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Answers
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Answer:
Hello............. ^_^
Q. (6)
Let Principal be 'P'
and Rate of interest be 'R'
Time (n) = 2 years
Simple Interest (S.I) = Rs. 400
Compound Interest (C.I) = Rs. 410
\begin{gathered}\begin{gathered}S.I = \frac{P \times R \times T}{100} \\ \\ = \frac{P \times R \times T}{100} = 400 \\ \\ = \frac{PR \times 2}{100} = 400 \\ \\ = \frac{PR}{50} = 400 \\ \\ = PR = 400 \times 50 \\ = PR = 20000\end{gathered}\end{gathered}
S.I=
100
P×R×T
=
100
P×R×T
=400
=
100
PR×2
=400
=
50
PR
=400
=PR=400×50
=PR=20000
S.I=100P×R×T=100P×R×T=400=100PR×2=400=50PR=400=PR=400×50=PR=20000
\begin{gathered}\begin{gathered}C.I = P( \: \: ( {1 + \frac{R}{100} })^{n} - 1) \\ \\ = P( \: \: ( {1 + \frac{R}{100} })^{n} - 1) = 410 \\ \\ = P( \: \: ( {1 + \frac{R}{100} })^{2} - 1) = 410 \\ \\ = P( \: \: {(1)}^{2} + 2 \times 1 \times \frac{R}{100} + { (\frac{R}{100}) }^{2} - 1 ) = 410 \\ \\ = P( \: \: 1 - 1 \times \frac{R}{50} + { (\frac{R}{100}) }^{2} \: \: \: ) = 410 \\ \\ = P( \: \frac{R}{100} + \frac{ {R}^{2} }{10000} \: \: ) = 410 \\ \\ (take \: \: common \: R \: ) \\ \\ = PR ( \frac{1}{50} + \frac{R}{10000} ) = 410 \\ (put \: the \: value \: of \: PR \: = 20000) \\ \\ = 20000( \frac{1}{50} + \frac{R}{10000} ) = 410 \\ \\ = 20000( \frac{200 + R }{10000} ) = 410 \\ \\ = \frac{R + 200}{10000} = \frac{410}{20000 } \\ \\ = \frac{R + 200}{10000} = \frac{41}{2000 } \\ \\ = R + 200 = \frac{41}{2000} \times 10000 \\ \\ = R + 200 = 41 \times 5 \\ = R + 200 = 205 \\ = R = 205 - 200 \\ = R = 5\end{gathered}\end{gathered}
C.I=P((1+
100
R
)
n
−1)
=P((1+
100
R
)
n
−1)=410
=P((1+
100
R
)
2
−1)=410
=P((1)
2
+2×1×
100
R
+(
100
R
)
2
−1)=410
=P(1−1×
50
R
+(
100
R
)
2
)=410
=P(
100
R
+
10000
R
2
)=410
(takecommonR)
=PR(
50
1
+
10000
R
)=410
(putthevalueofPR=20000)
=20000(
50
1
+
10000
R
)=410
=20000(
10000
200+R
)=410
=
10000
R+200
=
20000
410
=
10000
R+200
=
2000
41
=R+200=
2000
41
×10000
=R+200=41×5
=R+200=205
=R=205−200
=R=5
Rate of interest = 5%
Now Principal,
as PR
\begin{gathered}\begin{gathered}Now Principal, \\ as P \times R = 20000 \\ = > P \times 5 = 20000 \\ = > P = \frac{20000}{5} \\ \\ = > P = 4000\end{gathered}\end{gathered}
NowPrincipal,
asP×R=20000
=>P×5=20000
=>P=
5
20000
=>P=4000
NowPrincipal,asP×R=20000=>P×5=20000=>P=520000=>P=4000
Principal = Rs. 4000
Q. (7)
A man invested Rs. 1000 for 3 years at 11% Simple Interest .....................(CASE 1)
Principal (P) = Rs. 1000
Rate of interest (R) = 11% per annum
Time (n) = 3 years
\begin{gathered}\begin{gathered}S.I = \frac{P \times R \times T}{100} \\ \\ = \frac{1000 \times 11 \times 3}{100} \\ \\ = 10 \times 11 \times 3 \\ = 330\end{gathered}\end{gathered}
S.I=
100
P×R×T
=
100
1000×11×3
=10×11×3
=330
S.I=100P×R×T=1001000×11×3=10×11×3=330
S.I = Rs. 330
He also invested Rs. 1000 at 10% compound interest per annum compounded annually for 3 years .........................( CASE 2 )
Principal (P) = Rs. 1000
Rate of interest (R) = 10% per annum
Time (n) = 3 years
\begin{gathered}\begin{gathered}C.I = P( \: \: {(1 + \frac{R}{100}) }^{n} - 1) \\ \\ = 1000( \: \: {(1 + \frac{10}{100}) }^{3} - 1) \\ \\ = 1000( \: \: {(1 + \frac{1}{10}) }^{3} - 1) \\ \\ = 1000( \: \: { (\frac{10 + 1}{10} )}^{3} - 1 ) \\ \\ = 1000( \: { (\frac{11}{10}) }^{3} - 1 ) \\ \\ = 1000 \times (\frac{1331}{1000} - 1) \\ \\ = 1000 \times ( \frac{1331 - 1000}{1000} ) \\ \\ = 1000 \times \frac{331}{1000} \\ \\ = 331\end{gathered}\end{gathered}
C.I=P((1+
100
R
)
n
−1)
=1000((1+
100
10
)
3
−1)
=1000((1+
10
1
)
3
−1)
=1000((
10
10+1
)
3
−1)
=1000((
10
11
)
3
−1)
=1000×(
1000
1331
−1)
=1000×(
1000
1331−1000
)
=1000×
1000
331
=331
C.I=P((1+100R)n−1)=1000((1+10010)3−1)=1000((1+101)3−1)=1000((1010+1)3−1)=1000((1011)3−1)=1000×(10001331−1)=1000×(10001331−1000)=1000×1000331=331
C.I = Rs. 331
(S.I =330 < C.I = 331)
(CASE 1) < (CASE 2)
S.I < C.I
So, investment at compound Interest is better.
Q. (8)
Principal (P) = Rs. 400000
Rate of interest = 16% per annum
= 16/2 = 8% per half yearly (R)
Time (n) = 1 year = 2 half years
\begin{gathered}\begin{gathered}Amount = P( {1 + \frac{R}{100} ) }^{n} \\ \\ = 400000( {1 + \frac{8}{100} ) }^{2} \\ \\ = 400000( {1 + \frac{2}{25} ) }^{2} \\ \\ = 400000( {\frac{25 + 2}{25} ) }^{2} \\ \\ = 400000 \times { (\frac{27}{25}) }^{2} \\ \\ = 400000 \times \frac{27 \times 27}{25 \times 25} \\ \\ = 400000 \times \frac{729}{625} \\ \\ = 640 \times 729 \\ = 466560\end{gathered}\end{gathered}
Amount=P(1+
100
R
)
n
=400000(1+
100
8
)
2
=400000(1+
25
2
)
2
=400000(
25
25+2
)
2
=400000×(
25
27
)
2
=400000×
25×25
27×27
=400000×
625
729
=640×729
=466560
Amount=P(1+100R)n=400000(1+1008)2=400000(1+252)2=400000(2525+2)2=400000×(2527)2=400000×25×2527×27=400000×625729=640×729=466560
So, they earn Rs. 466560
...................^_^