RD Sharma solution class 8 chapter 14 Compound Interest Exercise 14.2

Exercise 14.2

Page-14.14

Question 1:

Compute the amount and the compound interest in each of the following by using the formulae when:
(i) Principal = Rs 3000, Rate = 5%, Time = 2 years
(ii) Principal = Rs 3000, Rate = 18%, Time = 2 years
(iii) Principal = Rs 5000, Rate = 10 paise per rupee per annum, Time = 2 years
(iv) Principal = Rs 2000, Rate = 4 paise per rupee per annum, Time = 3 years
(v) Principal = Rs 12800, Rate = $7\frac{1}{2}\%$, Time = 3 years
(vi) Principal = Rs 10000, Rate 20% per annum compounded half-yearly, Time = 2 years
(vii) Principal = Rs 160000, Rate = 10 paise per rupee per annum compounded half-yearly, Time = 2 years.

Answer 1:

$$\begin{gathered} \mathrm{Applying} \mathrm{the} \mathrm{rule} \mathrm{A}=\mathrm{P}{\left(1+\frac{\mathrm{R}}{100}\right)}^{\mathrm{n} }\mathrm{on} \mathrm{the} \mathrm{given} \mathrm{situations}, \mathrm{we} \mathrm{get}: \\ \left(\mathrm{i}\right) \\ \mathrm{A}=3,000{\left(1+\frac{5}{100}\right)}^2 \\ =3,000{\left(1.05\right)}^2 \\ =\mathrm{Rs} 3,307.50 \\ \mathrm{Now}, \\ \mathrm{CI}=\mathrm{A}-\mathrm{P} \\ =\mathrm{Rs} 3,307.50-\mathrm{Rs} 3,000 \\ =\mathrm{Rs} 307.50 \\ \left(\mathrm{ii}\right) \\ \mathrm{A}=3,000{\left(1+\frac{18}{100}\right)}^2 \\ =3,000{\left(1.18\right)}^2 \\ =\mathrm{Rs} 4,177.20 \\ \mathrm{Now}, \\ \mathrm{CI}=\mathrm{A}-\mathrm{P} \\ =\mathrm{Rs} 4,177.20-\mathrm{Rs} 3,000 \\ =\mathrm{Rs} 1,177.20 \\ \left(\mathrm{iii}\right) \\ \mathrm{A}=5,000{\left(1+\frac{10}{100}\right)}^2 \\ =5,000{\left(1.10\right)}^2 \\ =\mathrm{Rs} 6,050 \\ \mathrm{Now}, \\ \mathrm{CI}=\mathrm{A}-\mathrm{P} \\ =\mathrm{Rs} 6,050-\mathrm{Rs} 5,000 \\ =\mathrm{Rs} 1,050 \\ \left(\mathrm{iv}\right) \\ \mathrm{A}=2,000{\left(1+\frac{4}{100}\right)}^3 \\ =2,000{\left(1.04\right)}^3 \\ =\mathrm{Rs} 2,249.68 \\ \mathrm{Now}, \\ \mathrm{CI}=\mathrm{A}-\mathrm{P} \\ =\mathrm{Rs} 2,249.68-\mathrm{Rs} 2,000 \\ =\mathrm{Rs} 249.68 \\ \left(\mathrm{v}\right) \\ \mathrm{A}=12,800{\left(1+\frac{7.5}{100}\right)}^3 \\ =12,800{\left(1.075\right)}^3 \\ =\mathrm{Rs} 15,901.40 \\ \mathrm{Now}, \\ \mathrm{CI}=\mathrm{A}-\mathrm{P} \\ =\mathrm{Rs} 15,901.40-\mathrm{Rs} 12,800 \\ =\mathrm{Rs} 3,101.40 \\ \left(\mathrm{vi}\right) \\ \mathrm{A}=10,000{\left(1+\frac{20}{200}\right)}^4 \\ =10,000{\left(1.1\right)}^4 \\ =\mathrm{Rs} 14,641 \\ \mathrm{Now}, \\ \mathrm{CI}=\mathrm{A}-\mathrm{P} \\ =\mathrm{Rs} 14,641-\mathrm{Rs} 10,000 \\ =\mathrm{Rs} 4,641 \\ \left(\mathrm{vii}\right) \\ \mathrm{A}=16,000{\left(1+\frac{10}{200}\right)}^4 \\ =16,000{\left(1.05\right)}^4 \\ =\mathrm{Rs} 19,448.1 \\ \mathrm{Now}, \\ \mathrm{CI}=\mathrm{A}-\mathrm{P} \\ =\mathrm{Rs} 19,448.1-\mathrm{Rs} 16,000 \\ =\mathrm{Rs} 3,448.1 \end{gathered}$$

Question 2:

Find the amount of Rs 2400 after 3 years, when the interest is compounded annually at the rate of 20% per annum.

Answer 2:

$$\begin{gathered} \mathrm{Given}: \\ \mathrm{P}=\mathrm{Rs} 2,400 \\ \mathrm{R}=20\% \mathrm{p}.\mathrm{a}. \\ \mathrm{n}=3 \mathrm{years} \\ \mathrm{We} \mathrm{know} \mathrm{that} \mathrm{amount} \mathrm{A} \mathrm{at} \mathrm{the} \mathrm{end} \mathrm{of} \mathrm{n} \mathrm{years} \mathrm{at} \mathrm{the} \mathrm{rate} \mathrm{R}\% \mathrm{per} \mathrm{annum} \mathrm{when} \mathrm{the} \mathrm{interest} \mathrm{is} \\ \mathrm{compounded} \mathrm{annually} \mathrm{is} \mathrm{given} \mathrm{by} \mathrm{A}=\mathrm{P}{\left(1+\frac{\mathrm{R}}{100}\right)}^{\mathrm{n}}. \\ \therefore \mathrm{A}=2,400{\left(1+\frac{20}{100}\right)}^3 \\ =2,400{\left(1.2\right)}^3 \\ =4,147.20 \\ \mathrm{Thus}, \mathrm{the} \mathrm{required} \mathrm{amount} \mathrm{is} \mathrm{Rs} 4,147.20. \end{gathered}$$

Question 3:

Rahman lent Rs 16000 to Rasheed at the rate of $12\frac{1}{2}\%$ per annum compound interest. Find the amount payable by Rasheed to Rahman after 3 years.

Answer 3:

$$\begin{gathered} \mathrm{Given}: \\ \mathrm{P}=\mathrm{Rs} 16,000 \\ \mathrm{R}=12.5\% \mathrm{p}.\mathrm{a}. \\ \mathrm{n}=3 \mathrm{years} \\ \mathrm{We} \mathrm{know} \mathrm{that} \mathrm{amount} \mathrm{A} \mathrm{at} \mathrm{the} \mathrm{end} \mathrm{of} \mathrm{n} \mathrm{years} \mathrm{at} \mathrm{the} \mathrm{rate} \mathrm{R}\% \mathrm{per} \mathrm{annum} \mathrm{when} \mathrm{the} \mathrm{interest} \mathrm{is} \\ \mathrm{compounded} \mathrm{annually} \mathrm{is} \mathrm{given} \mathrm{by} \mathrm{A}=\mathrm{P}{\left(1+\frac{\mathrm{R}}{100}\right)}^{\mathrm{n}}. \\ \therefore \mathrm{A}=16,000{\left(1+\frac{12.5}{100}\right)}^3 \\ =16,000{\left(1.125\right)}^3 \\ =22,781.25 \\ \mathrm{Thus}, \mathrm{the} \mathrm{required} \mathrm{amount} \mathrm{is} \mathrm{Rs} 22,781.25. \end{gathered}$$

Question 4:

Meera borrowed a sum of Rs 1000 from Sita for two years. If the rate of interest is 10% compounded annually, find the amount that Meera has to pay back.

Answer 4:

$$\begin{gathered} \mathrm{Given}: \\ \mathrm{P}=\mathrm{Rs} 1,000 \\ \mathrm{R}=10\% \mathrm{p}.\mathrm{a}. \\ \mathrm{n}=2 \mathrm{years} \\ \mathrm{We} \mathrm{know} \mathrm{that} \mathrm{amount} \mathrm{A} \mathrm{at} \mathrm{the} \mathrm{end} \mathrm{of} \mathrm{n} \mathrm{years} \mathrm{at} \mathrm{the} \mathrm{rate} \mathrm{R}\% \mathrm{per} \mathrm{annum} \mathrm{when} \mathrm{the} \mathrm{interest} \mathrm{is} \\ \mathrm{compounded} \mathrm{annually} \mathrm{is} \mathrm{given} \mathrm{by} \mathrm{A}=\mathrm{P}\left(1+\frac{\mathrm{R}}{100}\right). \\ \therefore \mathrm{A}=1,000{\left(1+\frac{10}{100}\right)}^2 \\ =1,000{\left(1.1\right)}^2 \\ =1,210 \\ \mathrm{Thus}, \mathrm{the} \mathrm{required} \mathrm{amount} \mathrm{is} \mathrm{Rs} 1,210. \end{gathered}$$

Question 5:

Find the difference between the compound interest and simple interest. On a sum of Rs 50,000 at 10% per annum for 2 years.

Answer 5:

$$\begin{gathered} \mathrm{Given}: \\ \mathrm{P}=\mathrm{Rs} 50,000 \\ \mathrm{R}=10\% \mathrm{p}.\mathrm{a}. \\ \mathrm{n}=2 \mathrm{years} \\ \mathrm{We} \mathrm{know} \mathrm{that} \mathrm{amount} \mathrm{A} \mathrm{at} \mathrm{the} \mathrm{end} \mathrm{of} \mathrm{n} \mathrm{years} \mathrm{at} \mathrm{the} \mathrm{rate} \mathrm{R}\% \mathrm{per} \mathrm{annum} \mathrm{when} \mathrm{the} \mathrm{interest} \mathrm{is} \\ \mathrm{compounded} \mathrm{annually} \mathrm{is} \mathrm{given} \mathrm{by} \mathrm{A}=\mathrm{P}\left(1+\frac{\mathrm{R}}{100}\right). \\ \therefore \mathrm{A}=\mathrm{Rs} 50,000{\left(1+\frac{10}{100}\right)}^2 \\ =\mathrm{Rs} 50,000{\left(1.1\right)}^2 \\ =\mathrm{Rs} 60,500 \\ \mathrm{Also}, \\ \mathrm{CI}=\mathrm{A}-\mathrm{P} \\ =\mathrm{Rs} 60,500- \mathrm{Rs} 50,000 \\ =\mathrm{Rs} 10,500 \\ \mathrm{We} \mathrm{know} \mathrm{that}: \\ \mathrm{SI}=\frac{\mathrm{PRT}}{100} \\ =\frac{50,000\times 10\times 2}{100} \\ =\mathrm{Rs} 10,000 \\ \therefore \mathrm{Difference} \mathrm{between} \mathrm{CI} \mathrm{and} \mathrm{SI}=\mathrm{Rs} 10,500-\mathrm{Rs} 10,000 \\ =\mathrm{Rs} 500 \end{gathered}$$

Page-14.15

Question 6:

Amit borrowed Rs 16000 at $17\frac{1}{2}\%$ per annum simple interest. On the same day, he lent it to Ashu at the same rate but compounded annually. What does he gain at the end of 2 years?

Answer 6:

$$\begin{gathered} \mathrm{Amount} \mathrm{to} \mathrm{be} \mathrm{paid} \mathrm{by} \mathrm{Amit}: \\ \mathrm{SI}=\frac{\mathrm{PRT}}{100} \\ =\frac{16000\times 17.5\times 2}{100} \\ =\mathrm{Rs} 5,600 \\ \mathrm{Amount} \mathrm{gained} \mathrm{by} \mathrm{Amit}: \\ \mathrm{A}=\mathrm{P}{\left(1+\frac{\mathrm{R}}{100}\right)}^{\mathrm{n}} \\ =\mathrm{Rs} 16,000{\left(1+\frac{17.5}{100}\right)}^2 \\ =\mathrm{Rs} 16,000{\left(1.175\right)}^2 \\ =\mathrm{Rs} 22,090 \\ \mathrm{We} \mathrm{know} \mathrm{that}: \\ \mathrm{CI}=\mathrm{A}-\mathrm{P} \\ =\mathrm{Rs} 22,090-\mathrm{Rs} 16,000 \\ =\mathrm{Rs} 6090 \\ \mathrm{Amit}\text{'}\mathrm{s} \mathrm{gain} \mathrm{in} \mathrm{the} \mathrm{whole} \mathrm{transaction}=\mathrm{Rs} 6,090-\mathrm{Rs} 5,600 \\ =\mathrm{Rs} 490 \end{gathered}$$

Question 7:

Find the amount of Rs 4096 for 18 months at $12\frac{1}{2}\%$ per annum, the interest being compounded semi-annually.

Answer 7:

$$\begin{gathered} \mathrm{Given}: \\ \mathrm{P}=\mathrm{Rs} 4,096 \\ \mathrm{R}=12.5\% \mathrm{p}.\mathrm{a}. \\ \mathrm{n}=18 \mathrm{months}=1.5 \mathrm{years} \\ \mathrm{We} \mathrm{have}: \\ \mathrm{A}=\mathrm{P}{\left(1+\frac{\mathrm{R}}{100}\right)}^{\mathrm{n}} \\ \mathrm{When} \mathrm{the} \mathrm{interest} \mathrm{is} \mathrm{compounded} \mathrm{semi}-\mathrm{annually}, \mathrm{we} \mathrm{have}: \\ \mathrm{A}=\mathrm{P}{\left(1+\frac{\mathrm{R}}{200}\right)}^{2\mathrm{n}} \\ =\mathrm{Rs} 4,096{\left(1+\frac{12.5}{200}\right)}^3 \\ =\mathrm{Rs} 4,096{\left(1.0625\right)}^3 \\ =\mathrm{Rs} 4,913 \\ \mathrm{Thus}, \mathrm{the} \mathrm{required} \mathrm{amount} \mathrm{is} \mathrm{Rs} 4,913. \end{gathered}$$

Question 8:

Find the amount and the compound interest on Rs 8000 for $1\frac{1}{2}$ years at 10% per annum, compounded half-yearly.

Answer 8:

$$\begin{gathered} \mathrm{Given}: \\ \mathrm{P}=\mathrm{Rs} 8,000 \\ \mathrm{R}=10\% \mathrm{p}.\mathrm{a}. \\ \mathrm{n}=1.5 \mathrm{years} \\ \mathrm{When} \mathrm{compounded} \mathrm{half}-\mathrm{yearly}, \mathrm{we} \mathrm{have}: \\ \mathrm{A}=\mathrm{P}{\left(1+\frac{\mathrm{R}}{200}\right)}^{2\mathrm{n}} \\ =\mathrm{Rs} 8,000{\left(1+\frac{10}{200}\right)}^3 \\ =\mathrm{Rs} 8,000{\left(1.05\right)}^3 \\ =\mathrm{Rs} 9,261 \\ \mathrm{Also}, \\ \mathrm{CI}=\mathrm{A}-\mathrm{P} \\ =\mathrm{Rs} 9,261-\mathrm{Rs} 8,000 \\ =\mathrm{Rs} 1,261 \end{gathered}$$

Question 9:

Kamal borrowed Rs 57600 from LIC against her policy at $12\frac{1}{2}\%$ per annum to build a house. Find the amount that she pays to the LIC after $1\frac{1}{2}$ years if the interest is calculated half-yearly.

Answer 9:

$$\begin{gathered} \mathrm{Given}: \\ \mathrm{P}=\mathrm{Rs} 57,600 \\ \mathrm{R}=12.5\% \mathrm{p}.\mathrm{a}. \\ \mathrm{n}=1.5 \mathrm{years} \\ \mathrm{When} \mathrm{the} \mathrm{interest} \mathrm{is} \mathrm{compounded} \mathrm{half}-\mathrm{yearly}, \mathrm{we} \mathrm{have}: \\ \mathrm{A}=\mathrm{P}{\left(1+\frac{\mathrm{R}}{200}\right)}^{2\mathrm{n}} \\ =\mathrm{Rs} 57,600{\left(1+\frac{12.5}{200}\right)}^3 \\ =\mathrm{Rs} 57,600{\left(1.0625\right)}^3 \\ =\mathrm{Rs} 69,089.06 \\ \mathrm{Thus}, \mathrm{the} \mathrm{required} \mathrm{amount} \mathrm{is} \mathrm{Rs} 69,089.06. \end{gathered}$$

Question 10:

Abha purchased a house from Avas Parishad on credit. If the cost of the house is Rs 64000 and the rate of interest is 5% per annum compounded half-yearly, find the interest paid by Abha after one year and a half.

Answer 10:

$$\begin{gathered} \mathrm{Given}: \\ \mathrm{P}=\mathrm{Rs} 64,000 \\ \mathrm{R}=5\% \mathrm{p}.\mathrm{a}. \\ \mathrm{n}=1.5 \mathrm{years} \\ \mathrm{When} \mathrm{the} \mathrm{interest} \mathrm{is} \mathrm{compounded} \mathrm{half}-\mathrm{yearly}, \mathrm{we} \mathrm{have}: \\ \mathrm{A}=\mathrm{P}{\left(1+\frac{\mathrm{R}}{200}\right)}^{2\mathrm{n}} \\ =\mathrm{Rs} 64,000{\left(1+\frac{5}{200}\right)}^3 \\ =\mathrm{Rs} 64,000{\left(1.025\right)}^3 \\ =\mathrm{Rs} 68,921 \\ \mathrm{Also}, \\ \mathrm{CI}=\mathrm{A}-\mathrm{P} \\ =\mathrm{Rs} 68,921-\mathrm{Rs} 64,000 \\ =\mathrm{Rs} 4,921 \\ \mathrm{Thus}, \mathrm{the} \mathrm{required} \mathrm{interest} \mathrm{is} \mathrm{Rs} 4,921. \end{gathered}$$

Question 11:

Rakesh lent out Rs 10000 for 2 years at 20% per annum, compounded annually. How much more he could earn if the interest be compounded half-yearly?

Answer 11:

$$\begin{gathered} \mathrm{Given}: \\ \mathrm{P}=\mathrm{Rs} 10,000 \\ \mathrm{R}=20\% \mathrm{p}.\mathrm{a}. \\ \mathrm{n}=2 \mathrm{years} \\ \mathrm{A}=\mathrm{P}{\left(1+\frac{\mathrm{R}}{100}\right)}^{\mathrm{n}} \\ = \mathrm{Rs} 10,000{\left(1+\frac{20}{100}\right)}^2 \\ =\mathrm{Rs} 10,000{\left(1.2\right)}^2 \\ =\mathrm{Rs} 14,400 \\ \mathrm{When} \mathrm{the} \mathrm{interest} \mathrm{is} \mathrm{compounded} \mathrm{half}-\mathrm{yearly}, \mathrm{we} \mathrm{have}: \\ \mathrm{A}=\mathrm{P}{\left(1+\frac{\mathrm{R}}{200}\right)}^{2\mathrm{n}} \\ =\mathrm{Rs} 10,000{\left(1+\frac{20}{200}\right)}^4 \\ =\mathrm{Rs} 10,000{\left(1.1\right)}^4 \\ =\mathrm{Rs} 14,641 \\ \mathrm{Difference}=\mathrm{Rs} 14,641-\mathrm{Rs} 14,400 \\ =\mathrm{Rs} 241 \end{gathered}$$

Question 12:

Romesh borrowed a sum of Rs 245760 at 12.5% per annum, compounded annually. On the same day, he lent out his money to Ramu at the same rate of interest, but compounded semi-annually. Find his gain after 2 years.

Answer 12:

$$\begin{gathered} \mathrm{Given}: \\ \mathrm{P}=\mathrm{Rs} 245,760 \\ \mathrm{R}=12.5\% \mathrm{p}.\mathrm{a}. \\ \mathrm{n}=2 \mathrm{years} \\ \mathrm{When} \mathrm{compounded} \mathrm{annually}, \mathrm{we} \mathrm{have}: \\ \mathrm{A}=\mathrm{P}{\left(1+\frac{\mathrm{R}}{100}\right)}^{\mathrm{n}} \\ =\mathrm{Rs} 245,760{\left(1+\frac{12.5}{100}\right)}^2 \\ =\mathrm{Rs} 311,040 \\ \mathrm{When} \mathrm{compounded} \mathrm{semi}-\mathrm{annually}, \mathrm{we} \mathrm{have}: \\ \mathrm{A}=\mathrm{P}{\left(1+\frac{\mathrm{R}}{200}\right)}^{2\mathrm{n}} \\ =\mathrm{Rs} 245,760{\left(1+\frac{12.5}{200}\right)}^4 \\ =\mathrm{Rs} 245,760{\left(1.0625\right)}^4 \\ =\mathrm{Rs} 313,203.75 \\ \mathrm{Romesh}\text{'}\mathrm{s} \mathrm{gain}=\mathrm{Rs} 313,203.75-Rs 311,040 \\ =\mathrm{Rs} 2,163.75 \end{gathered}$$

Question 13:

Find the amount that David would receive if he invests Rs 8192 for 18 months at $12\frac{1}{2}\%$ per annum, the interest being compounded half-yearly.

Answer 13:

$$\begin{gathered} \mathrm{Given}: \\ \mathrm{P}=\mathrm{Rs} 8,192 \\ \mathrm{R}=12.5\% \mathrm{p}.\mathrm{a}. \\ \mathrm{n}=1.5 \mathrm{years} \\ \mathrm{When} \mathrm{the} \mathrm{interest} \mathrm{is} \mathrm{compounded} \mathrm{half}-\mathrm{yearly}, \mathrm{we} \mathrm{have}: \\ \mathrm{A}=\mathrm{P}{\left(1+\frac{\mathrm{R}}{200}\right)}^{2\mathrm{n}} \\ =\mathrm{Rs} 8,192{\left(1+\frac{12.5}{200}\right)}^3 \\ =\mathrm{Rs} 8,192{\left(1.0625\right)}^3 \\ =\mathrm{Rs} 9,826 \\ \mathrm{Thus}, \mathrm{the} \mathrm{required} \mathrm{amount} \mathrm{is} \mathrm{Rs} 9,826. \end{gathered}$$

Question 14:

Find the compound interest on Rs 15625 for 9 months, at 16% per annum, compounded quarterly.

Answer 14:

$$\begin{gathered} \mathrm{Given}: \\ \mathrm{P}=\mathrm{Rs} 15,625 \\ \mathrm{R}=16\%=\frac{16}{4}=4\% \mathrm{quarterly} \\ \mathrm{n}=9 \mathrm{months}=3 \mathrm{quarters} \\ \mathrm{We} \mathrm{know} \mathrm{that}: \\ \mathrm{A}=\mathrm{P}{\left(1+\frac{\mathrm{R}}{100}\right)}^{\mathrm{n}} \\ =\mathrm{Rs} 15,625{\left(1+\frac{4}{100}\right)}^3 \\ =\mathrm{Rs} 15,625{\left(1.04\right)}^3 \\ =\mathrm{Rs} 17,576 \\ \mathrm{Also}, \\ \mathrm{CI}=\mathrm{A}-\mathrm{P} \\ =\mathrm{Rs} 17,576-\mathrm{Rs} 15,625 \\ =\mathrm{Rs} 1,951 \\ \mathrm{Thus}, \mathrm{the} \mathrm{required} \mathrm{compound} \mathrm{interest} \mathrm{is} \mathrm{Rs} 1,951. \end{gathered}$$

Question 15:

Rekha deposited Rs 16000 in a foreign bank which pays interest at the rate of 20% per annum compounded quarterly, find the interest received by Rekha after one year.

Answer 15:

$$\begin{gathered} \mathrm{Given}: \\ \mathrm{P}=\mathrm{Rs} 16,000 \\ \mathrm{R}=20\% \mathrm{p}.\mathrm{a}. \\ \mathrm{n}=1 \mathrm{year} \\ \mathrm{We} \mathrm{know} \mathrm{that}: \\ \mathrm{A}=\mathrm{P}{\left(1+\frac{\mathrm{R}}{100}\right)}^{\mathrm{n}} \\ \mathrm{When} \mathrm{compounded} \mathrm{quarterly}, \mathrm{we} \mathrm{have}: \\ \mathrm{A}=\mathrm{P}{\left(1+\frac{\mathrm{R}}{400}\right)}^{4\mathrm{n}} \\ =\mathrm{Rs} 16,000{\left(1+\frac{20}{400}\right)}^4 \\ =\mathrm{Rs} 16,000{\left(1.05\right)}^4 \\ =\mathrm{Rs} 19,448.10 \\ \mathrm{Also}, \\ \mathrm{CI}=\mathrm{A}-\mathrm{P} \\ =\mathrm{Rs} 19,448.1-\mathrm{Rs} 16,000 \\ =\mathrm{Rs} 3,448.10 \\ \mathrm{Thus}, \mathrm{the} \mathrm{interest} \mathrm{received} \mathrm{by} \mathrm{Rekha} \mathrm{after} \mathrm{one} \mathrm{year} \mathrm{is} \mathrm{Rs} 3,448.10. \end{gathered}$$

Question 16:

Find the amount of Rs 12500 for 2 years compounded annually, the rate of interest being 15% for the first year and 16% for the second year.

Answer 16:

$$\begin{gathered} \mathrm{Given}: \\ \mathrm{P}=\mathrm{Rs} 12,500 \\ {\mathrm{R}}_1=15\% \mathrm{p}.\mathrm{a}. \\ {\mathrm{R}}_2=16\% \mathrm{p}.\mathrm{a}. \\ \therefore \mathrm{Amount} \mathrm{after} \mathrm{two} \mathrm{years}=\mathrm{P}\left(1+\frac{{\mathrm{R}}_1}{100}\right)\left(1+\frac{{\mathrm{R}}_2}{100}\right) \\ =\mathrm{Rs} 12,500\left(1+\frac{15}{100}\right)\left(1+\frac{16}{100}\right) \\ =\mathrm{Rs} 12,500\left(1.15\right)\left(1.16\right) \\ =\mathrm{Rs} 16,675 \\ \mathrm{Thus}, \mathrm{the} \mathrm{required} \mathrm{amount} \mathrm{is} \mathrm{Rs} 16,675. \end{gathered}$$

Question 17:

Ramu borrowed Rs 15625 from a finance company to buy a scooter. If the rate of interest be 16% per annum compounded annually, what payment will he have to make after $2\frac{1}{4}$ years?

Answer 17:

$$\begin{gathered} \mathrm{Given}: \\ \mathrm{P}=\mathrm{Rs} 15,625 \\ \mathrm{R}=16\% \mathrm{p}.\mathrm{a}. \\ \mathrm{n}=2\frac{1}{4} \mathrm{years} \\ \therefore \mathrm{Amount} \mathrm{after} 2\frac{1}{4} \mathrm{years}=\mathrm{P}{\left(1+\frac{\mathrm{R}}{100}\right)}^2\left(1+\frac{{\displaystyle \frac{1}{4}}\left(\mathrm{R}\right)}{100}\right) \\ =\mathrm{Rs} 15,625{\left(1+\frac{16}{100}\right)}^2\left(1+\frac{{\displaystyle \frac{16}{4}}}{100}\right) \\ =\mathrm{Rs} 15,625{\left(1.16\right)}^2\left(1.04\right) \\ =\mathrm{Rs} 21,866 \\ \mathrm{Thus}, \mathrm{the} \mathrm{required} \mathrm{amount} \mathrm{is} \mathrm{Rs} 21,866. \end{gathered}$$

Question 18:

What will Rs 125000 amount to at the rate of 6%, if the interest is calculated after every 3 months?

Answer 18:

$$\begin{gathered} \mathrm{Because} \mathrm{interest} \mathrm{is} \mathrm{calculated} \mathrm{after} \mathrm{every} 3 \mathrm{months}, \mathrm{it} \mathrm{is} \mathrm{compounded} \mathrm{quarterly}. \\ \mathrm{Given}: \\ \mathrm{P}=\mathrm{Rs} 125,000 \\ \mathrm{R}=6\% \mathrm{p}.\mathrm{a}.=\frac{6}{4}\% \mathrm{quarterly}=1.5\% \mathrm{quarterly} \\ \mathrm{n}=4 \\ \mathrm{So}, \\ \mathrm{A}=\mathrm{P}{\left(1+\frac{\mathrm{R}}{100}\right)}^{\mathrm{n}} \\ =125,000{\left(1+\frac{1.5}{100}\right)}^4 \\ =125,000{\left(1.015\right)}^4 \\ =132,670 \left(\mathrm{approx}\right) \\ \mathrm{Thus}, \mathrm{the} \mathrm{required} \mathrm{amount} \mathrm{is} \mathrm{Rs} 132,670. \end{gathered}$$

Question 19:

Find the compound interest at the rate of 5% for three years on that principal which in three years at the rate of 5% per annum gives Rs 12000 as simple interest.

Answer 19:

$$\begin{gathered} \mathrm{P}=\frac{\mathrm{SI}\times 100}{\mathrm{RT}} \\ \mathrm{According} \mathrm{to} \mathrm{the} \mathrm{given} \mathrm{values}, \mathrm{we} \mathrm{have}: \\ =\frac{12,000\times 100}{5\times 3} \\ =80,000 \\ \mathrm{The} \mathrm{principal} \mathrm{is} \mathrm{to} \mathrm{be} \mathrm{compounded} \mathrm{annually}. \\ \mathrm{So}, \\ \mathrm{A}=\mathrm{P}{\left(1+\frac{\mathrm{R}}{100}\right)}^{\mathrm{n}} \\ =80,000{\left(1+\frac{5}{100}\right)}^3 \\ =80,000{\left(1.05\right)}^3 \\ =92,610 \\ \mathrm{Now}, \\ \mathrm{CI}=\mathrm{A}-\mathrm{P} \\ =92,610-80,000 \\ =12,610 \\ \mathrm{Thus}, \mathrm{the} \mathrm{required} \mathrm{compound} \mathrm{interest} \mathrm{is} \mathrm{Rs} 12,610. \end{gathered}$$  

Question 20:

A sum of money was lent for 2 years at 20% compounded annually. If the interest is payable half-yearly instead of yearly, then the interest is Rs 482 more. Find the sum.

Answer 20:

$$\begin{gathered} \mathrm{A}=\mathrm{P}{\left(1+\frac{\mathrm{R}}{100}\right)}^{\mathrm{n}} \\ \mathrm{Also}, \\ \mathrm{P}=\mathrm{A}-\mathrm{CI} \\ \mathrm{Let} \mathrm{the} \mathrm{sum} \mathrm{of} \mathrm{money} \mathrm{be} \mathrm{Rs} \mathrm{x}. \\ \mathrm{If} \mathrm{the} \mathrm{interest} \mathrm{is} \mathrm{compounded} \mathrm{annually}, \mathrm{then}: \\ {\mathrm{A}}_1=\mathrm{x}{\left(1+\frac{20}{100}\right)}^2 \\ =1.44\mathrm{x} \\ \therefore \mathrm{CI}=1.44\mathrm{x}-\mathrm{x} \\ =0.44\mathrm{x} ...\left(1\right) \\ \mathrm{If} \mathrm{the} \mathrm{interest} \mathrm{is} \mathrm{compounded} \mathrm{half}-\mathrm{yearly}, \mathrm{then}: \\ {\mathrm{A}}_2=\mathrm{x}{\left(1+\frac{10}{100}\right)}^4 \\ =1.4641\mathrm{x} \\ \therefore \mathrm{CI}=1.4641\mathrm{x}-\mathrm{x} \\ =0.4641\mathrm{x} ...\left(2\right) \\ \mathrm{It} \mathrm{is} \mathrm{given} \mathrm{that} \mathrm{if} \mathrm{interest} \mathrm{is} \mathrm{compounded} \mathrm{half}-\mathrm{yearly}, \mathrm{then} \mathrm{it} \mathrm{will} \mathrm{be} \mathrm{Rs} 482 \mathrm{more}. \\ \therefore 0.4641\mathrm{x}=0.44\mathrm{x}+482 [\mathrm{From} (1) \mathrm{and} (2\left)\right] \\ 0.4641\mathrm{x}-0.44\mathrm{x}=482 \\ 0.0241\mathrm{x}=482 \\ \mathrm{x}=\frac{482}{0.0241} \\ =20,000 \\ \mathrm{Thus}, \mathrm{the} \mathrm{required} \mathrm{sum} \mathrm{is} \mathrm{Rs} 20,000. \end{gathered}$$  

Question 21:

Simple interest on a sum of money for 2 years at $6\frac{1}{2}\%$ per annum is Rs 5200. What will be the compound interest on the sum at the same rate for the same period?

Answer 21:

$$\begin{gathered} \mathrm{P}=\frac{\mathrm{SI}\times 100}{\mathrm{RT}} \\ \therefore \mathrm{P}=\frac{5,200\times 100}{6.5\times 2} \\ =40,000 \\ \mathrm{Now}, \\ \mathrm{A}=\mathrm{P}{\left(1+\frac{\mathrm{R}}{100}\right)}^{\mathrm{n}} \\ =40,000{\left(1+\frac{6.5}{100}\right)}^2 \\ =40,000{\left(1.065\right)}^2 \\ =45,369 \\ \mathrm{Also}, \\ \mathrm{CI}=\mathrm{A}-\mathrm{P} \\ =45,369-40,000 \\ =5,369 \\ \mathrm{Thus}, \mathrm{the} \mathrm{required} \mathrm{compound} \mathrm{interest} \mathrm{is} \mathrm{Rs} 5,369. \end{gathered}$$  

Question 22:

Find the compound interest at the rate of 5% per annum for 3 years on that principal which in 3 years at the rate of 5% per annum gives Rs 1200 as simple interest.

Answer 22:

$$\begin{gathered} \mathrm{We} \mathrm{know} \mathrm{that}: \\ \mathrm{P}=\frac{\mathrm{SI}\times 100}{\mathrm{RT}} \\ \therefore \mathrm{P}=\frac{1200\times 100}{5\times 3} \\ =8,000 \\ \mathrm{Now}, \\ \mathrm{A}=\mathrm{P}{\left(1+\frac{\mathrm{R}}{100}\right)}^{\mathrm{n}} \\ =8,000{\left(1+\frac{5}{100}\right)}^3 \\ =8,000{\left(1.05\right)}^3 \\ =9,261 \\ \mathrm{Now}, \\ \mathrm{CI}=\mathrm{A}-\mathrm{P} \\ =9,261-8,000 \\ =1,261 \\ \mathrm{Thus}, \mathrm{the} \mathrm{required} \mathrm{compound} \mathrm{interest} \mathrm{is} \mathrm{Rs} 1,261. \end{gathered}$$