Identify the correct relationship for annual sinking fund. Where, I = Annual sinking fund, S = Total amount of sinking fund invested on that i = Rate of interest n = number of years required to get 'S'
- ASinking fund (I) = Si / [(1 + i)ⁿ - 1]Correct
- BSinking fund (I) = Si / [(1 + n)ⁱ - 1]
- CSinking fund (I) = ni / [(1 + n)ˢ - 1]
- DSinking fund (I) = ni / [(1 + s)ⁿ - 1]
Solution
Concept
A sinking fund is a fixed yearly deposit that grows at compound interest to a total amount S after n years.
Formula
I = S × i / [(1 + i)n - 1]
Step-by-step solution
1. Let I be deposited at the end of every year for n years at rate i. 2. The total amount after n years is S = I × [(1 + i)n - 1] / i. 3. Rearrange for I: I = S × i / [(1 + i)n - 1]. 4. This matches option (A).
Explanation
Each yearly deposit earns compound interest. The sum of all deposits with interest gives S = I[(1 + i)n - 1]/i. Solving for I gives option (A).
Why the other options are wrong
B: The power should be n on (1 + i), not i on (1 + n). C: The numerator must be S×i, and S cannot be a power. D: The numerator must be S×i, and the base must be (1 + i), not (1 + S).
Shortcut
Remember: interest rate i goes inside the bracket and years n go in the power, so (1 + i)n.
Common mistake
Mixing up i and n in the bracket and the power.
Correct answer: A) Sinking fund (I) = Si / [(1 + i)ⁿ - 1]