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Question 1: If the interest is compounded half-yearly, calculate the amount when principal is Rs.7400; the rate of interest is 5% per annum and the duration is one year. [2005]

$P=7400\ Rs.; \ r=5\%; Compounded \ half \ yearly \ n=1 \ year$

$A=P(1+\frac{r}{2 \times 100})^{n \times 2} = 12000(1+\frac{5}{2 \times 100})^{1 \times 2} = 7774.63 \ Rs.$

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Question 2: Find the difference between the compound interest compounded yearly and half-yearly on Rs.10000 for 18 months at 10% per annum.

Compounded Yearly

$P=10000\ Rs.; \ r=10\%; Compounded \ yearly \ n=\frac{3}{2} \ year$

$A=P(1+\frac{r}{1 \times 100})^{1}.(1+\frac{r}{2 \times 100})^{\frac{1}{2} \times 2}$

$A=10000(1+\frac{10}{1 \times 100})^{1}.(1+\frac{10}{2 \times 100})^{\frac{1}{2} \times 2} = 11550 \ Rs.$

Compounded Half Yearly

$P=10000\ Rs.; \ r=10\%; Compounded \ half \ yearly \ n=\frac{3}{2} \ year$

$A=P(1+\frac{r}{2 \times 100})^{\frac{3}{2} \times 2} = 10000(1+\frac{10}{2 \times 100})^{\frac{3}{2} \times 2} = 11576.25 \ Rs.$

Difference $11576.25-11550 = 26.50 \ Rs.$

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Question 3: A man borrowed Rs.16000 for 3 years under the following terms:

1. 20% simple interest for the first 2 years;
2. 20% C.I. for the remaining one year on the amount due after 2 years, the interest being compounded semi-annually. Find the total amount to be paid at the end of the three years.

Simple interest for the first two years

$S.I. = 16000 \times \frac{20}{100} \times 2 = 6400 \ Rs.$

$Amount = 16000+6400 = 22400 \ Rs.$

Compound interest for the remainder of the term

$P=10000\ Rs.; \ r=20\%; Compounded \ half \ yearly \ n=1 \ year$

$A=P(1+\frac{r}{2 \times 100})^{1 \times 2} = 22400(1+\frac{20}{2 \times 100})^{1 \times 2} = 27104 \ Rs.$

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Question 4: What sum of money will amount to Rs.27783 in one and half years at 10% per annum compounded half yearly?

$P=x\ Rs.; \ r=10\%; Compounded \ half \ yearly \ n=\frac{3}{2} \ year; A=27783 \ Rs.$

$A=P(1+\frac{r}{2 \times 100})^{n \times 2}$

$27783=x(1+\frac{10}{2 \times 100})^{\frac{3}{2} \times 2} \Rightarrow 27783 = 1.157625x \Rightarrow x= 2400 \ Rs.$

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Question 5: A  invests a certain sum of money at 20% per annum, interest compounded yearly. B invests an equal amount of money at the same rate of interest per annum compounded half-yearly. If B gets Rs.33 more than A in 18 months, calculate the money invested by each.

A’s investment: Compounded Yearly

$P=x\ Rs.; \ r=20\%; Compounded \ yearly \ n=\frac{3}{2} \ year$

$A=P(1+\frac{r}{1 \times 100})^{1}.(1+\frac{r}{2 \times 100})^{\frac{1}{2} \times 2}$

$A=x(1+\frac{20}{1 \times 100})^{1}.(1+\frac{20}{2 \times 100})^{\frac{1}{2} \times 2} = 1.32x \ Rs.$

Compounded Half Yearly

$P=x\ Rs.; \ r=20\%; Compounded \ half \ yearly \ n=\frac{3}{2} \ year$

$A=P(1+\frac{r}{2 \times 100})^{\frac{3}{2} \times 2} = x(1+\frac{20}{2 \times 100})^{\frac{3}{2} \times 2} = 1.331x \ Rs.$

Difference $1.331x-1.32x=33 \Rightarrow x= 3000 \ Rs.$

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Question 6: At what rate of interest per annum will a sum of Rs.62500 earn a compound interest of Rs.5100 in one year? The interest is to be compounded half-yearly.

Compounded Half Yearly

$P=62500\ Rs.; A=(62500 + 5100) = 67600 \ Rs.; \ r=x\%; Compounded \ half \ yearly \ n=\frac{2}{2} \ year$

$67600=62500(1+\frac{x}{2 \times 100})^{\frac{2}{2} \times 2} \Rightarrow 1.0816 = (1+\frac{x}{200})^2 \Rightarrow x=8\%$

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Question 7: In what time will rs.1500 yield Rs.496.50 as compound interest at 20% per year compounded semi-annually?

Compounded Half Yearly

$P=1500\ Rs.; A=(1500 + 496.50) = 1996.50 \ Rs.; \ r=20\%; Compounded \ half \ yearly \ n=n \ year$

$1996.50=1500(1+\frac{20}{2 \times 100})^{n \times 2} \Rightarrow 1.331 = (1+\frac{20}{200})^{2n} \Rightarrow n=\frac{3}{2} years$

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Question 8: Calculate the C.I. on Rs.3500 at 6% per annum for 3 years, the interest being compounded half-yearly.

Compounded Half Yearly

$P=3500\ Rs.; \ r=6\%; Compounded \ half \ yearly \ n=3 \ year$

$A=3500(1+\frac{6}{2 \times 100})^{3 \times 2} \Rightarrow A= 4179.18 \ Rs.$

$C.I. = 4179.18-3500 = 679.18 \ Rs.$

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Question 9: Find the difference between compound interest and simple interest on Rs.12,000 and in $1 \frac{1}{2}$ at 10% compounded yearly.

Compounded Yearly

$P=12000\ Rs.; \ r=10\%; Compounded \ yearly \ n=\frac{3}{2} \ year$

$A=P(1+\frac{r}{1 \times 100})^{1}.(1+\frac{r}{2 \times 100})^{\frac{1}{2} \times 2}$

$A=12000(1+\frac{10}{1 \times 100})^{1}.(1+\frac{10}{2 \times 100})^{\frac{1}{2} \times 2} = 13860 \ Rs.$

Simple interest for 1.5 years

$S.I. = 12000 \times \frac{10}{100} \times \frac{3}{2} = 1800 \ Rs.$

$Amount = 16000+6400 = 22400 \ Rs.$

$Difference = (13860-13000)-1800 = 60 \ Rs.$

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Question 10: The simple interest on a sum of money for 3 years at 5% per annum is Rs.900. Find:

1. The sum of money and
2. The compound interest on this sum for 1.5 years payable half-yearly at double the rate per annum.

Simple interest for 3 years

$900 = x \times \frac{5}{100} \times 3 \Rightarrow x= 6000 \ Rs.$

$Amount = 16000+6400 = 22400 \ Rs.$

Compounded Half Yearly

$P=6000\ Rs.; \ r=10\%; Compounded \ half \ yearly \ n=\frac{3}{2} \ year$

$A=6000(1+\frac{10}{2 \times 100})^{\frac{3}{2} \times 2} \Rightarrow A= 6945.75 \ Rs.$

$Compound \ interest = 6945.75-6000 = 945.75 \ Rs.$

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Question 11: The compound interest in one year on a certain sum of money at 10% per annum compounded half-yearly exceeds the simple interest on the same sum at the same rate and for the same period by Rs.30. Calculate the sum.

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Simple interest for 1 years

$S.I. = x \times \frac{10}{100} \times 1 = 0.1x \ Rs.$

$Amount = 16000+6400 = 22400 \ Rs.$

$Difference = (13860-13000)-1800 = 60 \ Rs.$

Compounded Half Yearly

$P=x\ Rs.; \ r=10\%; Compounded \ half \ yearly \ n=1 \ year$

$A=x(1+\frac{10}{2 \times 100})^{1 \times 2} \Rightarrow A= 1.1025x \ Rs.$

$Compound \ interest = 1.1025x-x = 0.1025x \ Rs.$

Difference $0.1025x-0.1x=30 \Rightarrow x= 12000 \ Rs.$