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प्रश्न
A person sets up a sinking fund in order to have ₹ 1,00,000 after 10 years. What amount should be deposited bi-annually in the account that pays him 5% p.a. compounded semi-annually? [Given (1.025)20 = 1.675]
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उत्तर
Given A = ₹ 1,00,000
Amount is deposited bi-annually for 10 years.
∴ n = 10 × 2 = 20
Rate of intererst is 5% p.a.
∴ r = `(5)/(2)`% = 2.5%
i = `"r"/(100) = (2.5)/(100)` = 0.025
Now, A = `"C"/"i"[(1 + "i")^"n" - 1]`
∴ 1,00,000 = `"C"/(0.025)[(1 + 0.025)^20 - 1]`
∴ 1,00,000 = `"C"/(0.025)[(1.025)^20 - 1]`
∴ (1,00,000)(0.025) = C[(1.025)20 – 1]
∴ 2,500 = C[1.675 – 1]
∴ 2,500 = C(0.675)
∴ C = `(2,500)/(0.675)` = 3,703.70
∴ Amount of ₹ 3,703.70 should be deposited bi-annually into the account.
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[Given (1.1)4 = 1.4641]
For annuity due,
C = ₹ 20,000, n = 3, I = 0.1, (1.1)–3 = 0.7513
Therefore, P = `square/0.1 xx [1 - (1 + 0.1)^square]`
= 2,00,000 [1 – 0.7513]
= ₹ `square`
The future amount, A = ₹ 10,00,000
Period, n = 20, r = 5%, (1.025)20 = 1.675
A = `"C"/"I" [(1 + "i")^"n" - 1]`
I = `5/200` = `square` as interest is calculated semi-annually
A = 10,00,000 = `"C"/"I" [(1 + "i")^"n" - 1]`
10,00,000 = `"C"/0.025 [(1 + 0.025)^square - 1]`
= `"C"/0.025 [1.675 - 1]`
10,00,000 = `("C" xx 0.675)/0.025`
C = ₹ `square`
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
