Class 10: Compound Interest

$\displaystyle \textbf{Question 1: } \text{If the interest is compounded half-yearly, calculate the amount}$
$\displaystyle \text{when the principal is Rs. }7400,\text{ the rate is }5\%\text{ p.a. for }1\text{ year.} \hfill \text{[ICSE 2005]}$
$\displaystyle \text{Answer:}$
$\displaystyle P=\text{Rs. }7400,\ r=5\%\text{ p.a.,}\ n=1\text{ year}$
$\displaystyle \text{Rate per half-year}=\frac{5}{2}\%=2.5\%,\ \text{Number of half-years}=2$
$\displaystyle A=P\left(1+\frac{r}{2\times100}\right)^{2n}$
$\displaystyle =7400\left(1+\frac{5}{2\times100}\right)^2$
$\displaystyle =7400(1.025)^2$
$\displaystyle =\text{Rs. }7774.63$
$\displaystyle \therefore\ \text{Amount}=\text{Rs. }7774.63$
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$\displaystyle \textbf{Question 2: } \text{Find the difference between the compound interest compounded}$
$\displaystyle \text{yearly and half-yearly on Rs. }10000\text{ for }18\text{ months at }10\%\text{ p.a.}$
$\displaystyle \text{Answer:}$
$\displaystyle \textbf{Compounded yearly}$
$\displaystyle P=\text{Rs. }10000,\ r=10\%,\ n=\frac{3}{2}\text{ years}$
$\displaystyle A=10000\left(1+\frac{10}{100}\right)\left(1+\frac{10}{2\times100}\right)$
$\displaystyle =10000\times1.10\times1.05=\text{Rs. }11550$
$\displaystyle \text{Compound Interest}=11550-10000=\text{Rs. }1550$
$\displaystyle \textbf{Compounded half-yearly}$
$\displaystyle \text{Rate per half-year}=5\%,\ \text{Number of half-years}=3$
$\displaystyle A=10000\left(1+\frac{10}{2\times100}\right)^3$
$\displaystyle =10000(1.05)^3=\text{Rs. }11576.25$
$\displaystyle \text{Compound Interest}=11576.25-10000=\text{Rs. }1576.25$
$\displaystyle \text{Difference}=1576.25-1550=\text{Rs. }26.25$
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$\displaystyle \textbf{Question 3: }\text{A man borrowed Rs. }16000\text{ for }3\text{ years under the following terms:}$
$\displaystyle \text{20}\%\text{ simple interest for the first }2\text{ years and }20\%\text{ C.I. for the remaining}$
$\displaystyle 1\text{ year, compounded semi-annually. Find the total amount to be paid.}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Simple interest for the first }2\text{ years}$
$\displaystyle S.I.=16000\times\frac{20}{100}\times2=\text{Rs. }6400$
$\displaystyle \text{Amount after }2\text{ years}=16000+6400=\text{Rs. }22400$
$\displaystyle \text{Now, }P=\text{Rs. }22400,\ r=20\%,\ n=1\text{ year}$
$\displaystyle \text{Since interest is compounded semi-annually, rate per half-year}=10\%$
$\displaystyle \text{Number of half-years}=2$
$\displaystyle A=22400\left(1+\frac{20}{2\times100}\right)^2$
$\displaystyle =22400(1.1)^2$
$\displaystyle =\text{Rs. }27104$
$\displaystyle \therefore\ \text{Total amount to be paid}=\text{Rs. }27104$
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$\displaystyle \textbf{Question 4: }\text{What sum of money will amount to Rs. }27783\text{ in }1\frac{1}{2}\text{ years}$
$\displaystyle \text{at }10\%\text{ per annum compounded half-yearly?}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Let the principal be Rs. }x$
$\displaystyle A=\text{Rs. }27783,\ r=10\%,\ n=1\frac{1}{2}\text{ years}$
$\displaystyle \text{Rate per half-year}=5\%,\ \text{Number of half-years}=3$
$\displaystyle A=P\left(1+\frac{r}{2\times100}\right)^{2n}$
$\displaystyle 27783=x\left(1+\frac{10}{2\times100}\right)^3$
$\displaystyle 27783=x(1.05)^3$
$\displaystyle 27783=1.157625x$
$\displaystyle x=\frac{27783}{1.157625}=\text{Rs. }24000$
$\displaystyle \therefore\ \text{Required sum}=\text{Rs. }24000$
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$\displaystyle \textbf{Question 5: }\text{A invests a certain sum at }20\%\text{ p.a., compounded yearly.}$
$\displaystyle B\text{ invests an equal amount at the same rate, compounded half-yearly. If }B$
$\displaystyle \text{gets Rs. }33\text{ more than }A\text{ in }18\text{ months, calculate the money invested by each.}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Let the money invested by each be Rs. }x$
$\displaystyle \textbf{For A: Compounded yearly}$
$\displaystyle P=\text{Rs. }x,\ r=20\%,\ n=1\frac{1}{2}\text{ years}$
$\displaystyle A=x\left(1+\frac{20}{100}\right)\left(1+\frac{20}{2\times100}\right)$
$\displaystyle =x(1.2)(1.1)=1.32x$
$\displaystyle \textbf{For B: Compounded half-yearly}$
$\displaystyle \text{Rate per half-year}=10\%,\ \text{Number of half-years}=3$
$\displaystyle A=x\left(1+\frac{20}{2\times100}\right)^3$
$\displaystyle =x(1.1)^3=1.331x$
$\displaystyle \text{Given, }B\text{ gets Rs. }33\text{ more than }A$
$\displaystyle 1.331x-1.32x=33$
$\displaystyle 0.011x=33$
$\displaystyle x=\frac{33}{0.011}=\text{Rs. }3000$
$\displaystyle \therefore\ \text{Money invested by each}=\text{Rs. }3000$
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$\displaystyle \textbf{Question 6: }\text{At what rate of interest per annum will Rs. }62500$
$\displaystyle \text{earn a compound interest of Rs. }5100\text{ in }1\text{ year, compounded half-yearly?}$
$\displaystyle \text{Answer:}$
$\displaystyle P=\text{Rs. }62500,\ \text{Compound Interest}=\text{Rs. }5100$
$\displaystyle \therefore\ A=62500+5100=\text{Rs. }67600$
$\displaystyle \text{Let the rate be }x\%\text{ per annum.}$
$\displaystyle \text{Rate per half-year}=\frac{x}{2}\%,\ \text{Number of half-years}=2$
$\displaystyle A=P\left(1+\frac{x}{2\times100}\right)^2$
$\displaystyle 67600=62500\left(1+\frac{x}{200}\right)^2$
$\displaystyle \left(1+\frac{x}{200}\right)^2=\frac{67600}{62500}=1.0816=(1.04)^2$
$\displaystyle 1+\frac{x}{200}=1.04$
$\displaystyle \frac{x}{200}=0.04$
$\displaystyle x=8$
$\displaystyle \therefore\ \text{Required rate}=8\%\text{ p.a.}$
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$\displaystyle \textbf{Question 7: }\text{In what time will Rs. }1500\text{ yield Rs. }496.50\text{ as compound}$
$\displaystyle \text{interest at }20\%\text{ per annum, compounded semi-annually?}$
$\displaystyle \text{Answer:}$
$\displaystyle P=\text{Rs. }1500,\ \text{Compound Interest}=\text{Rs. }496.50$
$\displaystyle \therefore\ A=1500+496.50=\text{Rs. }1996.50$
$\displaystyle \text{Rate per half-year}=10\%,\ \text{Time}=n\text{ years}$
$\displaystyle A=P\left(1+\frac{20}{2\times100}\right)^{2n}$
$\displaystyle 1996.50=1500(1.1)^{2n}$
$\displaystyle \frac{1996.50}{1500}=(1.1)^{2n}$
$\displaystyle 1.331=(1.1)^{2n}=(1.1)^3$
$\displaystyle 2n=3$
$\displaystyle n=\frac{3}{2}\text{ years}=1\frac{1}{2}\text{ years}$
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$\displaystyle \textbf{Question 8: }\text{Calculate the compound interest on Rs. }3500\text{ at }6\%\text{ per annum}$
$\displaystyle \text{for }3\text{ years, the interest being compounded half-yearly.}$
$\displaystyle \text{Answer:}$
$\displaystyle P=\text{Rs. }3500,\ r=6\%\text{ p.a.},\ n=3\text{ years}$
$\displaystyle \text{Rate per half-year}=3\%,\ \text{Number of half-years}=6$
$\displaystyle A=P\left(1+\frac{6}{2\times100}\right)^{3\times2}$
$\displaystyle =3500(1.03)^6$
$\displaystyle =\text{Rs. }4179.18$
$\displaystyle \text{Compound Interest}=4179.18-3500=\text{Rs. }679.18$
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$\displaystyle \textbf{Question 9: }\text{Find the difference between compound interest and simple}$
$\displaystyle \text{interest on Rs. }12000\text{ for }1\frac{1}{2}\text{ years at }10\%\text{ p.a.,}$
$\displaystyle \text{the interest being compounded yearly.}$
$\displaystyle \text{Answer:}$
$\displaystyle \textbf{Compound Interest}$
$\displaystyle P=\text{Rs. }12000,\ r=10\%\text{ p.a.},\ n=1\frac{1}{2}\text{ years}$
$\displaystyle A=12000\left(1+\frac{10}{100}\right)\left(1+\frac{10}{2\times100}\right)$
$\displaystyle =12000(1.1)(1.05)=\text{Rs. }13860$
$\displaystyle \therefore\ \text{Compound Interest}=13860-12000=\text{Rs. }1860$
$\displaystyle \textbf{Simple Interest}$
$\displaystyle S.I.=12000\times\frac{10}{100}\times\frac{3}{2}=\text{Rs. }1800$
$\displaystyle \text{Difference}=1860-1800=\text{Rs. }60$
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$\displaystyle \textbf{Question 10: }\text{The simple interest on a sum of money for }3\text{ years}$
$\displaystyle \text{at }5\%\text{ per annum is Rs. }900.\text{ Find:}$
$\displaystyle \text{(i) the sum of money \qquad (ii) the compound interest for }1\frac{1}{2}\text{ years}$
$\displaystyle \text{payable half-yearly at double the rate per annum.}$
$\displaystyle \text{Answer:}$
$\displaystyle \textbf{(i) Finding the principal}$
$\displaystyle 900=P\times\frac{5}{100}\times3$
$\displaystyle P=\frac{900\times100}{5\times3}=\text{Rs. }6000$
$\displaystyle \textbf{(ii) Compound Interest}$
$\displaystyle P=\text{Rs. }6000,\ r=10\%\text{ p.a.},\ n=1\frac{1}{2}\text{ years}$
$\displaystyle \text{Rate per half-year}=5\%,\ \text{Number of half-years}=3$
$\displaystyle A=6000\left(1+\frac{10}{2\times100}\right)^3$
$\displaystyle =6000(1.05)^3=\text{Rs. }6945.75$
$\displaystyle \therefore\ \text{Compound Interest}=6945.75-6000=\text{Rs. }945.75$
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$\displaystyle \textbf{Question 11: }\text{The compound interest in }1\text{ year on a certain sum}$
$\displaystyle \text{at }10\%\text{ p.a., compounded half-yearly, exceeds the simple interest}$
$\displaystyle \text{on the same sum by Rs. }30.\text{ Calculate the sum.}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Let the principal be Rs. }x$
$\displaystyle \textbf{Simple Interest}$
$\displaystyle S.I.=x\times\frac{10}{100}\times1=0.1x$
$\displaystyle \textbf{Compound Interest}$
$\displaystyle \text{Rate per half-year}=5\%,\ \text{Number of half-years}=2$
$\displaystyle A=x\left(1+\frac{10}{2\times100}\right)^2$
$\displaystyle =x(1.05)^2=1.1025x$
$\displaystyle \therefore\ \text{Compound Interest}=1.1025x-x=0.1025x$
$\displaystyle \text{Given, Compound Interest}-\text{Simple Interest}=30$
$\displaystyle 0.1025x-0.1x=30$
$\displaystyle 0.0025x=30$
$\displaystyle x=\frac{30}{0.0025}=\text{Rs. }12000$
$\displaystyle \therefore\ \text{Required sum}=\text{Rs. }12000$
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