Solution:
$$\eqalign{ {I_1} = \frac{{P{T_1}{R_1}}}{{100}} \cr {I_1} = \frac{{ {500 \times 3 \times 5} }}{{100}} \cr \,\,\,\,\,\,\, = Rs.{\kern 1pt} 75 \cr {I_2} = \frac{{P{T_2}{R_2}}}{{100}} \cr {I_2} = \frac{{ {500 \times 4 \times 4} }}{{100}} \cr \,\,\,\,\,\,\,\, = Rs.{\kern 1pt} 80 \cr {\text{Difference}} = 80 - 75 \cr \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,5 \cr} $$

Alternatively
The interest is calculated simply and then it will have a rise of 15% in 1^st case and 16% in 2^nd case.
Difference = 1% on 500 = Rs. 5

Otherway,
500 == 15% $$ \uparrow $$ ⇒ 575 (1^st case)
500 == 16% $$ \uparrow $$ ⇒ 580 (2^nd case)
We can see clear difference of Rs. 5

Solution:
Let Asif lent Rs. X at 14% per year.
Hence, Money lent at 12% = (1500 - x);
Given, total interest = Rs. 186
$$ {\frac{{\left( {x \times 14 \times 1} \right)}}{{100}}} \, + $$    $$ {\frac{{\left[ {\left( {1500 - x} \right) \times 12 \times 1} \right]}}{{100}}} $$      = 186
$$\eqalign{ \frac{{14x}}{{100}} + \frac{{ {18000 - 12x} }}{{100}} = 186 \cr 14x + 18000 - 12x = 186 \times 100 \cr 2x = 18600 - 18000 \cr x = \frac{{600}}{2} = {\text{Rs}}{\text{. }}300 \cr {\text{Hence, money lent}}{\kern 1pt} {\text{at }}12\% \cr = 1500 - 300 \cr = {\text{Rs}}{\text{.}}\,1200 \cr} $$

Solution:
$$\eqalign{ {\text{Amount}},\,A = Rs.\,1440 \cr {\text{Principal}},\,P = Rs.\,1200 \cr {\text{Interest}},\,I = Rs.\,\left( {1440 - 1200} \right) = 240 \cr R = \frac{{ {240 \times 100} }}{{ {1200 \times 2} }} = 10\% \cr \cr {\text{Alternatively}}, \cr {\text{We}}\,{\text{can}}\,{\text{go}}\,{\text{through}}\,{\text{a}}\,{\text{thought}}\,{\text{process}}\,i.e. \cr 1200\, == 20\% \uparrow \left( {240\,{\text{in}}\,2\,{\text{years}}} \right) ⇒ 1400 \cr {\text{That}}\,{\text{means}}\,10\% \,{\text{rise}}\,{\text{in}}\,{\text{each}}\,{\text{year}} \cr} $$

Solution:
$$\eqalign{ {\text{A's,}}\,{\text{Principal}} = {\text{Rs}}{\text{. }}90,000 \cr {\text{Rate}} = 5\% \cr {\text{Time}} = 4{\text{ years}} \cr \therefore {\text{S}}{\text{.I}} = \frac{{90,000 \times 5 \times 4}}{{100}} = {\text{Rs}}{\text{. }}18,000 \cr {\text{B's,}}\,{\text{Principal}} = {\text{Rs}}{\text{. }}90,000 \cr {\text{Rate}} = 7\% \cr {\text{Time}} = 4{\text{ years}} \cr \therefore {\text{S}}{\text{.I}} = \frac{{90,000 \times 7 \times 4}}{{100}} = {\text{Rs}}{\text{. }}25,200 \cr \therefore {\text{Profit of A}} \cr = \left( {25,200 - 18,000} \right) = {\text{Rs}}{\text{. }}7,200 \cr} $$

Solution:
Let the principal be P and rate of interest be R%.
Therefore Required ratio
$$\eqalign{ = \frac{{ {\frac{{P \times R \times 6}}{{100}}} }}{{ {\frac{{P \times R \times 9}}{{100}}} }} \cr = \frac{{6PR}}{{6PR}} \cr = \frac{6}{9} \cr = 2:3 \cr} $$

Solution:
$${\text{Gain in 2 years}}$$
$$ = {\text{Rs}}{\text{.}}\,\left[ {\left( {5000 \times \frac{{25}}{4} \times \frac{2}{{100}}} \right) - \left( {\frac{{5000 \times 4 \times 2}}{{100}}} \right)} \right]$$
$$\eqalign{ = {\text{Rs}}{\text{.}}\,\left( {625 - 400} \right) \cr = {\text{Rs}}.225 \cr \therefore {\text{Gain in 1 year}} \cr = {\text{Rs}}{\text{.}}\,\left( {\frac{{225}}{2}} \right) \cr = {\text{Rs}}{\text{.}}\,112.50 \cr} $$

Solution:
$$\eqalign{ {\text{Given,}}\,{\text{Principal}},\,P = Rs.\,1000 \cr {\text{Compoind}}\,{\text{rate}},\,R = 20\% \,{\text{per}}\,{\text{annum}} \cr = \frac{{20}}{2} = 10\% \,{\text{half - yearly}} \cr {\text{Time}} = 18\,{\text{month}} = 3\,{\text{half - years}} \cr {\text{Amount}}, \cr A = \left\{ {P \times {{\left[ {1 + \left( {\frac{R}{{100}}} \right)} \right]}^n}} \right\} \cr = \left\{ {1000 \times {{\left[ {1 + \left( {\frac{{10}}{{100}}} \right)} \right]}^3}} \right\} \cr = { {\frac{{1000 \times 11 \times 11 \times 11}}{{10 \times 10 \times 10}}} } \cr A = Rs.\,1331 \cr {\text{Hence,}}\,{\text{compound}}\,{\text{interest}} = Rs.\,331 \cr} $$