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A sum of 72820 is due to repaid at the end of three years if it has to be repaid in three equal annual installment the installment being paid at the beginning of each year at 10% p.a compound annually, find the value of installment?
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A sum of 72820 is due to repaid at the end of three years if it has to...
To find the value of the annual installment, we can use the formula for the present value of an annuity:
PV = A * (1 - (1 + r)^(-n)) / r

Where PV is the present value of the annuity (the sum that needs to be repaid), A is the annual installment, r is the interest rate (10% in this case), and n is the number of years (3 in this case).

Substituting the given values, we get:

PV = A * (1 - (1 + 0.1)^(-3)) / 0.1

Solving for A, we get:

A = PV / ((1 - (1 + r)^(-n)) / r)

= 72820 / ((1 - (1 + 0.1)^(-3)) / 0.1)
= 72820 / (1 - (1.1)^(-3)) / 0.1
= 72820 / (1 - (1/1.1)^3) / 0.1
= 72820 / (1 - (1/1.1)^3) / 0.1
= 72820 / (1 - (0.9091)^3) / 0.1
= 72820 / (1 - 0.756) / 0.1
= 72820 / 0.244 / 0.1
= 72820 / 0.0244
= 29816.39
Therefore, the annual installment is approximately 29816.39.

This question is part of UPSC exam. View all CAT courses
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A sum of 72820 is due to repaid at the end of three years if it has to...
Given:
Principal amount (P) = 72820
Number of years (n) = 3
Rate of interest (r) = 10% p.a. compounded annually

To Find:
Value of installment

Solution:

Step 1: Calculate the compound interest for 3 years on the principal amount of 72820.

The formula to calculate compound interest is:
A = P(1 + r/n)^(nt)

Where,
A = Final amount
P = Principal amount
r = Rate of interest
n = Number of times interest is compounded per year
t = Number of years

In this case, the interest is compounded annually, so n = 1.

Plugging in the values,
A = 72820(1 + 0.10/1)^(1*3)
A = 72820(1 + 0.10)^3
A = 72820(1.10)^3
A = 72820(1.331)
A = 96953.72

Step 2: Calculate the value of each installment.

Since the installment is paid at the beginning of each year, the present value of the installments should be equal to the principal amount.

Let x be the value of each installment.

The present value of the first installment is x.
The present value of the second installment is x/(1 + r).
The present value of the third installment is x/(1 + r)^2.

Adding these present values, we get:
x + x/(1 + r) + x/(1 + r)^2 = P
x(1 + 1/(1 + r) + 1/(1 + r)^2) = P

Plugging in the values,
x(1 + 1/(1 + 0.10) + 1/(1 + 0.10)^2) = 72820
x(1 + 1/1.10 + 1/1.21) = 72820
x(1 + 0.909 + 0.826) = 72820
x(2.735) = 72820
2.735x = 72820
x = 72820/2.735
x = 26621.57

Answer:
The value of each installment is approximately 26621.57.
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A sum of 72820 is due to repaid at the end of three years if it has to be repaid in three equal annual installment the installment being paid at the beginning of each year at 10% p.a compound annually, find the value of installment?
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