Answer: Option B. -> 5%
$$\eqalign{
& {\text{Principal}}\,\,\,\,\,{\text{Interest}} \cr
& \underbrace {\,\,\,\,\,\,\,4{\text{P}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{P}}\,\,\,\,\,\,\,\,\,\,}_{} \cr
& {\text{Time = Rate }}\% {\text{ (given)}} \cr
& {\text{Now by using formula , }} \cr
& {\text{P = }}\frac{{4{\text{P}} \times {\text{R}} \times {\text{R}}}}{{100}} \cr
& \Rightarrow {{\text{R}}^2} = \frac{{100}}{4} \cr
& \Rightarrow {\text{R = }}\frac{{10}}{2} \cr
& \Rightarrow {\text{R = 5}}\% \cr} $$

Answer: Option A. -> Rs. 1500
$$\eqalign{
& {\text{By using formula,}} \cr
& {\text{Installment}} \cr
& {\text{ = }}\frac{{6450 \times 100}}{{4 \times 100 + \left( {3 + 2 + 1} \right) \times 5}} \cr
& {\text{ = }}\frac{{6450 \times 100}}{{4 \times 100 + \left( 6 \right) \times 5}} \cr
& = \frac{{6450 \times 100}}{{4 \times 100 + 30}} \cr
& = \frac{{6450 \times 100}}{{430}} \cr
& = {\text{Rs}}{\text{. 1500}} \cr
& {\text{Hence value of installment}} \cr
& {\text{ = Rs}}{\text{. 1500}} \cr} $$

Answer: Option D. -> 10%
$$\eqalign{
& {\text{Let the rate }}\% {\text{ = R}} \cr
& {\text{According to the question,}} \cr
& \frac{{5000 \times 2 \times {\text{R}}}}{{100}} + \frac{{3000 \times 4 \times {\text{R}}}}{{100}} = 2200 \cr
& \Rightarrow 100{\text{R}} + 120{\text{R}} = 2200 \cr
& \Rightarrow 220{\text{R}} = 2200 \cr
& \Rightarrow {\text{R}} = 10\% \cr
& {\text{Hence required rate}}\% \cr
& = 10\% \cr} $$

Answer: Option B. -> Rs. 600
$$\eqalign{
& {\text{S}}{\text{.I}}{\text{. for 5 years}} \cr
& = {\text{Rs}}{\text{.}}\left( {1020 - 720} \right) \cr
& = {\text{Rs}}{\text{. 300}} \cr
& {\text{S}}{\text{.I}}{\text{. for 2 years}} \cr
& = {\text{Rs}}{\text{.}}\left( {\frac{{300}}{5} \times 2} \right) \cr
& = {\text{Rs}}{\text{. }}120 \cr
& \therefore {\text{Principal}} \cr
& = {\text{Rs}}{\text{.}}\left( {{\text{720}} - 120} \right) \cr
& = {\text{Rs}}{\text{. }}600 \cr} $$

Answer: Option A. -> 6.25%, Rs. 18600
$$\eqalign{
& {\text{S}}{\text{.I}}{\text{. }}{\text{for 3 years}} \cr
& = {\text{Rs}}{\text{.}}\left( {24412.50 - 20925} \right) \cr
& = {\text{Rs}}{\text{. }}3487.50 \cr
& {\text{S}}{\text{.I}}{\text{. }}{\text{for 2 years}} \cr
& = {\text{Rs}}{\text{.}}\left( {\frac{{3487.50}}{3} \times 2} \right) \cr
& = {\text{Rs}}{\text{. }}2325 \cr
& \therefore \text{Principal} \cr
& = {\text{Rs}}{\text{.}}\left( {20925 - 2325} \right) \cr
& = {\text{Rs}}{\text{. }}18600 \cr
& {\text{Hence,}} \cr
& {\text{rate}} = \left( {\frac{{100 - 2325}}{{18600 \times 2}}} \right)\% \cr
& = 6.25\% \cr} $$

Answer: Option B. -> 5 years 4 months
$$\eqalign{
& {\text{Let sum = Rs}}{\text{. }}x{\text{}} \cr
& {\text{Then,}} \cr
& {\text{S}}{\text{.I}}{\text{. = Rs}}{\text{. }}x{\text{}} \cr
& \therefore \text{Time} = \left( {\frac{{100 \times {\text{S}}{\text{.I}}{\text{.}}}}{{{\text{P}} \times {\text{R}}}}} \right) \cr
& = \left( {\frac{{100 \times x}}{{x \times 18.75}}} \right){\text{years}} \cr
& = \frac{{26}}{3}{\text{years}} \cr
& = 5\frac{1}{3}{\text{years}} \cr
& = {\text{5 years 4 months}}{\text{}} \cr} $$

Answer: Option B. -> 21 years
$$\eqalign{
& {\text{Let sum}} = {\text{Rs}}{\text{. }}x \cr
& {\text{Then,}} \cr
& {\text{S}}{\text{.I}}{\text{.}} = {\text{Rs}}{\text{.}}\,x \cr
& \therefore \text{Rate}\,\% \cr
& = \left( {\frac{{100 \times x}}{{x \times 7}}} \right)\% \cr
& = \frac{{100}}{7}\% \cr
& {\text{Now, sum}} = {\text{Rs}}{\text{. }}x \cr
& {\text{S}}{\text{.I}}. = {\text{Rs}}{\text{. }}3x \cr
& \text{Rate} = \frac{{100}}{7}\% \cr
& \therefore {\text{Total Time}} \cr
& = \left( {\frac{{100 \times 3x}}{{x \times \frac{{100}}{7}}}} \right){\text{years}} \cr
& = 21\,{\text{years}} \cr} $$

Answer: Option C. -> 6 years
$$\eqalign{
& {\text{16}}\frac{2}{3} = \frac{{1 \to {\text{ Interest}}}}{{6 \to {\text{ Principal }}}} \cr
& {\text{Let principal = 6}} \cr
& {\text{Interest = 6}} \cr
& {\text{Time = t years}} \cr
& {\text{By using formula }} \cr
& {\text{6}} = \frac{{6 \times 50 \times {\text{t}}}}{{3 \times 100}} \cr
& \Rightarrow {\text{t}} = 6\,{\text{years}} \cr} $$
Alternate
Note : In such type of questions to save your valuable time think like the given way.
$$\eqalign{
& {\text{Rate}}\% \cr
& {\text{ = 16}}\frac{2}{3}\% = \frac{{1 \to {\text{ Interest}}}}{{6 \to {\text{ Principal }}}} \cr
& {\text{Represent for 1 years}} \cr
& {\text{According to the question,}} \cr
& {\text{Principal = Interest}} \cr
& {\text{6 = 1}} \times {\text{6}} \cr
& {\text{Hence,}} \cr
& {\text{Time = 1}} \times {\text{6}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,{\text{ = 6 years}} \cr} $$
Note : If interest will be six times then time will also be six times.

Answer: Option B. -> 10%
Let the total amount = Rs. 6
Total average rate of interest
$$\eqalign{
& {\text{ = }}\frac{{\left( {3 \times 10\% } \right) + \left( {2 \times 9\% } \right) + \left( {1 \times 12\% } \right)}}{6} \cr
& = \frac{{\left( {30 + 18 + 12} \right)}}{6} \% \cr
& = 10\% \cr} $$

Answer: Option A. -> Rs. 1200
More interest paid in 2 years
$$\eqalign{
& {\text{ = 2}} \times {\text{1}} = {\text{2}}\% \cr
& {\text{According to the question, }} \cr
& {\text{2}}\% {\text{ of sum = Rs}}{\text{. 24}} \cr
& {\text{1}}\% {\text{ of sum = Rs}}{\text{.}}\frac{{24}}{2} \cr
& {\text{Total sum}} \cr
& {\text{ = Rs}}{\text{. }}\frac{{24}}{2} \times 100 \cr
& = {\text{Rs}}{\text{. }}1200 \cr} $$