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

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

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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.

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.

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?

$P=x\ Rs.; \ r=10\%$; 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.

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.

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?

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.

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.

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.

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.

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