The simple interest and compound interest of a certain sum of money for 2 years are Rs 840 and Rs 869.40 respectively. Let us calculate that sum of money and the rate of interest.
Answers
Answer:
Given:
Simple interest (S.I) = Rs. 840
Compound Interest (C.I) = Rs. 869.40
Time = 2 years
To find:
The Rate of Interest (R) and the Principal (P)
The formula to find Simple Interest (S.I) is
\boxed{\bf {S.I=\frac{PRT}{100}}}
S.I=
100
PRT
By putting the value of Time = 2 years,
S.I=\frac{P\times R \times 2}{100}S.I=
100
P×R×2
\frac{PR}{50}= Rs\ 840
50
PR
=Rs 840 ___(i)
Now,
The formula to find the Compound Interest (C.I) is
\boxed{\bf C.I = P[(1+\frac{R}{100})^{n}-1]}
C.I=P[(1+
100
R
)
n
−1]
Putting the value of time = 2 years,
C.I = P[(1+\frac{R}{100})^{2}-1]C.I=P[(1+
100
R
)
2
−1]
\implies P[(1+\frac{R}{100})^{2}-1]= Rs\ 869.40⟹P[(1+
100
R
)
2
−1]=Rs 869.40
\implies P[\{(1)^{2}+(\frac{R}{100})^{2}+2(1)(\frac{R}{100}) \}-1]= Rs\ 869.40⟹P[{(1)
2
+(
100
R
)
2
+2(1)(
100
R
)}−1]=Rs 869.40
[ As (a+b)² = a² + b² + 2ab ]
\implies P[\red{1}+\frac{R^{2}}{10000}+\frac{R}{50}\red{-1}]= Rs\ 869.40⟹P[1+
10000
R
2
+
50
R
−1]=Rs 869.40
\implies P[\frac{R^{2}}{10000}+\frac{R}{50}]= Rs\ 869.40⟹P[
10000
R
2
+
50
R
]=Rs 869.40
Here (R) can be taken as common in LHS
\implies PR[\frac{R}{10000}+\frac{1}{50}]= Rs\ 869.40⟹PR[
10000
R
+
50
1
]=Rs 869.40
[Taking LCM]
\implies PR[\frac{R+200}{10000}]= Rs\ 869.40⟹PR[
10000
R+200
]=Rs 869.40 ___(ii)
Now,
Dividing equation (ii) by equation (i)
\boxed{\implies \dfrac{PR[\frac{R+200}{10000}]= Rs\ 869.40}{\frac{PR}{50}= Rs\ 840}}
⟹
50
PR
=Rs 840
PR[
10000
R+200
]=Rs 869.40
Here (PR) will be cancelled in both denominator and numerator.
\implies \dfrac{[\frac{R+200}{10000}]}{\frac{1}{50}} = \dfrac{ 869.40}{ 840}⟹
50
1
[
10000
R+200
]
=
840
869.40
\implies \dfrac{R+200}{10000}\times 50 = \dfrac{ 869.40}{ 840}⟹
10000
R+200
×50=
840
869.40
\implies \dfrac{R+200}{200} = \dfrac{ 869.40}{ 840}⟹
200
R+200
=
840
869.40
By doing cross multiplication.
⇒ 200(869.40) = 840(R+200)
⇒ 173880 = 840R + 168000
⇒ 173880 - 168000 = 840R
[By transporting 168000 to LHS]
⇒ 5880 = 840R
⇒ 5880 ÷ 840 = R
⇒ 7 = R
∴ R% = 7%
Now,
\implies \frac{PR}{50} = 840⟹
50
PR
=840
Putting the value of R,
\implies \frac{P\times 7}{50} = 840⟹
50
P×7
=840
⇒ 7P = 840 × 50
⇒ 7P = 42000
⇒ P = 42000 ÷ 7
∴ Principal (P) = Rs. 6000
Hence,
The principal (P) = Rs. 6000
Rate of Interest (R) = 7%