A: Calculate the difference between the simple interest and the compound interest on ${\rm{Rs}}{\rm{.}}\;{\rm{4000}}$ in ${\rm{2}}$ years at ${\rm{8\% }}$ per annum compounded yearly.
B: On a certain sum of money, the difference between the compound interest for a year, payable half yearly and the simple interest for a year is ${\rm{Rs}}{\rm{.}}\;180{\rm{/ - }}$. Find the sum lent out, if the rate of interest in both the cases is ${\rm{10\% }}$ per annum. Then, find ${\rm{10}}A + B$.
A) $72256$
B) $720256$
C) $722056$
D) $722506$

Hint: Formula for simple interest is,
$SI = \dfrac{{P \times n \times r}}{{100}}$.
Formula for compound interest is,
$CI = P{\left( {1 + \dfrac{r}{{100}}} \right)^n} - P$.
 Formula for compound interest for half yearly is,
$CI = P{\left[ {1 + \dfrac{{\left( {\dfrac{R}{2}} \right)}}{{100}}} \right]^{2n}} - P$

Complete step by step answer:
In case A,
Given, principle amount, $P = 4000$
Time, $n = 2\;{\rm{years}}$
Rate of interest, $r = 8\% $
Step I:
Now, formula for simple interest is,
$SI = \dfrac{{P \times n \times r}}{{100}}$
Now, place the values of P, n,and r in the above equation we get,
$\begin{array}{c}SI = \dfrac{{P \times n \times r}}{{100}}\\ = \dfrac{{4000 \times 2 \times 8}}{{100}}\\ = 640\;{\rm{Rs}}{\rm{.}}\end{array}$

Therefore, simple interest is $640\;{\rm{Rs}}{\rm{.}}$
Step II
Again, formula for compound interest is,
$CI = P{\left( {1 + \dfrac{r}{{100}}} \right)^n} - P$
Now, place the values of P, n,and r in the above equation we get, $\begin{array}{c}CI = P{\left( {1 + \dfrac{r}{{100}}} \right)^n} - P\\ = 4000 \times {\left( {1 + \dfrac{8}{{100}}} \right)^2} - 4000\\ = \left( {4000 \times \dfrac{{27}}{{25}} \times \dfrac{{27}}{{25}}} \right) - 4000\\ = 4665 - 4000\\ = 665\end{array}$
Therefore, the compound interest is $665.6$ Rs.
Now, difference between the simple interest and the compound interest
$\begin{array}{c} = {\rm{665}}{\rm{.6}} - {\rm{640}}\\ = {\rm{25}}{\rm{.6}}\;{\rm{Rs}}{\rm{.}}\end{array}$
Hence, $A = 25.6$
Part B:
Given, rate of interest, $r = 10\% $
Difference between compound interest and simple interest,
$CI - SI = 180$
For simple interest,
$n = 1$, $r = 10\% $
Therefore, simple interest,
$\begin{array}{c}SI = \dfrac{{Pnr}}{{100}}\\ = \dfrac{{P \times 1 \times 10}}{{100}}\\ = \dfrac{P}{{10}}\end{array}$
Formula for half-yearly compound interest,
$CI = P{\left[ {1 + \dfrac{{\left( {\dfrac{R}{2}} \right)}}{{100}}} \right]^{2n}} - P$
For compound interest,
$n = 1$, $r = 10\% $
Therefore, compound interest,
$\begin{array}{c}CI = P{\left[ {1 + \dfrac{{\left( {\dfrac{R}{2}} \right)}}{{100}}} \right]^{2n}} - P\\ = P{\left[ {1 + \dfrac{{\left( {\dfrac{{10}}{2}} \right)}}{{100}}} \right]^2} - P\\ = P{\left[ {1 + \dfrac{5}{{100}}} \right]^2} - P\\ = P{\left[ {\dfrac{{105}}{{100}}} \right]^2} - P\\ = P{\left[ {\dfrac{{21}}{{20}}} \right]^2} - P\\ = \dfrac{{441P}}{{400}} - P\\ = \dfrac{{41P}}{{400}}\end{array}$
Difference between compound interest and simple interest,
$\begin{array}{c}CI - SI = 180\\\dfrac{{41P}}{{400}} - \dfrac{P}{{10}} = 180\\\dfrac{{41P - 40P}}{{400}} = 180\\\dfrac{P}{{400}} = 180\\P = 72000\end{array}$
Hence, the principal amount is $72000$ Rs.
Hence, $B = 72000$
Now, calculate the value of ${\rm{10}}A + B$
Place the value of $A = 25.6$ and $B = 72000$ in the above equation.
$\begin{array}{c}{\rm{10}}A + B = \left( {10 \times 25.6} \right) + 72000\\ = 256 + 72000\\ = 72256\end{array}$

Note: Simple Interest: Simple interest is a convenient and fast way to measure the interest rate on a loan. Simple interest is measured by the calculation by the principal of the nominal interest rate by the amount of days between payments that pass. Formula for simple interest is, $SI = \dfrac{{P \times n \times r}}{{100}}$.
Compound interest: Compound interest is applying debt to the principal balance of a loan or savings, or return on equity, in other words. That is the product of reinvesting interest, rather than taking it off, such the interest on the principal balance and previously accrued interest is then paid in the next cycle. Formula for compound interest is,
$CI = P{\left( {1 + \dfrac{r}{{100}}} \right)^n} - P$.
 Formula for compound interest for half yearly is,
$CI = P{\left[ {1 + \dfrac{{\left( {\dfrac{R}{2}} \right)}}{{100}}} \right]^{2n}} - P$

Hence the correct answer is option A.