\displaystyle \textbf{Exercise 2(A)}

\displaystyle \textbf{Question 1: }\text{Calculate the amount and the compound interest on:}
\displaystyle \text{(i) Rs. }3,500\text{ at }10\%\text{ per annum in }2\text{ years.}
\displaystyle \text{(ii) Rs. }6,000\text{ in }3\text{ years at }5\%\text{ per year.}
\displaystyle \text{Answer:}
\displaystyle \text{(i) Principal }=Rs.\ 3,500,\ \text{Rate }=10\%\text{ p.a., Time }=2\text{ years}
\displaystyle \text{Interest for first year}=\frac{3500\times10}{100}=Rs.\ 350
\displaystyle \text{Amount at the end of first year}=3500+350=Rs.\ 3,850
\displaystyle \text{Interest for second year}=\frac{3850\times10}{100}=Rs.\ 385
\displaystyle \text{Amount at the end of second year}=3850+385=Rs.\ 4,235
\displaystyle \therefore \text{Amount}=Rs.\ 4,235
\displaystyle \text{Compound Interest}=4235-3500=Rs.\ 735
\displaystyle \\

\displaystyle \text{(ii) Principal }=Rs.\ 6,000,\ \text{Rate }=5\%\text{ p.a., Time }=3\text{ years}
\displaystyle \text{Interest for first year}=\frac{6000\times5}{100}=Rs.\ 300
\displaystyle \text{Amount at the end of first year}=6000+300=Rs.\ 6,300
\displaystyle \text{Interest for second year}=\frac{6300\times5}{100}=Rs.\ 315
\displaystyle \text{Amount at the end of second year}=6300+315=Rs.\ 6,615
\displaystyle \text{Interest for third year}=\frac{6615\times5}{100}=Rs.\ 330.75
\displaystyle \text{Amount at the end of third year}=6615+330.75=Rs.\ 6,945.75
\displaystyle \therefore \text{Amount}=Rs.\ 6,945.75
\displaystyle \text{Compound Interest}=6945.75-6000=Rs.\ 945.75
\displaystyle \\

\displaystyle \textbf{Question 2: }\text{Calculate the amount and the compound interest on:}
\displaystyle \text{(i) Rs. }8,000\text{ in }2\frac{1}{2}\text{ years at }15\%\text{ per annum.}
\displaystyle \text{(ii) Rs. }20,000\text{ in }2\frac{1}{4}\text{ years at }10\%\text{ per annum.}
\displaystyle \text{Answer:}
\displaystyle \text{(i) Principal }=Rs.\ 8,000,\ \text{Rate }=15\%\text{ p.a., Time }=2\frac{1}{2}\text{ years}
\displaystyle \text{Interest for first year}=\frac{8000\times15}{100}=Rs.\ 1,200
\displaystyle \text{Amount at the end of first year}=8000+1200=Rs.\ 9,200
\displaystyle \text{Interest for second year}=\frac{9200\times15}{100}=Rs.\ 1,380
\displaystyle \text{Amount at the end of second year}=9200+1380=Rs.\ 10,580
\displaystyle \text{Interest for next }6\text{ months}=\frac{10580\times15\times1}{100\times2}=Rs.\ 793.50
\displaystyle \text{Amount at the end of }2\frac{1}{2}\text{ years}=10580+793.50=Rs.\ 11,373.50
\displaystyle \therefore \text{Amount}=Rs.\ 11,373.50
\displaystyle \text{Compound Interest}=11373.50-8000=Rs.\ 3,373.50
\displaystyle \\

\displaystyle \text{(ii) Principal }=Rs.\ 20,000,\ \text{Rate }=10\%\text{ p.a., Time }=2\frac{1}{4}\text{ years}
\displaystyle \text{Interest for first year}=\frac{20000\times10}{100}=Rs.\ 2,000
\displaystyle \text{Amount at the end of first year}=20000+2000=Rs.\ 22,000
\displaystyle \text{Interest for second year}=\frac{22000\times10}{100}=Rs.\ 2,200
\displaystyle \text{Amount at the end of second year}=22000+2200=Rs.\ 24,200
\displaystyle \text{Interest for next }3\text{ months}=\frac{24200\times10\times3}{100\times12}=Rs.\ 605
\displaystyle \text{Amount at the end of }2\frac{1}{4}\text{ years}=24200+605=Rs.\ 24,805
\displaystyle \therefore \text{Amount}=Rs.\ 24,805
\displaystyle \text{Compound Interest}=24805-20000=Rs.\ 4,805
\displaystyle \\

\displaystyle \textbf{Question 3: }\text{Calculate the amount and the compound interest on:}
\displaystyle \text{(i) Rs. }4,600\text{ in }2\text{ years when the rates of interest for}
\displaystyle \text{the successive years are }10\%\text{ and }12\%\text{ respectively.}
\displaystyle \text{(ii) Rs. }16,000\text{ in }3\text{ years when the rates of interest for}
\displaystyle \text{the successive years are }10\%,14\%\text{ and }15\%\text{ respectively.}
\displaystyle \text{Answer:}
\displaystyle \text{(i) Principal }=Rs.\ 4,600
\displaystyle \text{Interest for first year}=\frac{4600\times10}{100}=Rs.\ 460
\displaystyle \text{Amount at the end of first year}=4600+460=Rs.\ 5,060
\displaystyle \text{Interest for second year}=\frac{5060\times12}{100}=Rs.\ 607.20
\displaystyle \text{Amount at the end of second year}=5060+607.20=Rs.\ 5,667.20
\displaystyle \therefore \text{Amount}=Rs.\ 5,667.20
\displaystyle \text{Compound Interest}=5667.20-4600=Rs.\ 1,067.20
\displaystyle \\

\displaystyle \text{(ii) Principal }=Rs.\ 16,000
\displaystyle \text{Interest for first year}=\frac{16000\times10}{100}=Rs.\ 1,600
\displaystyle \text{Amount at the end of first year}=16000+1600=Rs.\ 17,600
\displaystyle \text{Interest for second year}=\frac{17600\times14}{100}=Rs.\ 2,464
\displaystyle \text{Amount at the end of second year}=17600+2464=Rs.\ 20,064
\displaystyle \text{Interest for third year}=\frac{20064\times15}{100}=Rs.\ 3,009.60
\displaystyle \text{Amount at the end of third year}=20064+3009.60=Rs.\ 23,073.60
\displaystyle \therefore \text{Amount}=Rs.\ 23,073.60
\displaystyle \text{Compound Interest}=23073.60-16000=Rs.\ 7,073.60
\displaystyle \\

\displaystyle \textbf{Question 4: }\text{Find the compound interest, correct to the nearest}
\displaystyle \text{rupee, on Rs. }2,400\text{ for }2\frac{1}{2}\text{ years at }5\%\text{ per annum.}
\displaystyle \text{Answer:}
\displaystyle \text{Principal }=Rs.\ 2,400,\ \text{Rate }=5\%\text{ p.a., Time }=2\frac{1}{2}\text{ years}
\displaystyle \text{Interest for first year}=\frac{2400\times5}{100}=Rs.\ 120
\displaystyle \text{Amount at the end of first year}=2400+120=Rs.\ 2,520
\displaystyle \text{Interest for second year}=\frac{2520\times5}{100}=Rs.\ 126
\displaystyle \text{Amount at the end of second year}=2520+126=Rs.\ 2,646
\displaystyle \text{Interest for next }6\text{ months}=\frac{2646\times5\times1}{100\times2}=Rs.\ 66.15
\displaystyle \text{Amount at the end of }2\frac{1}{2}\text{ years}=2646+66.15=Rs.\ 2,712.15
\displaystyle \text{Compound Interest}=2712.15-2400=Rs.\ 312.15
\displaystyle \therefore \text{Compound Interest}\approx Rs.\ 312\text{ (nearest rupee)}
\displaystyle \\

\displaystyle \textbf{Question 5: }\text{Calculate the compound interest for the second year}
\displaystyle \text{on Rs. }8,000\text{ invested for }3\text{ years at }10\%\text{ per annum.}
\displaystyle \text{Answer:}
\displaystyle \text{Principal }=Rs.\ 8,000,\ \text{Rate }=10\%\text{ p.a.}
\displaystyle \text{Interest for first year}=\frac{8000\times10}{100}=Rs.\ 800
\displaystyle \text{Amount at the end of first year}=8000+800=Rs.\ 8,800
\displaystyle \text{Interest for the second year}=\frac{8800\times10}{100}=Rs.\ 880
\displaystyle \therefore \text{Compound Interest for the second year}=Rs.\ 880
\displaystyle \\

\displaystyle \textbf{Question 6: }A\text{ borrowed Rs. }2,500\text{ from }B\text{ at }12\%\text{ per}
\displaystyle \text{annum compound interest. After }2\text{ years, }A\text{ gave Rs. }2,936\text{ and a}
\displaystyle \text{watch to }B\text{ to clear the account. Find the cost of the watch.}
\displaystyle \text{Answer:}
\displaystyle \text{Principal}=Rs.\ 2,500,\quad \text{Rate}=12\%\text{ p.a.}
\displaystyle \text{Interest for first year}=\frac{2500\times12}{100}=Rs.\ 300
\displaystyle \text{Amount at the end of first year}=2500+300=Rs.\ 2,800
\displaystyle \text{Interest for second year}=\frac{2800\times12}{100}=Rs.\ 336
\displaystyle \text{Amount at the end of second year}=2800+336=Rs.\ 3,136
\displaystyle \text{Amount paid in cash}=Rs.\ 2,936
\displaystyle \text{Cost of the watch}=3136-2936=Rs.\ 200
\displaystyle \therefore \text{The cost of the watch is Rs. }200.
\displaystyle \\

\displaystyle \textbf{Question 7: }\text{How much will Rs. }50,000\text{ amount to in }3\text{ years,}
\displaystyle \text{compounded yearly, if the rates for the successive years are }6\%,
\displaystyle 8\%\text{ and }10\%\text{ respectively?}
\displaystyle \text{Answer:}
\displaystyle \text{Principal}=Rs.\ 50,000
\displaystyle \text{Interest for first year}=\frac{50000\times6}{100}=Rs.\ 3,000
\displaystyle \text{Amount at the end of first year}=50000+3000=Rs.\ 53,000
\displaystyle \text{Interest for second year}=\frac{53000\times8}{100}=Rs.\ 4,240
\displaystyle \text{Amount at the end of second year}=53000+4240=Rs.\ 57,240
\displaystyle \text{Interest for third year}=\frac{57240\times10}{100}=Rs.\ 5,724
\displaystyle \text{Amount at the end of third year}=57240+5724=Rs.\ 62,964
\displaystyle \therefore \text{The amount after }3\text{ years is Rs. }62,964.
\displaystyle \\

\displaystyle \textbf{Question 8: }\text{Meenal lends Rs. }75,000\text{ at C.I. for }3\text{ years. If the}
\displaystyle \text{rate of interest for the first two years is }15\%\text{ per year and for}
\displaystyle \text{the third year it is }16\%,\text{ calculate the sum Meenal will get at}
\displaystyle \text{the end of the third year.}
\displaystyle \text{Answer:}
\displaystyle \text{Principal}=Rs.\ 75,000
\displaystyle \text{Interest for first year}=\frac{75000\times15}{100}=Rs.\ 11,250
\displaystyle \text{Amount at the end of first year}=75000+11250=Rs.\ 86,250
\displaystyle \text{Interest for second year}=\frac{86250\times15}{100}=Rs.\ 12,937.50
\displaystyle \text{Amount at the end of second year}=86250+12937.50
\displaystyle =Rs.\ 99,187.50
\displaystyle \text{Interest for third year}=\frac{99187.50\times16}{100}=Rs.\ 15,870
\displaystyle \text{Amount at the end of third year}=99187.50+15870
\displaystyle =Rs.\ 1,15,057.50
\displaystyle \therefore \text{Meenal will get Rs. }1,15,057.50\text{ at the end of the third year.}
\displaystyle \\

\displaystyle \textbf{Question 9: }\text{Govind borrows Rs. }18,000\text{ at }10\%\text{ simple interest.}
\displaystyle \text{He immediately invests the money at }10\%\text{ compound interest}
\displaystyle \text{compounded half-yearly. How much money does Govind gain in one year?}
\displaystyle \text{Answer:}
\displaystyle \text{Money borrowed}=Rs.\ 18,000
\displaystyle \text{Simple Interest for one year}=\frac{18000\times10\times1}{100}=Rs.\ 1,800
\displaystyle \text{Amount to be repaid after one year}=18000+1800=Rs.\ 19,800
\displaystyle \text{The money is invested at }10\%\text{ compounded half-yearly.}
\displaystyle \text{Rate per half-year}=\frac{10}{2}=5\%
\displaystyle \text{Interest for first half-year}=\frac{18000\times5}{100}=Rs.\ 900
\displaystyle \text{Amount at the end of first half-year}=18000+900=Rs.\ 18,900
\displaystyle \text{Interest for second half-year}=\frac{18900\times5}{100}=Rs.\ 945
\displaystyle \text{Amount at the end of one year}=18900+945=Rs.\ 19,845
\displaystyle \text{Gain}=19845-19800=Rs.\ 45
\displaystyle \therefore \text{Govind gains Rs. }45.
\displaystyle \\

\displaystyle \textbf{Question 10: }\text{Find the compound interest on Rs. }4,000\text{ accrued in}
\displaystyle \text{three years, when the rate of interest is }8\%\text{ for the first year}
\displaystyle \text{and }10\%\text{ per year for the second and third years.}
\displaystyle \text{Answer:}
\displaystyle \text{Principal}=Rs.\ 4,000
\displaystyle \text{Interest for first year}=\frac{4000\times8}{100}=Rs.\ 320
\displaystyle \text{Amount at the end of first year}=4000+320=Rs.\ 4,320
\displaystyle \text{Interest for second year}=\frac{4320\times10}{100}=Rs.\ 432
\displaystyle \text{Amount at the end of second year}=4320+432=Rs.\ 4,752
\displaystyle \text{Interest for third year}=\frac{4752\times10}{100}=Rs.\ 475.20
\displaystyle \text{Amount at the end of third year}=4752+475.20=Rs.\ 5,227.20
\displaystyle \text{Compound Interest}=5227.20-4000=Rs.\ 1,227.20
\displaystyle \therefore \text{The compound interest is Rs. }1,227.20.
\displaystyle \\

\displaystyle \textbf{Exercise 2(B)}

\displaystyle \textbf{Question 1: }\text{Calculate the difference between the simple}
\displaystyle \text{interest and the compound interest on Rs. }4,000\text{ in }2\text{ years}
\displaystyle \text{at }8\%\text{ per annum compounded yearly.}
\displaystyle \text{Answer:}
\displaystyle \text{Principal}=Rs.\ 4,000,\quad \text{Rate}=8\%\text{ p.a.}
\displaystyle \text{Simple Interest for }2\text{ years}=\frac{4000\times8\times2}{100}
\displaystyle =Rs.\ 640
\displaystyle \text{Interest for first year}=\frac{4000\times8}{100}=Rs.\ 320
\displaystyle \text{Amount at the end of first year}=4000+320=Rs.\ 4,320
\displaystyle \text{Interest for second year}=\frac{4320\times8}{100}=Rs.\ 345.60
\displaystyle \text{Compound Interest}=320+345.60=Rs.\ 665.60
\displaystyle \text{Required difference}=665.60-640=Rs.\ 25.60
\displaystyle \therefore \text{The difference between C.I. and S.I. is Rs. }25.60.
\displaystyle \\

\displaystyle \textbf{Question 2: }A\text{ man lends Rs. }12,500\text{ at }12\%\text{ for the first}
\displaystyle \text{year, at }15\%\text{ for the second year and at }18\%\text{ for the third}
\displaystyle \text{year. If the rates of interest are compounded yearly, find the}
\displaystyle \text{difference between the C.I. of the first year and the C.I. of the}
\displaystyle \text{third year.}
\displaystyle \text{Answer:}
\displaystyle \text{Principal}=Rs.\ 12,500
\displaystyle \text{Interest for first year}=\frac{12500\times12}{100}=Rs.\ 1,500
\displaystyle \text{Amount at the end of first year}=12500+1500=Rs.\ 14,000
\displaystyle \text{Interest for second year}=\frac{14000\times15}{100}=Rs.\ 2,100
\displaystyle \text{Amount at the end of second year}=14000+2100=Rs.\ 16,100
\displaystyle \text{Interest for third year}=\frac{16100\times18}{100}=Rs.\ 2,898
\displaystyle \text{Required difference}=2898-1500=Rs.\ 1,398
\displaystyle \therefore \text{The required difference is Rs. }1,398.
\displaystyle \\

\displaystyle \textbf{Question 3: }A\text{ sum of money is lent at }8\%\text{ per annum}
\displaystyle \text{compound interest. If the interest for the second year exceeds that}
\displaystyle \text{for the first year by Rs. }96,\text{ find the sum of money.}
\displaystyle \text{Answer:}
\displaystyle \text{Let the sum of money be Rs. }P.
\displaystyle \text{Interest for the first year}=\frac{P\times8}{100}=\frac{2P}{25}
\displaystyle \text{Amount at the end of the first year}=P+\frac{2P}{25}
\displaystyle =\frac{27P}{25}
\displaystyle \text{Interest for the second year}=\frac{27P}{25}\times\frac{8}{100}
\displaystyle =\frac{54P}{625}
\displaystyle \text{According to the question,}
\displaystyle \frac{54P}{625}-\frac{2P}{25}=96
\displaystyle \frac{54P-50P}{625}=96
\displaystyle \frac{4P}{625}=96
\displaystyle P=\frac{96\times625}{4}
\displaystyle P=15,000
\displaystyle \therefore \text{The sum of money is Rs. }15,000.
\displaystyle \\

\displaystyle \textbf{Question 4: }A\text{ man borrows Rs. }6,000\text{ at }5\%\text{ C.I. per}
\displaystyle \text{annum. If he repays Rs. }1,200\text{ at the end of each year, find the}
\displaystyle \text{amount of the loan outstanding at the beginning of the third year.}
\displaystyle \text{Answer:}
\displaystyle \text{Original loan}=Rs.\ 6,000
\displaystyle \text{Interest for the first year}=\frac{6000\times5}{100}=Rs.\ 300
\displaystyle \text{Amount due at the end of the first year}=6000+300
\displaystyle =Rs.\ 6,300
\displaystyle \text{Amount repaid at the end of the first year}=Rs.\ 1,200
\displaystyle \text{Outstanding loan after the first repayment}=6300-1200
\displaystyle =Rs.\ 5,100
\displaystyle \text{Interest for the second year}=\frac{5100\times5}{100}=Rs.\ 255
\displaystyle \text{Amount due at the end of the second year}=5100+255
\displaystyle =Rs.\ 5,355
\displaystyle \text{Amount repaid at the end of the second year}=Rs.\ 1,200
\displaystyle \text{Outstanding loan after the second repayment}=5355-1200
\displaystyle =Rs.\ 4,155
\displaystyle \therefore \text{The loan outstanding at the beginning of the third year}
\displaystyle \text{is Rs. }4,155.
\displaystyle \\

\displaystyle \textbf{Question 5: }A\text{ man borrows Rs. }5,000\text{ at }12\%\text{ compound}
\displaystyle \text{interest payable every six months. He repays Rs. }1,800\text{ at the end}
\displaystyle \text{of every six months. Calculate the third payment he has to make at}
\displaystyle \text{the end of }18\text{ months in order to clear the entire loan.}
\displaystyle \text{Answer:}
\displaystyle \text{Principal}=Rs.\ 5,000
\displaystyle \text{Rate per annum}=12\%
\displaystyle \text{Rate per half-year}=\frac{12}{2}\%=6\%
\displaystyle \text{Interest for the first six months}=\frac{5000\times6}{100}
\displaystyle =Rs.\ 300
\displaystyle \text{Amount due at the end of the first six months}=5000+300
\displaystyle =Rs.\ 5,300
\displaystyle \text{Balance after the first payment}=5300-1800
\displaystyle =Rs.\ 3,500
\displaystyle \text{Interest for the next six months}=\frac{3500\times6}{100}
\displaystyle =Rs.\ 210
\displaystyle \text{Amount due at the end of }12\text{ months}=3500+210
\displaystyle =Rs.\ 3,710
\displaystyle \text{Balance after the second payment}=3710-1800
\displaystyle =Rs.\ 1,910
\displaystyle \text{Interest for the next six months}=\frac{1910\times6}{100}
\displaystyle =Rs.\ 114.60
\displaystyle \text{Amount due at the end of }18\text{ months}=1910+114.60
\displaystyle =Rs.\ 2,024.60
\displaystyle \therefore \text{The third payment required to clear the loan is Rs. }2,024.60.
\displaystyle \\

\displaystyle \textbf{Question 6: }\text{On a certain sum of money, the difference between}
\displaystyle \text{the compound interest for a year, payable half-yearly, and the simple}
\displaystyle \text{interest for a year is Rs. }180.\text{ Find the sum lent out, if the rate}
\displaystyle \text{of interest in both cases is }10\%\text{ per annum.}
\displaystyle \text{Answer:}
\displaystyle \text{Let the sum lent out be Rs. }P.
\displaystyle \text{Simple Interest for one year}=\frac{P\times10\times1}{100}
\displaystyle =\frac{P}{10}
\displaystyle \text{For compound interest, rate per half-year}=\frac{10}{2}\%=5\%
\displaystyle \text{Interest for the first half-year}=\frac{P\times5}{100}
\displaystyle =\frac{P}{20}
\displaystyle \text{Amount at the end of the first half-year}=P+\frac{P}{20}
\displaystyle =\frac{21P}{20}
\displaystyle \text{Interest for the second half-year}=\frac{21P}{20}\times\frac{5}{100}
\displaystyle =\frac{21P}{400}
\displaystyle \text{Compound Interest for one year}=\frac{P}{20}+\frac{21P}{400}
\displaystyle =\frac{20P+21P}{400}
\displaystyle =\frac{41P}{400}
\displaystyle \text{According to the question,}
\displaystyle \frac{41P}{400}-\frac{P}{10}=180
\displaystyle \frac{41P-40P}{400}=180
\displaystyle \frac{P}{400}=180
\displaystyle P=180\times400
\displaystyle P=72,000
\displaystyle \therefore \text{The sum lent out is Rs. }72,000.
\displaystyle \\

\displaystyle \textbf{Question 7: }A\text{ manufacturer estimates that his machine depreciates}
\displaystyle \text{by }15\%\text{ of its value at the beginning of the year. Find the}
\displaystyle \text{original value (cost) of the machine, if it depreciates by Rs. }5,355
\displaystyle \text{during the second year.}
\displaystyle \text{Answer:}
\displaystyle \text{Let the original value of the machine be Rs. }P.
\displaystyle \text{Value at the beginning of the second year}=P-\frac{15P}{100}
\displaystyle =\frac{85P}{100}=\frac{17P}{20}
\displaystyle \text{Depreciation during the second year}=15\%\text{ of }\frac{17P}{20}
\displaystyle =\frac{15}{100}\times\frac{17P}{20}
\displaystyle =\frac{51P}{400}
\displaystyle \text{According to the question,}
\displaystyle \frac{51P}{400}=5,355
\displaystyle P=\frac{5355\times400}{51}
\displaystyle P=42,000
\displaystyle \therefore \text{The original value (cost) of the machine is Rs. }42,000.
\displaystyle \\

\displaystyle \textbf{Question 8: }A\text{ man invests Rs. }5,600\text{ at }14\%\text{ per annum}
\displaystyle \text{compound interest for }2\text{ years. Calculate:}
\displaystyle \text{(i) the interest for the first year.}
\displaystyle \text{(ii) the amount at the end of the first year.}
\displaystyle \text{(iii) the interest for the second year, correct to the nearest rupee.}
\displaystyle \text{Answer:}
\displaystyle \text{Principal}=Rs.\ 5,600,\quad \text{Rate}=14\%\text{ p.a.}
\displaystyle \text{(i) Interest for the first year}=\frac{5600\times14}{100}
\displaystyle =Rs.\ 784
\displaystyle \therefore \text{Interest for the first year}=Rs.\ 784
\displaystyle \\

\displaystyle \text{(ii) Amount at the end of the first year}
\displaystyle =5600+784
\displaystyle =Rs.\ 6,384
\displaystyle \therefore \text{Amount at the end of the first year}=Rs.\ 6,384
\displaystyle \\

\displaystyle \text{(iii) Interest for the second year}=\frac{6384\times14}{100}
\displaystyle =Rs.\ 893.76
\displaystyle \therefore \text{Interest for the second year}\approx Rs.\ 894
\displaystyle \text{(correct to the nearest rupee)}
\displaystyle \\

\displaystyle \textbf{Question 9:}
\displaystyle \text{(i) Find the difference between the compound interest of the second}
\displaystyle \text{year and the compound interest of the third year on Rs. }48,000
\displaystyle \text{invested for }5\text{ years at }10\%\text{ per annum compounded yearly.}
\displaystyle \text{(ii) A sum of Rs. }50,000\text{ is invested for }8\text{ years at compound}
\displaystyle \text{interest, the rate of interest being }10\%,12\%,14\%\text{ and }16\%
\displaystyle \text{respectively for the first }4\text{ consecutive years. Find the total}
\displaystyle \text{of the interests earned during the first and third years.}
\displaystyle \text{Answer:}
\displaystyle \text{(i) Principal}=Rs.\ 48,000,\quad \text{Rate}=10\%\text{ p.a.}
\displaystyle \text{Interest for the first year}=\frac{48000\times10}{100}
\displaystyle =Rs.\ 4,800
\displaystyle \text{Amount at the end of the first year}=48000+4800
\displaystyle =Rs.\ 52,800
\displaystyle \text{Compound interest for the second year}=\frac{52800\times10}{100}
\displaystyle =Rs.\ 5,280
\displaystyle \text{Amount at the end of the second year}=52800+5280
\displaystyle =Rs.\ 58,080
\displaystyle \text{Compound interest for the third year}=\frac{58080\times10}{100}
\displaystyle =Rs.\ 5,808
\displaystyle \text{Required difference}=5808-5280
\displaystyle =Rs.\ 528
\displaystyle \therefore \text{The required difference is Rs. }528.
\displaystyle \\

\displaystyle \text{(ii) Principal}=Rs.\ 50,000
\displaystyle \text{Interest for the first year}=\frac{50000\times10}{100}
\displaystyle =Rs.\ 5,000
\displaystyle \text{Amount at the end of the first year}=50000+5000
\displaystyle =Rs.\ 55,000
\displaystyle \text{Interest for the second year}=\frac{55000\times12}{100}
\displaystyle =Rs.\ 6,600
\displaystyle \text{Amount at the end of the second year}=55000+6600
\displaystyle =Rs.\ 61,600
\displaystyle \text{Interest for the third year}=\frac{61600\times14}{100}
\displaystyle =Rs.\ 8,624
\displaystyle \text{Total interest earned during the first and third years}
\displaystyle =5000+8624
\displaystyle =Rs.\ 13,624
\displaystyle \therefore \text{The required total interest is Rs. }13,624.
\displaystyle \\

\displaystyle \textbf{Question 10: }A\text{ man saves Rs. }3,000\text{ every year and invests}
\displaystyle \text{it at the end of the year at }10\%\text{ compound interest. Calculate}
\displaystyle \text{the total amount of his savings at the end of the third year.}
\displaystyle \text{Answer:}
\displaystyle \text{The first sum of Rs. }3,000\text{ is invested at the end of the first year.}
\displaystyle \text{Interest on the first sum during the second year}
\displaystyle =\frac{3000\times10}{100}=Rs.\ 300
\displaystyle \text{Amount of the first sum at the end of the second year}
\displaystyle =3000+300=Rs.\ 3,300
\displaystyle \text{Interest on this amount during the third year}
\displaystyle =\frac{3300\times10}{100}=Rs.\ 330
\displaystyle \text{Amount of the first sum at the end of the third year}
\displaystyle =3300+330=Rs.\ 3,630
\displaystyle \text{The second sum of Rs. }3,000\text{ is invested at the end of the second year.}
\displaystyle \text{Interest on the second sum during the third year}
\displaystyle =\frac{3000\times10}{100}=Rs.\ 300
\displaystyle \text{Amount of the second sum at the end of the third year}
\displaystyle =3000+300=Rs.\ 3,300
\displaystyle \text{The third sum of Rs. }3,000\text{ is invested at the end of the third year.}
\displaystyle \text{Total savings at the end of the third year}
\displaystyle =3630+3300+3000
\displaystyle =Rs.\ 9,930
\displaystyle \therefore \text{The total amount of his savings is Rs. }9,930.
\displaystyle \\

\displaystyle \textbf{Question 11: }A\text{ man borrows Rs. }10,000\text{ at }5\%\text{ per annum}
\displaystyle \text{compound interest. He repays }35\%\text{ of the sum borrowed at the}
\displaystyle \text{end of the first year and }42\%\text{ of the sum borrowed at the end}
\displaystyle \text{of the second year. How much must he pay at the end of the third}
\displaystyle \text{year in order to clear the debt?}
\displaystyle \text{Answer:}
\displaystyle \text{Original loan}=Rs.\ 10,000
\displaystyle \text{First repayment}=35\%\text{ of Rs. }10,000
\displaystyle =\frac{35}{100}\times10000=Rs.\ 3,500
\displaystyle \text{Second repayment}=42\%\text{ of Rs. }10,000
\displaystyle =\frac{42}{100}\times10000=Rs.\ 4,200
\displaystyle \text{Interest for the first year}=\frac{10000\times5}{100}
\displaystyle =Rs.\ 500
\displaystyle \text{Amount due at the end of the first year}=10000+500
\displaystyle =Rs.\ 10,500
\displaystyle \text{Balance after the first repayment}=10500-3500
\displaystyle =Rs.\ 7,000
\displaystyle \text{Interest for the second year}=\frac{7000\times5}{100}
\displaystyle =Rs.\ 350
\displaystyle \text{Amount due at the end of the second year}=7000+350
\displaystyle =Rs.\ 7,350
\displaystyle \text{Balance after the second repayment}=7350-4200
\displaystyle =Rs.\ 3,150
\displaystyle \text{Interest for the third year}=\frac{3150\times5}{100}
\displaystyle =Rs.\ 157.50
\displaystyle \text{Amount due at the end of the third year}=3150+157.50
\displaystyle =Rs.\ 3,307.50
\displaystyle \therefore \text{He must pay Rs. }3,307.50\text{ to clear the debt.}
\displaystyle \\

\displaystyle \textbf{Question 12: }\text{Mr. Mehta invested Rs. }8,000\text{ every year at the}
\displaystyle \text{beginning of the year, at }10\%\text{ per annum compound interest.}
\displaystyle \text{Calculate his total savings at the beginning of the third year.}
\displaystyle \text{Answer:}
\displaystyle \text{The first sum of Rs. }8,000\text{ is invested at the beginning of the}
\displaystyle \text{first year and earns interest for two years.}
\displaystyle \text{Interest on the first sum during the first year}
\displaystyle =\frac{8000\times10}{100}=Rs.\ 800
\displaystyle \text{Amount at the end of the first year}=8000+800
\displaystyle =Rs.\ 8,800
\displaystyle \text{Interest on this amount during the second year}
\displaystyle =\frac{8800\times10}{100}=Rs.\ 880
\displaystyle \text{Amount of the first sum at the beginning of the third year}
\displaystyle =8800+880=Rs.\ 9,680
\displaystyle \text{The second sum of Rs. }8,000\text{ is invested at the beginning of the}
\displaystyle \text{second year and earns interest for one year.}
\displaystyle \text{Interest on the second sum during the second year}
\displaystyle =\frac{8000\times10}{100}=Rs.\ 800
\displaystyle \text{Amount of the second sum at the beginning of the third year}
\displaystyle =8000+800=Rs.\ 8,800
\displaystyle \text{The third sum of Rs. }8,000\text{ is invested at the beginning of the}
\displaystyle \text{third year and has not yet earned any interest.}
\displaystyle \text{Total savings at the beginning of the third year}
\displaystyle =9680+8800+8000
\displaystyle =Rs.\ 26,480
\displaystyle \therefore \text{Mr. Mehta's total savings are Rs. }26,480.
\displaystyle \\

\displaystyle \textbf{Exercise 2(C)}

\displaystyle \textbf{Question 1: }A\text{ sum is invested at compound interest compounded}
\displaystyle \text{yearly. If the interest for two successive years are Rs. }5,700\text{ and}
\displaystyle \text{Rs. }7,410,\text{ calculate the rate of interest.}
\displaystyle \text{Answer:}
\displaystyle \text{Interest for the first year}=Rs.\ 5,700
\displaystyle \text{Interest for the second year}=Rs.\ 7,410
\displaystyle \text{Increase in interest}=7410-5700=Rs.\ 1,710
\displaystyle \text{Under compound interest, the increase in interest during the second}
\displaystyle \text{year is the interest on the first year's interest.}
\displaystyle \therefore \frac{5700\times R}{100}=1710
\displaystyle R=\frac{1710\times100}{5700}
\displaystyle R=30
\displaystyle \therefore \text{The rate of interest is }30\%\text{ per annum.}
\displaystyle \\

\displaystyle \textbf{Question 2: }A\text{ certain sum of money is put at compound interest,}
\displaystyle \text{compounded half-yearly. If the interest for two successive half-years}
\displaystyle \text{are Rs. }650\text{ and Rs. }760.50,\text{ find the rate of interest.}
\displaystyle \text{Answer:}
\displaystyle \text{Interest for the first half-year}=Rs.\ 650
\displaystyle \text{Interest for the second half-year}=Rs.\ 760.50
\displaystyle \text{Increase in interest}=760.50-650=Rs.\ 110.50
\displaystyle \text{Since the interest is compounded half-yearly,}
\displaystyle \text{the increase in interest is the interest on the first half-year's interest.}
\displaystyle \therefore \frac{650\times R}{100}=110.50
\displaystyle R=\frac{110.50\times100}{650}
\displaystyle R=17\%
\displaystyle \text{This is the rate for one half-year.}
\displaystyle \therefore \text{Annual rate}=2\times17\%=34\%
\displaystyle \therefore \text{The rate of interest is }34\%\text{ per annum.}
\displaystyle \\

\displaystyle \textbf{Question 3: }A\text{ certain sum amounts to Rs. }5,292\text{ in two}
\displaystyle \text{years and Rs. }5,556.60\text{ in three years, interest being}
\displaystyle \text{compounded annually. Find:}
\displaystyle \text{(i) the rate of interest}
\displaystyle \text{(ii) the original sum.}
\displaystyle \text{Answer:}
\displaystyle \text{Amount at the end of the second year}=Rs.\ 5,292
\displaystyle \text{Amount at the end of the third year}=Rs.\ 5,556.60
\displaystyle \text{Interest for the third year}=5556.60-5292
\displaystyle =Rs.\ 264.60
\displaystyle \text{(i) Let the rate of interest be }R\%\text{ per annum.}
\displaystyle \frac{5292\times R}{100}=264.60
\displaystyle R=\frac{264.60\times100}{5292}
\displaystyle R=5
\displaystyle \therefore \text{The rate of interest is }5\%\text{ per annum.}
\displaystyle \\

\displaystyle \text{(ii) Let the amount at the end of the first year be Rs. }A.
\displaystyle \text{Interest for the second year}=\frac{5A}{100}
\displaystyle A+\frac{5A}{100}=5292
\displaystyle \frac{105A}{100}=5292
\displaystyle A=\frac{5292\times100}{105}
\displaystyle A=5,040
\displaystyle \text{Let the original sum be Rs. }P.
\displaystyle P+\frac{5P}{100}=5040
\displaystyle \frac{105P}{100}=5040
\displaystyle P=\frac{5040\times100}{105}
\displaystyle P=4,800
\displaystyle \therefore \text{The original sum is Rs. }4,800.
\displaystyle \\

\displaystyle \textbf{Question 4: }\text{The compound interest, calculated yearly, on a}
\displaystyle \text{certain sum of money for the second year is Rs. }1,089\text{ and for}
\displaystyle \text{the third year it is Rs. }1,197.90.\text{ Calculate the rate of interest}
\displaystyle \text{and the sum of money.}
\displaystyle \text{Answer:}
\displaystyle \text{Interest for the second year}=Rs.\ 1,089
\displaystyle \text{Interest for the third year}=Rs.\ 1,197.90
\displaystyle \text{Increase in interest}=1197.90-1089
\displaystyle =Rs.\ 108.90
\displaystyle \text{The increase is the interest on the second year's interest.}
\displaystyle \text{Let the rate of interest be }R\%\text{ per annum.}
\displaystyle \frac{1089\times R}{100}=108.90
\displaystyle R=\frac{108.90\times100}{1089}
\displaystyle R=10
\displaystyle \therefore \text{The rate of interest is }10\%\text{ per annum.}
\displaystyle \\

\displaystyle \text{Let the amount at the end of the first year be Rs. }A.
\displaystyle \frac{A\times10}{100}=1089
\displaystyle A=\frac{1089\times100}{10}
\displaystyle A=10,890
\displaystyle \text{Let the original sum be Rs. }P.
\displaystyle P+\frac{10P}{100}=10890
\displaystyle \frac{110P}{100}=10890
\displaystyle P=\frac{10890\times100}{110}
\displaystyle P=9,900
\displaystyle \therefore \text{The sum of money is Rs. }9,900.
\displaystyle \\

\displaystyle \textbf{Question 5: }\text{Mohit invests Rs. }8,000\text{ for }3\text{ years at a}
\displaystyle \text{certain rate of interest, compounded annually. At the end of one}
\displaystyle \text{year it amounts to Rs. }9,440.\text{ Calculate:}
\displaystyle \text{(i) the rate of interest per annum.}
\displaystyle \text{(ii) the amount at the end of the second year.}
\displaystyle \text{(iii) the interest accrued in the third year.}
\displaystyle \text{Answer:}
\displaystyle \text{Principal}=Rs.\ 8,000
\displaystyle \text{Amount at the end of the first year}=Rs.\ 9,440
\displaystyle \text{Interest for the first year}=9440-8000
\displaystyle =Rs.\ 1,440
\displaystyle \text{(i) Let the rate of interest be }R\%\text{ per annum.}
\displaystyle \frac{8000\times R}{100}=1440
\displaystyle R=\frac{1440\times100}{8000}
\displaystyle R=18
\displaystyle \therefore \text{The rate of interest is }18\%\text{ per annum.}
\displaystyle \\

\displaystyle \text{(ii) Interest for the second year}=\frac{9440\times18}{100}
\displaystyle =Rs.\ 1,699.20
\displaystyle \text{Amount at the end of the second year}=9440+1699.20
\displaystyle =Rs.\ 11,139.20
\displaystyle \therefore \text{The amount at the end of the second year is Rs. }11,139.20.
\displaystyle \\

\displaystyle \text{(iii) Interest accrued in the third year}
\displaystyle =\frac{11139.20\times18}{100}
\displaystyle =Rs.\ 2,005.056
\displaystyle \therefore \text{The interest accrued in the third year is Rs. }2,005.06.
\displaystyle \\

\displaystyle \textbf{Question 6: }\text{Geeta borrowed Rs. }15,000\text{ for }18\text{ months at a}
\displaystyle \text{certain rate of interest compounded semi-annually. If at the end of}
\displaystyle \text{six months it amounted to Rs. }15,600,\text{ calculate:}
\displaystyle \text{(i) the rate of interest per annum.}
\displaystyle \text{(ii) the total amount of money that Geeta must pay at the end of}
\displaystyle 18\text{ months in order to clear the account.}
\displaystyle \text{Answer:}
\displaystyle \text{Principal}=Rs.\ 15,000
\displaystyle \text{Amount at the end of the first six months}=Rs.\ 15,600
\displaystyle \text{Interest for the first six months}=15600-15000
\displaystyle =Rs.\ 600
\displaystyle \text{(i) Let the rate of interest per half-year be }R\%.
\displaystyle \frac{15000\times R}{100}=600
\displaystyle R=\frac{600\times100}{15000}
\displaystyle R=4
\displaystyle \text{Rate of interest per annum}=2\times4\%
\displaystyle =8\%
\displaystyle \therefore \text{The rate of interest is }8\%\text{ per annum.}
\displaystyle \\

\displaystyle \text{(ii) Interest for the second six months}
\displaystyle =\frac{15600\times4}{100}
\displaystyle =Rs.\ 624
\displaystyle \text{Amount at the end of }12\text{ months}=15600+624
\displaystyle =Rs.\ 16,224
\displaystyle \text{Interest for the third six months}
\displaystyle =\frac{16224\times4}{100}
\displaystyle =Rs.\ 648.96
\displaystyle \text{Amount at the end of }18\text{ months}=16224+648.96
\displaystyle =Rs.\ 16,872.96
\displaystyle \therefore \text{Geeta must pay Rs. }16,872.96\text{ to clear the account.}
\displaystyle \\

\displaystyle \textbf{Question 7: }\text{Ramesh invests Rs. }12,800\text{ for }3\text{ years at the}
\displaystyle \text{rate of }10\%\text{ per annum compound interest. Find:}
\displaystyle \text{(i) the sum due to Ramesh at the end of the first year.}
\displaystyle \text{(ii) the interest he earns for the second year.}
\displaystyle \text{(iii) the total amount due to him at the end of the third year.}
\displaystyle \text{Answer:}
\displaystyle \text{Principal}=Rs.\ 12,800,\quad \text{Rate}=10\%\text{ p.a.}
\displaystyle \text{(i) Interest for the first year}=\frac{12800\times10}{100}
\displaystyle =Rs.\ 1,280
\displaystyle \text{Amount at the end of the first year}=12800+1280
\displaystyle =Rs.\ 14,080
\displaystyle \therefore \text{The sum due at the end of the first year is Rs. }14,080.
\displaystyle \\

\displaystyle \text{(ii) Interest for the second year}=\frac{14080\times10}{100}
\displaystyle =Rs.\ 1,408
\displaystyle \therefore \text{The interest for the second year is Rs. }1,408.
\displaystyle \\

\displaystyle \text{(iii) Amount at the end of the second year}=14080+1408
\displaystyle =Rs.\ 15,488
\displaystyle \text{Interest for the third year}=\frac{15488\times10}{100}
\displaystyle =Rs.\ 1,548.80
\displaystyle \text{Amount at the end of the third year}=15488+1548.80
\displaystyle =Rs.\ 17,036.80
\displaystyle \therefore \text{The total amount due at the end of the third year is}
\displaystyle \text{Rs. }17,036.80.
\displaystyle \\

\displaystyle \textbf{Question 8: }\text{The compound interest, calculated yearly, on a}
\displaystyle \text{certain sum of money for the second year is Rs. }864\text{ and for}
\displaystyle \text{the third year it is Rs. }933.12.\text{ Calculate the rate of interest}
\displaystyle \text{and the compound interest on the same sum and at the same rate}
\displaystyle \text{for the fourth year.}
\displaystyle \text{Answer:}
\displaystyle \text{Interest for the second year}=Rs.\ 864
\displaystyle \text{Interest for the third year}=Rs.\ 933.12
\displaystyle \text{Increase in interest}=933.12-864
\displaystyle =Rs.\ 69.12
\displaystyle \text{Let the rate of interest be }R\%\text{ per annum.}
\displaystyle \frac{864\times R}{100}=69.12
\displaystyle R=\frac{69.12\times100}{864}
\displaystyle R=8
\displaystyle \therefore \text{The rate of interest is }8\%\text{ per annum.}
\displaystyle \\

\displaystyle \text{Amount at the end of the third year}
\displaystyle =933.12+\frac{933.12\times100}{8}
\displaystyle =933.12+11664
\displaystyle =Rs.\ 12,597.12
\displaystyle \text{Compound interest for the fourth year}
\displaystyle =\frac{12597.12\times8}{100}
\displaystyle =Rs.\ 1,007.7696
\displaystyle \therefore \text{Compound interest for the fourth year}
\displaystyle \approx Rs.\ 1,007.77.
\displaystyle \\

\displaystyle \textbf{Question 9: }A\text{ sum of money placed out at compound interest}
\displaystyle \text{amounts to Rs. }20,160\text{ in }3\text{ years and to Rs. }24,192\text{ in}
\displaystyle 4\text{ years. Calculate:}
\displaystyle \text{(i) the rate of interest.}
\displaystyle \text{(ii) amount in }2\text{ years.}
\displaystyle \text{(iii) amount in }5\text{ years.}
\displaystyle \text{Answer:}
\displaystyle \text{Amount at the end of the third year}=Rs.\ 20,160
\displaystyle \text{Amount at the end of the fourth year}=Rs.\ 24,192
\displaystyle \text{Interest for the fourth year}=24192-20160
\displaystyle =Rs.\ 4,032
\displaystyle \text{(i) Let the rate of interest be }R\%\text{ per annum.}
\displaystyle \frac{20160\times R}{100}=4032
\displaystyle R=\frac{4032\times100}{20160}
\displaystyle R=20
\displaystyle \therefore \text{The rate of interest is }20\%\text{ per annum.}
\displaystyle \\

\displaystyle \text{(ii) Let the amount at the end of the second year be Rs. }A.
\displaystyle A+\frac{20A}{100}=20160
\displaystyle \frac{120A}{100}=20160
\displaystyle A=\frac{20160\times100}{120}
\displaystyle A=16,800
\displaystyle \therefore \text{The amount at the end of }2\text{ years is Rs. }16,800.
\displaystyle \\

\displaystyle \text{(iii) Interest for the fifth year}=\frac{24192\times20}{100}
\displaystyle =Rs.\ 4,838.40
\displaystyle \text{Amount at the end of the fifth year}=24192+4838.40
\displaystyle =Rs.\ 29,030.40
\displaystyle \therefore \text{The amount at the end of }5\text{ years is Rs. }29,030.40.
\displaystyle \\

\displaystyle \textbf{Question 10: }\text{Rs. }8,000\text{ is lent out at }7\%\text{ compound}
\displaystyle \text{interest for }2\text{ years. At the end of the first year Rs. }3,560\text{ are}
\displaystyle \text{returned. Calculate:}
\displaystyle \text{(i) the interest paid for the second year.}
\displaystyle \text{(ii) the total interest paid in two years.}
\displaystyle \text{(iii) the total amount of money paid in two years to clear the debt.}
\displaystyle \text{Answer:}
\displaystyle \text{Original loan}=Rs.\ 8,000
\displaystyle \text{Interest for the first year}=\frac{8000\times7}{100}
\displaystyle =Rs.\ 560
\displaystyle \text{Amount due at the end of the first year}=8000+560
\displaystyle =Rs.\ 8,560
\displaystyle \text{Amount returned at the end of the first year}=Rs.\ 3,560
\displaystyle \text{Outstanding balance}=8560-3560
\displaystyle =Rs.\ 5,000
\displaystyle \text{(i) Interest for the second year}=\frac{5000\times7}{100}
\displaystyle =Rs.\ 350
\displaystyle \therefore \text{The interest paid for the second year is Rs. }350.
\displaystyle \\

\displaystyle \text{(ii) Total interest paid in two years}=560+350
\displaystyle =Rs.\ 910
\displaystyle \therefore \text{The total interest paid in two years is Rs. }910.
\displaystyle \\

\displaystyle \text{Amount payable at the end of the second year}=5000+350
\displaystyle =Rs.\ 5,350
\displaystyle \text{(iii) Total amount paid in two years}=3560+5350
\displaystyle =Rs.\ 8,910
\displaystyle \therefore \text{The total amount paid to clear the debt is Rs. }8,910.
\displaystyle \\

\displaystyle \textbf{Question 11: }\text{The cost of a machine depreciated by Rs. }4,000
\displaystyle \text{during the first year and by Rs. }3,600\text{ during the second year.}
\displaystyle \text{Calculate:}
\displaystyle \text{(i) the rate of depreciation.}
\displaystyle \text{(ii) the original cost of the machine.}
\displaystyle \text{(iii) its cost at the end of the third year.}
\displaystyle \text{Answer:}
\displaystyle \text{Depreciation during the first year}=Rs.\ 4,000
\displaystyle \text{Depreciation during the second year}=Rs.\ 3,600
\displaystyle \text{Decrease in yearly depreciation}=4000-3600
\displaystyle =Rs.\ 400
\displaystyle \text{(i) The decrease of Rs. }400\text{ is the depreciation on Rs. }4,000.
\displaystyle \text{Let the rate of depreciation be }R\%\text{ per annum.}
\displaystyle \frac{4000\times R}{100}=400
\displaystyle R=\frac{400\times100}{4000}
\displaystyle R=10
\displaystyle \therefore \text{The rate of depreciation is }10\%\text{ per annum.}
\displaystyle \\

\displaystyle \text{(ii) Let the original cost of the machine be Rs. }P.
\displaystyle \frac{P\times10}{100}=4000
\displaystyle P=\frac{4000\times100}{10}
\displaystyle P=40,000
\displaystyle \therefore \text{The original cost of the machine is Rs. }40,000.
\displaystyle \\

\displaystyle \text{Value of the machine at the end of the first year}
\displaystyle =40000-4000
\displaystyle =Rs.\ 36,000
\displaystyle \text{Value of the machine at the end of the second year}
\displaystyle =36000-3600
\displaystyle =Rs.\ 32,400
\displaystyle \text{Depreciation during the third year}=\frac{32400\times10}{100}
\displaystyle =Rs.\ 3,240
\displaystyle \text{(iii) Value of the machine at the end of the third year}
\displaystyle =32400-3240
\displaystyle =Rs.\ 29,160
\displaystyle \therefore \text{The cost of the machine at the end of the third year is}
\displaystyle \text{Rs. }29,160.
\displaystyle \\

\displaystyle \textbf{Question 12: }\text{The cost of a machine is Rs. }32,000.\text{ Its value}
\displaystyle \text{depreciates at the rate of }5\%\text{ every year. Find the total}
\displaystyle \text{depreciation in its value by the end of }2\text{ years.}
\displaystyle \text{Answer:}
\displaystyle \text{Original cost of the machine}=Rs.\ 32,000
\displaystyle \text{Rate of depreciation}=5\%\text{ per annum}
\displaystyle \text{Depreciation during the first year}=\frac{32000\times5}{100}
\displaystyle =Rs.\ 1,600
\displaystyle \text{Value of the machine at the end of the first year}
\displaystyle =32000-1600
\displaystyle =Rs.\ 30,400
\displaystyle \text{Depreciation during the second year}=\frac{30400\times5}{100}
\displaystyle =Rs.\ 1,520
\displaystyle \text{Total depreciation in }2\text{ years}=1600+1520
\displaystyle =Rs.\ 3,120
\displaystyle \therefore \text{The total depreciation by the end of }2\text{ years is}
\displaystyle \text{Rs. }3,120.
\displaystyle \\

\displaystyle \textbf{Question 13: }\text{Find the sum, invested at }10\%\text{ compounded}
\displaystyle \text{annually, on which the interest for the third year exceeds the}
\displaystyle \text{interest of the first year by Rs. }252.
\displaystyle \text{Answer:}
\displaystyle \text{Let the sum invested be Rs. }P.
\displaystyle \text{Interest for the first year}=\frac{P\times10}{100}
\displaystyle =\frac{P}{10}
\displaystyle \text{Amount at the end of the first year}=P+\frac{P}{10}
\displaystyle =\frac{11P}{10}
\displaystyle \text{Interest for the second year}=\frac{11P}{10}\times\frac{10}{100}
\displaystyle =\frac{11P}{100}
\displaystyle \text{Amount at the end of the second year}
\displaystyle =\frac{11P}{10}+\frac{11P}{100}
\displaystyle =\frac{121P}{100}
\displaystyle \text{Interest for the third year}=\frac{121P}{100}\times\frac{10}{100}
\displaystyle =\frac{121P}{1000}
\displaystyle \text{According to the question,}
\displaystyle \frac{121P}{1000}-\frac{P}{10}=252
\displaystyle \frac{121P-100P}{1000}=252
\displaystyle \frac{21P}{1000}=252
\displaystyle P=\frac{252\times1000}{21}
\displaystyle P=12,000
\displaystyle \therefore \text{The sum invested is Rs. }12,000.
\displaystyle \\

\displaystyle \textbf{Question 14: }A\text{ man borrows Rs. }10,000\text{ at }10\%\text{ compound}
\displaystyle \text{interest compounded yearly. At the end of each year, he pays back}
\displaystyle 30\%\text{ of the sum borrowed. How much money is left unpaid just after}
\displaystyle \text{the second year?}
\displaystyle \text{Answer:}
\displaystyle \text{Principal}=Rs.\ 10,000
\displaystyle \text{Interest for the first year}=\frac{10000\times10}{100}=Rs.\ 1,000
\displaystyle \text{Amount due at the end of the first year}=10000+1000
\displaystyle =Rs.\ 11,000
\displaystyle \text{Amount repaid at the end of the first year}
\displaystyle =30\%\text{ of }Rs.\ 10,000=Rs.\ 3,000
\displaystyle \text{Outstanding balance after the first repayment}
\displaystyle =11000-3000=Rs.\ 8,000
\displaystyle \text{Interest for the second year}=\frac{8000\times10}{100}
\displaystyle =Rs.\ 800
\displaystyle \text{Amount due at the end of the second year}=8000+800
\displaystyle =Rs.\ 8,800
\displaystyle \text{Amount repaid at the end of the second year}
\displaystyle =30\%\text{ of }Rs.\ 10,000=Rs.\ 3,000
\displaystyle \text{Money left unpaid after the second year}=8800-3000
\displaystyle =Rs.\ 5,800
\displaystyle \therefore \text{The money left unpaid just after the second year is}
\displaystyle \text{Rs. }5,800.
\displaystyle \\

\displaystyle \textbf{Question 15: }A\text{ man borrows Rs. }10,000\text{ at }10\%\text{ compound}
\displaystyle \text{interest compounded yearly. At the end of each year, he pays back}
\displaystyle 20\%\text{ of the amount for that year. How much money is left unpaid}
\displaystyle \text{just after the second year?}
\displaystyle \text{Answer:}
\displaystyle \text{Principal}=Rs.\ 10,000
\displaystyle \text{Interest for the first year}=\frac{10000\times10}{100}=Rs.\ 1,000
\displaystyle \text{Amount due at the end of the first year}=10000+1000
\displaystyle =Rs.\ 11,000
\displaystyle \text{Amount repaid at the end of the first year}
\displaystyle =20\%\text{ of }11000=Rs.\ 2,200
\displaystyle \text{Outstanding balance after the first repayment}
\displaystyle =11000-2200=Rs.\ 8,800
\displaystyle \text{Interest for the second year}=\frac{8800\times10}{100}
\displaystyle =Rs.\ 880
\displaystyle \text{Amount due at the end of the second year}=8800+880
\displaystyle =Rs.\ 9,680
\displaystyle \text{Amount repaid at the end of the second year}
\displaystyle =20\%\text{ of }9680=Rs.\ 1,936
\displaystyle \text{Money left unpaid after the second year}=9680-1936
\displaystyle =Rs.\ 7,744
\displaystyle \therefore \text{The money left unpaid just after the second year is}
\displaystyle \text{Rs. }7,744.
\displaystyle \\

\displaystyle \textbf{Exercise 2(D)}

\displaystyle \textbf{Question 1: }\text{What sum will amount to Rs. }6,593.40\text{ in}
\displaystyle 2\text{ years at C.I., if the rates are }10\%\text{ and }11\%\text{ for the two}
\displaystyle \text{successive years?}
\displaystyle \text{Answer:}
\displaystyle \text{Let the original sum be Rs. }P.
\displaystyle \text{Interest for the first year}=\frac{P\times10}{100}
\displaystyle =\frac{P}{10}
\displaystyle \text{Amount at the end of the first year}=P+\frac{P}{10}
\displaystyle =\frac{11P}{10}
\displaystyle \text{Interest for the second year}=\frac{11P}{10}\times\frac{11}{100}
\displaystyle =\frac{121P}{1000}
\displaystyle \text{Amount at the end of the second year}
\displaystyle =\frac{11P}{10}+\frac{121P}{1000}
\displaystyle =\frac{1100P+121P}{1000}
\displaystyle =\frac{1221P}{1000}
\displaystyle \text{According to the question,}
\displaystyle \frac{1221P}{1000}=6593.40
\displaystyle P=\frac{6593.40\times1000}{1221}
\displaystyle P=5,400
\displaystyle \therefore \text{The original sum is Rs. }5,400.
\displaystyle \\

\displaystyle \textbf{Question 2: }\text{The value of a machine depreciated by }10\%
\displaystyle \text{per year during the first two years and }15\%\text{ per year during}
\displaystyle \text{the third year. Express the total depreciation of the machine, as}
\displaystyle \text{percent, during the three years.}
\displaystyle \text{Answer:}
\displaystyle \text{Let the original value of the machine be Rs. }100.
\displaystyle \text{Depreciation during the first year}=\frac{100\times10}{100}
\displaystyle =Rs.\ 10
\displaystyle \text{Value at the end of the first year}=100-10
\displaystyle =Rs.\ 90
\displaystyle \text{Depreciation during the second year}=\frac{90\times10}{100}
\displaystyle =Rs.\ 9
\displaystyle \text{Value at the end of the second year}=90-9
\displaystyle =Rs.\ 81
\displaystyle \text{Depreciation during the third year}=\frac{81\times15}{100}
\displaystyle =Rs.\ 12.15
\displaystyle \text{Value at the end of the third year}=81-12.15
\displaystyle =Rs.\ 68.85
\displaystyle \text{Total depreciation in three years}=100-68.85
\displaystyle =Rs.\ 31.15
\displaystyle \text{Depreciation percent}=\frac{31.15}{100}\times100\%
\displaystyle =31.15\%
\displaystyle \therefore \text{The total depreciation during the three years is }31.15\%.
\displaystyle \\

\displaystyle \textbf{Question 3: }\text{Rachna borrows Rs. }12,000\text{ at }10\%\text{ per}
\displaystyle \text{annum interest compounded half-yearly. She repays Rs. }4,000\text{ at}
\displaystyle \text{the end of every six months. Calculate the third payment she has}
\displaystyle \text{to make at the end of }18\text{ months to clear the entire loan.}
\displaystyle \text{Answer:}
\displaystyle \text{Principal}=Rs.\ 12,000
\displaystyle \text{Rate per half-year}=\frac{10}{2}\%=5\%
\displaystyle \text{Interest for the first six months}=\frac{12000\times5}{100}
\displaystyle =Rs.\ 600
\displaystyle \text{Amount due at the end of the first six months}=12000+600
\displaystyle =Rs.\ 12,600
\displaystyle \text{Balance after the first payment}=12600-4000
\displaystyle =Rs.\ 8,600
\displaystyle \text{Interest for the next six months}=\frac{8600\times5}{100}
\displaystyle =Rs.\ 430
\displaystyle \text{Amount due at the end of }12\text{ months}=8600+430
\displaystyle =Rs.\ 9,030
\displaystyle \text{Balance after the second payment}=9030-4000
\displaystyle =Rs.\ 5,030
\displaystyle \text{Interest for the next six months}=\frac{5030\times5}{100}
\displaystyle =Rs.\ 251.50
\displaystyle \text{Amount due at the end of }18\text{ months}=5030+251.50
\displaystyle =Rs.\ 5,281.50
\displaystyle \therefore \text{The third payment required to clear the loan is Rs. }5,281.50.
\displaystyle \\

\displaystyle \textbf{Question 4: }\text{On a certain sum of money, invested at the rate}
\displaystyle \text{of }10\%\text{ per annum compounded annually, the interest for the}
\displaystyle \text{first year plus the interest for the third year is Rs. }2,652.
\displaystyle \text{Find the sum.}
\displaystyle \text{Answer:}
\displaystyle \text{Let the sum invested be Rs. }P.
\displaystyle \text{Interest for the first year}=\frac{P\times10}{100}
\displaystyle =\frac{P}{10}
\displaystyle \text{Amount at the end of the first year}=P+\frac{P}{10}
\displaystyle =\frac{11P}{10}
\displaystyle \text{Interest for the second year}=\frac{11P}{10}\times\frac{10}{100}
\displaystyle =\frac{11P}{100}
\displaystyle \text{Amount at the end of the second year}
\displaystyle =\frac{11P}{10}+\frac{11P}{100}
\displaystyle =\frac{121P}{100}
\displaystyle \text{Interest for the third year}=\frac{121P}{100}\times\frac{10}{100}
\displaystyle =\frac{121P}{1000}
\displaystyle \text{According to the question,}
\displaystyle \frac{P}{10}+\frac{121P}{1000}=2652
\displaystyle \frac{100P+121P}{1000}=2652
\displaystyle \frac{221P}{1000}=2652
\displaystyle P=\frac{2652\times1000}{221}
\displaystyle P=12,000
\displaystyle \therefore \text{The sum invested is Rs. }12,000.
\displaystyle \\

\displaystyle \textbf{Question 5: }\text{During every financial year, the value of a}
\displaystyle \text{machine depreciates by }12\%.\text{ Find the original cost of a machine}
\displaystyle \text{which depreciates by Rs. }2,640\text{ during the second financial year}
\displaystyle \text{of its purchase.}
\displaystyle \text{Answer:}
\displaystyle \text{Let the original cost of the machine be Rs. }P.
\displaystyle \text{Depreciation during the first year}=\frac{12P}{100}
\displaystyle \text{Value at the beginning of the second year}
\displaystyle =P-\frac{12P}{100}
\displaystyle =\frac{88P}{100}
\displaystyle \text{Depreciation during the second year}
\displaystyle =\frac{88P}{100}\times\frac{12}{100}
\displaystyle =\frac{264P}{2500}
\displaystyle \text{According to the question,}
\displaystyle \frac{264P}{2500}=2640
\displaystyle P=\frac{2640\times2500}{264}
\displaystyle P=25,000
\displaystyle \therefore \text{The original cost of the machine is Rs. }25,000.
\displaystyle \\

\displaystyle \textbf{Question 6: }\text{Find the sum on which the difference between}
\displaystyle \text{the simple interest and the compound interest at the rate of }8\%
\displaystyle \text{per annum compounded annually is Rs. }64\text{ in }2\text{ years.}
\displaystyle \text{Answer:}
\displaystyle \text{Let the required sum be Rs. }P.
\displaystyle \text{Simple Interest for the first year}=\frac{P\times8}{100}
\displaystyle =\frac{2P}{25}
\displaystyle \text{Amount at the end of the first year}=P+\frac{2P}{25}
\displaystyle =\frac{27P}{25}
\displaystyle \text{Compound Interest for the second year}
\displaystyle =\frac{27P}{25}\times\frac{8}{100}
\displaystyle =\frac{54P}{625}
\displaystyle \text{Simple Interest for the second year}=\frac{2P}{25}
\displaystyle \text{Difference between C.I. and S.I. for }2\text{ years}
\displaystyle =\frac{54P}{625}-\frac{2P}{25}
\displaystyle =\frac{54P-50P}{625}
\displaystyle =\frac{4P}{625}
\displaystyle \text{According to the question,}
\displaystyle \frac{4P}{625}=64
\displaystyle P=\frac{64\times625}{4}
\displaystyle P=10,000
\displaystyle \therefore \text{The required sum is Rs. }10,000.
\displaystyle \\

\displaystyle \textbf{Question 7: }A\text{ sum of Rs. }13,500\text{ is invested at }16\%\text{ per}
\displaystyle \text{annum compound interest for }5\text{ years. Calculate:}
\displaystyle \text{(i) the interest for the first year.}
\displaystyle \text{(ii) the amount at the end of the first year.}
\displaystyle \text{(iii) the interest for the second year, correct to the nearest rupee.}
\displaystyle \text{Answer:}
\displaystyle \text{Principal}=Rs.\ 13,500,\quad \text{Rate}=16\%\text{ p.a.}
\displaystyle \text{(i) Interest for the first year}=\frac{13500\times16}{100}
\displaystyle =Rs.\ 2,160
\displaystyle \therefore \text{The interest for the first year is Rs. }2,160.
\displaystyle \\

\displaystyle \text{(ii) Amount at the end of the first year}=13500+2160
\displaystyle =Rs.\ 15,660
\displaystyle \therefore \text{The amount at the end of the first year is Rs. }15,660.
\displaystyle \\

\displaystyle \text{(iii) Interest for the second year}=\frac{15660\times16}{100}
\displaystyle =Rs.\ 2,505.60
\displaystyle \therefore \text{The interest for the second year}\approx Rs.\ 2,506
\displaystyle \text{(correct to the nearest rupee).}
\displaystyle \\

\displaystyle \textbf{Question 8: }\text{Saurabh invests Rs. }48,000\text{ for }7\text{ years at }10\%
\displaystyle \text{per annum compound interest. Calculate:}
\displaystyle \text{(i) the interest for the first year.}
\displaystyle \text{(ii) the amount at the end of the second year.}
\displaystyle \text{(iii) the interest for the third year.}
\displaystyle \text{Answer:}
\displaystyle \text{Principal}=Rs.\ 48,000,\quad \text{Rate}=10\%\text{ p.a.}
\displaystyle \text{(i) Interest for the first year}=\frac{48000\times10}{100}
\displaystyle =Rs.\ 4,800
\displaystyle \therefore \text{The interest for the first year is Rs. }4,800.
\displaystyle \\

\displaystyle \text{Amount at the end of the first year}=48000+4800
\displaystyle =Rs.\ 52,800
\displaystyle \text{Interest for the second year}=\frac{52800\times10}{100}
\displaystyle =Rs.\ 5,280
\displaystyle \text{(ii) Amount at the end of the second year}=52800+5280
\displaystyle =Rs.\ 58,080
\displaystyle \therefore \text{The amount at the end of the second year is Rs. }58,080.
\displaystyle \\

\displaystyle \text{(iii) Interest for the third year}=\frac{58080\times10}{100}
\displaystyle =Rs.\ 5,808
\displaystyle \therefore \text{The interest for the third year is Rs. }5,808.
\displaystyle \\

\displaystyle \textbf{Question 9: }\text{Ashok borrowed Rs. }12,000\text{ at some rate per cent}
\displaystyle \text{compound interest. After a year, he paid back Rs. }4,000.\text{ If the}
\displaystyle \text{compound interest for the second year is Rs. }920,\text{ find:}
\displaystyle \text{(i) the rate of interest charged.}
\displaystyle \text{(ii) the amount of debt at the end of the second year.}
\displaystyle \text{Answer:}
\displaystyle \text{Let the rate of interest be }R\%\text{ per annum.}
\displaystyle \text{Interest for the first year}=\frac{12000R}{100}=120R
\displaystyle \text{Amount due at the end of the first year}=12000+120R
\displaystyle \text{Balance after paying Rs. }4,000=12000+120R-4000
\displaystyle =8000+120R
\displaystyle \text{Interest for the second year}=Rs.\ 920
\displaystyle \therefore \frac{(8000+120R)R}{100}=920
\displaystyle 8000R+120R^2=92000
\displaystyle 3R^2+200R-2300=0
\displaystyle 3R^2+230R-30R-2300=0
\displaystyle R(3R+230)-10(3R+230)=0
\displaystyle (R-10)(3R+230)=0
\displaystyle R=10\quad \text{or}\quad R=-\frac{230}{3}
\displaystyle \text{Since the rate of interest cannot be negative, }R=10.
\displaystyle \therefore \text{The rate of interest charged is }10\%\text{ per annum.}
\displaystyle \\

\displaystyle \text{(ii) Interest for the first year}=\frac{12000\times10}{100}
\displaystyle =Rs.\ 1,200
\displaystyle \text{Amount due at the end of the first year}=12000+1200
\displaystyle =Rs.\ 13,200
\displaystyle \text{Balance after paying Rs. }4,000=13200-4000
\displaystyle =Rs.\ 9,200
\displaystyle \text{Debt at the end of the second year}=9200+920
\displaystyle =Rs.\ 10,120
\displaystyle \therefore \text{The amount of debt at the end of the second year is}
\displaystyle \text{Rs. }10,120.
\displaystyle \\

\displaystyle \textbf{Question 10: }\text{On a certain sum of money lent out at C.I.,}
\displaystyle \text{the interests for the first, second and third years are Rs. }1,500,
\displaystyle \text{Rs. }1,725\text{ and Rs. }2,070\text{ respectively. Find the rate of interest}
\displaystyle \text{for the (i) second year and (ii) third year.}
\displaystyle \text{Answer:}
\displaystyle \text{Interest for the first year}=Rs.\ 1,500
\displaystyle \text{Interest for the second year}=Rs.\ 1,725
\displaystyle \text{Increase in interest during the second year}=1725-1500
\displaystyle =Rs.\ 225
\displaystyle \text{(i) Rate of interest for the second year}
\displaystyle =\frac{225}{1500}\times100\%
\displaystyle =15\%
\displaystyle \therefore \text{The rate of interest for the second year is }15\%.
\displaystyle \\

\displaystyle \text{Interest for the third year}=Rs.\ 2,070
\displaystyle \text{Increase in interest during the third year}=2070-1725
\displaystyle =Rs.\ 345
\displaystyle \text{(ii) Rate of interest for the third year}
\displaystyle =\frac{345}{1725}\times100\%
\displaystyle =20\%
\displaystyle \therefore \text{The rate of interest for the third year is }20\%.
\displaystyle \\

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