Substituting R = 4.5%, we get:
n = log (31.3712 × 4.5/100 + 1)/log (1 + 4.5/100)
n = log 1.34788/log 1.045
n = 19.91 years (approx.)
Therefore, the number of years is approximately 20 years.
Answer: Option B) 20 years (approx.)
Annuity = Rs. 100
Amount = Rs. 3137.12
Rate of interest = 4.5% p.a. (C.I.)
To find:
Number of years
Solution:
Step 1: Find the annual rate of interest (R)
Let the annual rate of interest be R. Then,
Amount = Annuity × [(1 + R/100)^n - 1]/(R/100)
Where n is the number of years.
Substituting the given values, we get:
3137.12 = 100 × [(1 + R/100)^n - 1]/(R/100)
Multiplying both sides by (R/100) and simplifying, we get:
31.3712R = (1 + R/100)^n - 1
Step 2: Apply logarithms
Taking logarithm on both sides, we get:
log (31.3712R + 1) = n log (1 + R/100)
Dividing both sides by log (1 + R/100), we get:
n = log (31.3712R + 1)/log (1 + R/100)
Step 3: Substitute values and solve
Substituting R = 4.5%, we get:
n = log (31.3712 × 4.5/100 + 1)/log (1 + 4.5/100)
n = log 1.34788/log 1.045
n = 19.91 years (approx.)
Therefore, the number of years is approximately 20 years.
Answer: Option B) 20 years (approx.)
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