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]

Answer:

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

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

Answer:

Compounded Yearly

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

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

A=10000 \Big(1+ \frac{10}{1 \times 100} \Big)^{1}. \Big(1+ \frac{10}{2 \times 100} \Big)^{\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 \Big(1+ \frac{r}{2 \times 100} \Big)^{\frac{3}{2} \times 2} = 10000 \Big(1+ \frac{10}{2 \times 100} \Big)^{\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.

Answer:

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 \Big(1+ \frac{r}{2 \times 100} \Big)^{1 \times 2} = 22400 \Big(1+ \frac{20}{2 \times 100} \Big)^{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?

Answer:

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

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

27783=x \Big(1+ \frac{10}{2 \times 100} \Big)^{\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.

Answer:

A’s investment: Compounded Yearly

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

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

A=x \Big(1+ \frac{20}{1 \times 100} \Big)^{1}. \Big(1+ \frac{20}{2 \times 100} \Big)^{\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 \Big(1+ \frac{r}{2 \times 100} \Big)^{\frac{3}{2} \times 2} = x \Big(1+ \frac{20}{2 \times 100} \Big)^{\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.

Answer:

Compounded Half Yearly

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

67600=62500 \Big(1+ \frac{x}{2 \times 100} \Big)^{\frac{2}{2} \times 2} \Rightarrow 1.0816 = \Big(1+ \frac{x}{200} \Big)^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?

Answer:

Compounded Half Yearly

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

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

Answer:

Compounded Half Yearly

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

A=3500 \Big(1+ \frac{6}{2 \times 100} \Big)^{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.

Answer:

Compounded Yearly

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

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

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

Answer:

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 \Big(1+ \frac{10}{2 \times 100} \Big)^{\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.

Answer:

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 \Big(1+ \frac{10}{2 \times 100} \Big)^{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.