[4 marks sum]

Question 14 Marks

Find the amount and compound interest on $Rs.50000$ onn $1 \frac{1}{2}$ years at $8\% p.a.$ compounded half$-$yearly.

Answer

Here $P=Rs. 50000, t=1 \frac{1}{2}$ years, $r=8 \%$
Since interest is compounded half$-$yearly, so
Now, Amount
$= P \left(1+\frac{ r }{200}\right)^{2 t }$
$=50000\left(1+\frac{8}{200}\right)^3$
$=50000\left(\frac{104}{100}\right)^3$
$=56243.20$
Hence, Amount $=Rs. 56243.20$
Also, $C.I.$
$ =A \cdot P .$
$=\text { Rs. } 56243.20-R s .50000$
$=\text { Rs. } 6243.20 . $

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Question 24 Marks

At what rate percent will $Rs.12000$ yield $Rs.13891.50$ as compound interest in $3$ years?

Answer

Given : $ A=\text { Rs. } 13891.50, P=R s .12000, N=3$ years
$13891.50=12000\left(1+\frac{ r }{100}\right)^3$
$\Rightarrow \frac{13891.50}{12000}=\left(1+\frac{ r }{100}\right)^3$
$\Rightarrow \frac{13891.50}{12000 \times 100}=\left(1+\frac{ r }{100}\right)^3$
$\Rightarrow \frac{9261}{8000}=\left(1+\frac{ r }{100}\right)^3$
$\Rightarrow \frac{21}{20}=1+\frac{ r }{100}$
$\Rightarrow \frac{21}{20}-1=\frac{ r }{100}$
$\Rightarrow \frac{1}{20}=\frac{ r }{100}$
$\Rightarrow r =5 \%$

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Question 34 Marks

Find the amount and the compound interest on the following :$Rs.16000$ for $3$ years at $10\%, 8\%$ and $6\%$ for successive years.

Answer

$Rs. 16000$ for $3$ years at $10 \%, 8 \%$ and $6 \%$ for successive years.
Here $P= Rs. 16000, t=3$ years, $r=10 \%, 8 \%, 6 \%$ successively.
Now, Amount
$ = P \left(1+\frac{ r _1}{100}\right)\left(1+\frac{ r _2}{100}\right)\left(1+\frac{ r _3}{100}\right)$
$=16000\left(1+\frac{10}{100}\right)\left(1+\frac{8}{100}\right)\left(1+\frac{6}{100}\right)$
$=16000\left(\frac{11}{10}\right)\left(\frac{108}{100}\right)\left(\frac{106}{100}\right)$
$=20148.48$
Hence, Amount $= Rs. 20148.48$
Also, $C.l.$
$= A - P $
$= Rs \cdot 20148.48- Rs .16000 $
$= Rs .4148 .48 .$

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Question 44 Marks

Find the amount and the compound interest on the following $:Rs.25000$ for $2$ years at $6\%$ per annum compounded semi$-$annually.

Answer

$Rs. 25000$ for $2$ years at $6 \%$ per annum compounded semi$-$annually.
Here $P=R s .25000, t=2$ years, $r=6 \%$
Since interest is compounded semi$-$annually, so
Amount
$= P \left(1+\frac{ r }{200}\right)^{2 t }$
$=25000\left(1+\frac{6}{200}\right)^4$
$=25000\left(\frac{103}{100}\right)^4$
$=28137.72$
Hence, Amount $= Rs. 28137.72$
Also, $C.I.$
$=A \cdot P$
$=R s .28137 .72-R s .25000$
$=R s .3137 .72 .$

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Question 54 Marks

The difference between $C.I.$ payable annually and $S.I.$ on $Rs.50,000$ for two years is $Rs.125$ at the same rate of interest per annum. Find the rate of interest.

Answer

Let the rate of interest per year be $r \%$.
$ \text { S.I.}$ in $2$ years $=\text { Rs. } \frac{50000 \times r \times 2}{100}$
$=\text { Rs. } 1000\ r$
And,$\text { C.I.}$ in $ 2$ years
$=\text { A }-P$
$=\text { Rs. } 50000\left(1+\frac{r}{100}\right)^2-\text { Rs. } 50000$
Given$,\text { C.I. }- \text { S.I. }=\text { Rs. } 125$
$\Rightarrow 50000\left(1+\frac{r}{100}\right)^2-50000-1000 r=125$
$\Rightarrow 50000\left(1+\frac{r^2}{10000}+\frac{2 r}{100}\right)-50000-1000 r=125$
$\Rightarrow 50000+5 r^2+1000 r-50000-1000 r=125$
$\Rightarrow 5 r^2=25$
$\Rightarrow r 2=25$
$\Rightarrow r= \pm 5 $
But the rater of interest cannot ne negative.
$\therefore$ Rate of interest is $5 \%$.

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Question 64 Marks

Find the amount and the compound interest on the following :$Rs.15000$ for $2$ years at $8\%$ per annum compounded semi$-$annually.

Answer

$Rs. 15000$ for $2$ years at $8 \%$ per annum compounded semi$-$annually.
Here $P=R s .15000, t=2$ years, $r=8 \%$
Since interest is compounded semi$-$annually, so
Amount
$= P \left(1+\frac{ r }{100}\right)^{2 t }$
$=15000\left(1+\frac{8}{200}\right)^4$
$=15000\left(\frac{26}{25}\right)^4$
$=15000 \times \frac{26}{25} \times \frac{26}{25} \times \frac{26}{25} \times \frac{26}{25}$
$=17547.88$
Hence, Amount $= Rs. 17547.88$
Also, $C.l.$
$= A - P$
$= Rs. 17547.88- Rs. 15000$
$= Rs. 2547.88 .$

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Question 74 Marks

Find the amount and the compound interest on the following:$Rs.8000$ for $3$ years at $10\%$ per annum compounded annually.

Answer

$Rs. 8000$ for $3$ years at $10 \%$ per annum compounded annually.
Here $P=R s .8000, t=3$ years, $r=10 \%$
Now, Amount
$ = P \left(1+\frac{ r }{100}\right)^{ t }$
$=8000\left(1+\frac{10}{100}\right)^3$
$=8000\left(\frac{11}{10}\right)^3$
$=8000 \times \frac{1331}{1000}$
$=10648 $
Hence, Amount $= Rs. 10648$
Also, $C.I.$
$= A - P$
$= Rs. 10648-R s .8000$
$= Rs. 2648 .$

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Question 84 Marks

The simple interest on a certain sum for $3$ years at $4\%$ is $Rs.600.$ Find the compound interest for the same sum at the same percent and in the same time.

Answer

Since, Simple interest
$ =\frac{ P \times r \times t }{100}$
$\Rightarrow 600=\frac{ P \times 4 \times 3}{100}$
$\Rightarrow P =\frac{60000^2}{12}$
$=5000 $
Now for $C.I. P= Rs. 5000, r=4 \%, t=3$ years
Amount
$ = P \left(1+\frac{ r }{100}\right)^{ t }$
$=5000\left(1+\frac{4}{100}\right)^3$
$=5000 \times\left(\frac{26}{25}\right)^3$
$=5624.32 $
Hence, Amount $= Rs. 5624.32$
Also, $C.I.$
$ =A \cdot P$
$=R s .5624 .32-R s .5000$
$=R s .624 .32 . $

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Question 94 Marks

How much will $Rs.14000$ amounts to $2$ years at compound interest, if the rates for the successive years be $5\%$ and $8\%$ respectively?

Answer

Here $P_1= Rs. 14000$ and $r=5 \%$
So, Amount after $1$ year
$= P \left(1+\frac{ r }{100}\right) $
$=14000\left(1+\frac{5}{100}\right) $
$=14000 \times \frac{105}{100} $
$=14700$
Thus, $P_2= Rs. 14700$ and $r=8 \%$
Amount after $2$ year
$= P \left(1+\frac{ r }{100}\right) $
$=14700\left(1+\frac{8}{100}\right) $
$=14700 \times \frac{108}{100} $
$=15876$
Hence, Amount $= Rs. 15876.$

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Question 104 Marks

Rakesh invests $Rs.25600$ at $5\%$ per annum compound interest payable annually for $3$ years. Find the amount standing to his credit at the end of the second year.

Answer

For $1^{st}$ year: $P=R s .25600, R=5 \%$ and $T=1$ year
$\therefore$ Interest $= Rs. \frac{25600 \times 5 \times 1}{100}$
$= Rs. 1280$
And, amount
$= Rs. 25600+R s .1280$
$= Rs. 26880$
For $2^{nd}$ year: $P=R s .26880, R=5 \%$ and $T=1$ year
$\therefore$ Interest $=\text { Rs. } \frac{26880 \times 5 \times 1}{00} $
$=\text { Rs. } 1344$
And, amount
$=\text { Rs. } 26880+\text { Rs. } 1344 $
$=\text { Rs. } 28224$
$\therefore$ Amount at the end of $2^{nd}$ year is $Rs. 28224 .$

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Question 114 Marks

What sum will amount to $Rs.10120$ in $2$ years at $C.I.$ payable annually, if the rates are $10\%$ and $15\%$ for the successive years?

Answer

For $1^{\text {st }}$ year: $P=R s .100, R=10 \%$ and $T=1$ year
Interest $=Rs \cdot \frac{100 \times 10 \times 1}{100}$
$= Rs. 10$
Amount
$= Rs. 100+ Rs. 10$
$= Rs. 110$
For $2^{\text {nd }}$ year: $P=R s .110, R=15 \%$ and $T=1$ year
Interest $= Rs. \frac{100 \times 15 \times 1}{100}$
$= Rs. 16.50$
Amount
$= Rs. 110+R s .16 .50$
$=\text { Rs. } 126.50$
When amount is $\text {Rs. } 126.50,$ Principal is $\text {Rs. } 100 \text {. }$
Hence, when amount is $\text {Rs. } 10120 \text {, }$
Principal $= \text {Rs. } \frac{10120 \times 100}{126.50}$
$=\text { Rs. } 8000 \text {. }$

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MATHEMATICS [4 marks sum]

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