Question
The compound interest payable annually on a certain sum for $2$ years is $Rs .40.80$ and the simple interest is $Rs. 40$. Find the sum and the rate percent.
✓
Answer
Let sum be $Rs. P$ and $r \%$ be the rate of interest.
We have $t=2$ years, $C.I. = Rs. 40.80$ and $S.I. = Rs. 40$
Since, Simple interest
$ =\frac{ P \times r \times t }{100}$
$\Rightarrow 40=\frac{ P \times r \times 2}{100}$
$\Rightarrow Pr =\frac{4000}{2}$
$=2000$
Now,
$ \text { C.I. }=A-P$
$=P\left(1+\frac{r}{100}\right)^t-P$
$\left.=P\left[\left(1+\frac{ r }{100}\right)\right]^{ t }-1\right]$
$\Rightarrow 40.80= P \left[\left(1+\frac{ r }{100}\right)^2-1\right]$
$\Rightarrow 40.80= P \left(1+\frac{ r ^2}{10000}+\frac{2 r }{100}-1\right)$
$\Rightarrow 40.80= P \left(\frac{ r ^2}{10000}+\frac{2 r }{100}\right)$
$\Rightarrow 40.80= Pr \left(\frac{ r }{10000}+\frac{2}{100}\right)$
$\Rightarrow 40.80=2000\left(\frac{ r +200}{10000}\right)$
$\Rightarrow 40.80=\frac{ r +200}{5}$
$\Rightarrow r =40.80 \times 5-200$
$=204-200$
$=4$
Hence, $r=4 \%$
Now, $Pr =2000$
$ \Rightarrow P=\frac{2000}{ r }$
$=\frac{2000}{4}$
$=500 . $
Thus, sum is $Rs. 500$ and rate of interest is $4 \%$.
We have $t=2$ years, $C.I. = Rs. 40.80$ and $S.I. = Rs. 40$
Since, Simple interest
$ =\frac{ P \times r \times t }{100}$
$\Rightarrow 40=\frac{ P \times r \times 2}{100}$
$\Rightarrow Pr =\frac{4000}{2}$
$=2000$
Now,
$ \text { C.I. }=A-P$
$=P\left(1+\frac{r}{100}\right)^t-P$
$\left.=P\left[\left(1+\frac{ r }{100}\right)\right]^{ t }-1\right]$
$\Rightarrow 40.80= P \left[\left(1+\frac{ r }{100}\right)^2-1\right]$
$\Rightarrow 40.80= P \left(1+\frac{ r ^2}{10000}+\frac{2 r }{100}-1\right)$
$\Rightarrow 40.80= P \left(\frac{ r ^2}{10000}+\frac{2 r }{100}\right)$
$\Rightarrow 40.80= Pr \left(\frac{ r }{10000}+\frac{2}{100}\right)$
$\Rightarrow 40.80=2000\left(\frac{ r +200}{10000}\right)$
$\Rightarrow 40.80=\frac{ r +200}{5}$
$\Rightarrow r =40.80 \times 5-200$
$=204-200$
$=4$
Hence, $r=4 \%$
Now, $Pr =2000$
$ \Rightarrow P=\frac{2000}{ r }$
$=\frac{2000}{4}$
$=500 . $
Thus, sum is $Rs. 500$ and rate of interest is $4 \%$.
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