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Question
Solve the following :
Find the least number of years for which an annuity of ₹3,000 per annum must run in order that its amount exceeds ₹60,000 at 10% compounded annually. [(1.1)11 = 2.8531, (1.1)12 = 3.1384]
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Solution
Given, C = ₹3,000, A = ₹60,000, r = 10% p.a.
∴ i = `"r"/(100) = (10)/(100)` = 0.1
Since, A = `"C"/"i"[(1 + "i")^"n" - 1]`
∴ 60,000 = `(3,000)/(0.1)[(1 + 0.1)^"n" - 1]`
∴ `(60,000 xx 0.1)/(3,000)` = (1.1)n – 1
∴ 2 = (1.1)n – 1
∴ (1.1)n = 2 + 1
∴ (1.1)n = 3
It is given that (1.1)11 = 2.8531 and (1.1)12 = 3.1384
∴ n will be between 11 years and 12 years.
Thus, the least number of years for which an annuity of ₹3,000 per annum must run is 12 years.
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[Given (1.1)4 = 1.4641]
For an annuity due, C = ₹ 2000, rate = 16% p.a. compounded quarterly for 1 year
∴ Rate of interest per quarter = `square/4` = 4
⇒ r = 4%
⇒ i = `square/100 = 4/100` = 0.04
n = Number of quarters
= 4 × 1
= `square`
⇒ P' = `(C(1 + i))/i [1 - (1 + i)^-n]`
⇒ P' = `(square(1 + square))/0.04 [1 - (square + 0.04)^-square]`
= `(2000(square))/square [1 - (square)^-4]`
= 50,000`(square)`[1 – 0.8548]
= ₹ 7,550.40
