Interest — Part 2 MCQs with Explanations

Let the original rate be R%. Then, new rate = (2R)%. Note: Here, original rate is for 1 year(s); the new rate is for only 4 months i.e. $$\frac{1}{3}$$ year(s). $$\eqalign{ & \therefore {\frac{{725 \times R \times 1}}{{100}}} + {\frac{{362.50 \times 2R \times 1}}{{100 \times 3}}} \cr & = 33.50 \cr & \Rightarrow \left( {2175 + 725} \right)R = 33.50 \times 100 \times 3 \cr & \Rightarrow \left( {2175 + 725} \right)R = 10050 \cr & \Rightarrow \left( {2900} \right)R = 10050 \cr & \Rightarrow R = \frac{{10050}}{{2900}} = 3.46 \cr & \therefore \text{Original rate} = 3.46\% \cr} $$

$$\eqalign{ & {\text{Principal}} = Rs.\,\left( {\frac{{100 \times 5400}}{{12 \times 3}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,15000 \cr} $$

$$\eqalign{ & {\text{S}}{\text{.I}}{\text{. }}{\kern 1pt} {\text{for 3 years}} \cr & = {\text{Rs}}{\text{.}}\left( {12005 - 9800} \right) \cr & = {\text{Rs}}{\text{. }}2205 \cr & {\text{S}}{\text{.I}}{\text{. for 5 years}} \cr & = {\text{Rs}}{\text{.}}\,\left( {\frac{{2205}}{3} \times 5} \right) \cr & = {\text{Rs}}{\text{.}}\,3675 \cr & \therefore {\text{Principal}} \cr & = {\text{Rs}}{\text{.}}\,(9800 - 3675) \cr & = {\text{Rs}}{\text{.}}\,6125 \cr & {\text{Hence,}}\,{\text{rate}} = \left( {\frac{{100 \times 3675}}{{6125 \times 5}}} \right)\% \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 12\% \cr} $$

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

We need to know the S.I., principal and time to find the rate. Since the principal is not given, so data is inadequate.

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

$$\eqalign{ & {\text{Let principal = 6P}} \cr & {\text{Hence Amount}} \cr & {\text{ = 6P}} \times \frac{7}{6} = 7{\text{P}} \cr & \therefore {\text{SI = 7P}} - {\text{6P = P}} \cr & {\text{Time = 3 years}} \cr & \boxed{{\text{SI = }}\frac{{{\text{P}} \times {\text{T}} \times {\text{R}}}}{{100}}} \cr & \Rightarrow {\text{P = }}\frac{{6{\text{P}} \times {\text{R}} \times {\text{3}}}}{{100}} \cr & \Rightarrow {\text{R = }}\frac{{100}}{{18}} \cr & \,\,\,\,\,\,\,\,\,\,\,\, = \frac{{50}}{9} \cr & \,\,\,\,\,\,\,\,\,\,\,\, = 5\frac{5}{9}\% \cr} $$ Alternate: Note: In such type of questions to save your valuable time try to think like that. \[\begin{gathered} \begin{array}{*{20}{c}} {{\text{Amount}}}&{{\text{Principal}}} \end{array} \hfill \\ {\text{ }}\underbrace {7\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{6}}}_{ + 1} \hfill \\ \end{gathered} \] $$\eqalign{ & {\text{Required rate % }} \cr & {\text{ = }}\frac{1}{6} \times \frac{{100}}{3} \cr & = 5\frac{5}{9}\% \cr} $$

Let the rate of interest for two different sources is r 1 and r 2 respectively. $$\eqalign{ & \frac{{1500 \times {{\text{r}}_1} \times 3}}{{100}} - \frac{{1500 \times {{\text{r}}_2} \times 3}}{{100}} \cr & = 13.50 \cr & 4500{{\text{r}}_1} - 4500{{\text{r}}_2} = 1350 \cr & \left( {{{\text{r}}_1} - {{\text{r}}_2}} \right) = \frac{{1350}}{{4500}} = 0.3\% \cr} $$ Hence required difference in rates = 0.3%

$$\eqalign{ & {\text{Time = 2 years 3 months}} \cr & {\text{ = 2 + }}\frac{3}{{12}}{\text{ = }}\frac{9}{2}{\text{ years}} \cr & {\text{We know }}\boxed{{\text{SI = }}\frac{{{\text{P}} \times {\text{T}} \times {\text{R}}}}{{100}}} \cr & {\text{P = Rs 1600,}} \cr & {\text{T = }}\frac{9}{4}{\text{years,}} \cr & {\text{SI = Rs 252}} \cr & {\text{Put values in the formula ,}} \cr & \Rightarrow 252 = \frac{{1600 \times {\text{R}} \times 9}}{{100}} \cr & \Rightarrow 252 = 36{\text{R}} \cr & \Rightarrow {\text{R = }}\frac{{252}}{{36}} = 7\% \cr} $$

$$\eqalign{ & {\text{P}} = {\text{Rs}}.\,520, \cr & {\text{R}} = 13\% \cr & {\text{T}} = \frac{1}{2}yr. \cr & \therefore {\text{S}}{\text{.I}}{\text{.}} = {\text{Rs}}{\text{.}}\left( {\frac{{520 \times 13}}{{100 \times 2}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. }}33.80 \cr & {\text{Hence, amount after 6 months}} \cr & = {\text{Rs}}{\text{. }}\left( {520 + 33.80} \right) \cr & = {\text{Rs}}{\text{. }}553.80 \cr} $$

$$\eqalign{ & {\text{P}} = {\text{Rs}}{\text{. }}16800 \cr & {\text{R}} = 6\frac{1}{4}\% = \frac{{25}}{4}\% \cr & {\text{T}} = 9\,{\text{mths}} = \frac{3}{4}{\text{yr}}{\text{.}} \cr & \therefore {\text{S}}{\text{.I}}{\text{.}} = {\text{Rs}}{\text{.}}\left( {16800 \times \frac{{25}}{4} \times \frac{3}{4} \times \frac{1}{{100}}} \right) \cr & = {\text{Rs}}{\text{.}}\,787.50 \cr} $$

$$\eqalign{ & {\text{Let the principal}} \cr & {\text{ = 9 units }} \cr & {\text{Hence simple interest}} \cr & {\text{ = }}\frac{4}{9} \times {\text{9 = 4 units}} \cr & {\text{Let,}} \cr & {\text{Rate of interest = R% }} \cr & {\text{R = T (given)}} \cr & {\text{By using formula}} \cr & \boxed{{\text{SI = }}\frac{{{\text{P}} \times {\text{T}} \times {\text{R}}}}{{100}}} \cr & 4 = \frac{{9 \times {\text{R}} \times {\text{R}}}}{{100}} \cr & {{\text{R}}^2} = \frac{{400}}{9} \cr & {\text{R = }}\frac{{20}}{3} = 6\frac{2}{3}\% \cr} $$

$$\eqalign{ & {\text{Let principal = 5 units}} \cr & {\text{Hence interest}} \cr & {\text{ = 5}} \times \frac{2}{5} \cr & {\text{ = 2 units}} \cr & {\text{Time = 10 years}} \cr & {\text{By using formula, }} \cr & {\text{Rate% = }}\frac{2}{5} \times \frac{{100}}{{10}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{ = 4% }} \cr} $$

$$\eqalign{ & {\text{Principal = Rs}}{\text{. 1750}} \cr & {\text{Let the first part = }}x \cr & {\text{Hence second part}} \cr & {\text{ = }}\left( {1750 - x} \right) \cr & {\text{According to the question,}} \cr & \Rightarrow x \times \frac{8}{{100}} \times 1 = \left( {1750 - x} \right) \times \frac{6}{{100}} \times 1 \cr & \Rightarrow 4x = 5250 - 3x \cr & \Rightarrow 7x = 5250 \cr & \Rightarrow x = 750 \cr & {\text{First part = Rs}}{\text{. 750}} \cr & \therefore {\text{ Second part}} \cr & {\text{ = Rs}}{\text{. }}\left( {1750 - 750} \right) \cr & {\text{ = Rs}}{\text{. 1000}} \cr & {\text{Required interest}} \cr & {\text{ = 750}} \times \frac{8}{{100}} \cr & {\text{ = Rs}}{\text{. 60}} \cr} $$ Alternate Note : In such type of questions to save your valuable time follow the given below method. Let, Principal = 100 units in both cases 1 st part 2 nd part Total Principal → 100 ×3 : 100 ×4 +→ 700 units Interest → 8 ×3 : 6 ×4 Note : Interest is same in both cases According to the question, $$\eqalign{ & {\text{700 units = Rs}}{\text{. 1750}} \cr & {\text{1 unit = Rs}}{\text{. }}\frac{{1750}}{{700}} \cr & {\text{24 units = Rs}}{\text{. }}\frac{{1750}}{{700}} \times 24 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. 60}} \cr & {\text{Hence, required interest}} \cr & {\text{ = Rs}}{\text{. 60}} \cr} $$

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

Let the amounts invested in schemes A and B be Rs. 5x and 4x respectively. then, 5x = 20000 ⇒ x = 4000 ∴ Amount invested in scheme B = Rs. 16000 Total interest received on maturity $$ = {\text{Rs}}{\text{.}}\left[ {{\text{120}}\% {\text{ of}}\left( {\frac{{{\text{20000}} \times {\text{8}} \times {\text{5}}}}{{{\text{100}}}}} \right) + \left( {\frac{{{\text{16000}} \times {\text{9}} \times {\text{5}}}}{{{\text{100}}}}} \right)} \right]$$ $$\eqalign{ & = {\text{Rs}}{\text{.}}\,\left( {{\text{120}}\% \,{\text{of}}\,{\text{8000}} + {\text{7200}}} \right) \cr & = {\text{Rs}}{\text{.}}\,\left( {{\text{9600}} + {\text{7200}}} \right) \cr & = {\text{Rs}}{\text{.}}\,{\text{16800}} \cr & \therefore {\text{Total amount}} \cr & = {\text{Rs}}{\text{.}}\,\left( {{\text{20000}} + {\text{16000}} + {\text{16800}}} \right) \cr & = {\text{Rs}}{\text{.}}\,{\text{52800}} \cr} $$

Let the sum at 8% be Rs. x Let the sum at 5% be 1550 - x According to the given condition, $$\eqalign{ & \frac{{{\text{x}} \times 8 \times 3}}{{100}} + \frac{{\left( {1550 - {\text{x}}} \right) \times 5 \times 3}}{{100}} = 300 \cr & \Rightarrow \frac{{24{\text{x}} + \left( {1550 - {\text{x}}} \right) \times 15}}{{100}} = 300 \cr & \Rightarrow \frac{{24{\text{x}} + 23250 - 15{\text{x}}}}{{100}} = 300 \cr & \Rightarrow 24{\text{x}} - 15{\text{x}} = 30000 - 23250 \cr & \Rightarrow 24{\text{x}} - 15{\text{x}} = 6750 \cr & \Rightarrow 9{\text{x}} = 6750 \cr & \Rightarrow {\text{x}} = \frac{{6750}}{9} = 750 \cr & \frac{{{\text{Money lent at 5}}\% }}{{{\text{Money lent at 8}}\% }} \cr & = \frac{{1550 - 750}}{{750}} \cr & = \frac{{800}}{{750}} \cr & = \frac{{16}}{{15}} \cr & = 16:15 \cr} $$

(Principal + Interest) in 5 years → 520 (Principal + Interest) in 7 years → 568 $$\eqalign{ & \therefore {\text{Interest in 2 years = 48}} \cr & \therefore {\text{I = }}\frac{{{\text{48}}}}{2}{\text{ }} \cr & \,\,\,\,\,\,\,\,\,{\text{ = Rs}}{\text{. 24 in one year}} \cr & \therefore {\text{P = 520 - 24}} \times {\text{5}} \cr & \,\,\,\,\,\,\,\,\,\,{\text{ = Rs}}{\text{. 400}} \cr} $$

Let initial capital = 100 units According to the question, $$\eqalign{ & \frac{1}{4}{\text{unit = Rs}}{\text{. 61}}{\text{.50}} \cr & {\text{1 unit = Rs}}{\text{. 61}}{\text{.50}} \times {\text{4}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{ = Rs}}{\text{. 246}} \cr & {\text{100 units = Rs}}{\text{. 24600}} \cr} $$ Hence, Required capital = Rs. 24600 Alternate Difference in percentage $$\eqalign{ & {\text{ = 8% }} - \frac{{31}}{4}\% \cr & \Leftrightarrow \frac{1}{4}\% = 61.50 \cr & \Leftrightarrow 100\% = 24600 \cr} $$

$$\eqalign{ & {\text{P}} = {\text{Rs}}{\text{. 9534}} \cr & {\text{S}}{\text{.I}}{\text{.}} = {\text{Rs}}{\text{.}}\left( {{\text{11442}} - {\text{9534}}} \right) \cr & \,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.1908}} \cr & {\text{R}} = {\text{4}}\% \cr & \therefore {\text{Time}} = \left( {\frac{{{\text{100}} \times {\text{1908}}}}{{{\text{9534}} \times {\text{4}}}}} \right){\text{years}} \cr & = \left( {\frac{{{\text{47700}}}}{{{\text{9534}}}}} \right){\text{years}} \approx {\text{5 years}} \cr} $$

$$\eqalign{ & {\text{Amount Rs 7000}} \cr & {\text{Total interest in 5 years}} \cr & {\text{ = 5}} \times \frac{{10}}{3}\% = \frac{{50}}{3}\% = \frac{1}{6} \cr} $$ Principal Amount 6 (6 + 1) ↓ × 1000 ↓ × 1000 6000 7000 Hence required principal = Rs. 6000

Let the amount invested = Rs. P According to the questions, $$\eqalign{ & \Leftrightarrow \frac{{{\text{P}} \times 9 \times {\text{2}}}}{{100}} + \frac{{{\text{P}} \times {\text{10}} \times {\text{2}}}}{{100}} = 760 \cr & \Leftrightarrow \frac{{ {18{\text{P + 20P}}} }}{{100}} = 760 \cr & \Leftrightarrow 38{\text{P = 76000}} \cr & \Leftrightarrow {\text{P = 2000}} \cr} $$ Alternate Total interest percent $$\eqalign{ & {\text{ = }}\left( {9 \times 2} \right)\% + \left( {10 \times 2} \right)\% \cr & \Rightarrow 38\% = 760 \cr & \Rightarrow 100\% = 2000 \cr} $$ Hence required principal = Rs. 2000

$$\eqalign{ & {{\text{T}}_1} = 15\operatorname{months} \cr & \,\,\,\,\,\, = \frac{{15}}{{12}}years \cr & {R_1} = 7\frac{1}{2}\% = \frac{{15}}{2}\% \cr & {{\text{T}}_2} = 8\operatorname{months} \cr & \,\,\,\,\,\,\, = \frac{8}{{12}}years \cr & {{\text{R}}_2} = 12\frac{1}{2}\% = \frac{{25}}{2}\% \cr & {\text{Let the principal}} = {\text{P}} \cr & {\text{According to the question,}} \cr & \Leftrightarrow \frac{{{\text{P}} \times {\text{15}} \times {\text{15}}}}{{12 \times 2 \times 100}} - \frac{{{\text{P}} \times 25 \times 8}}{{12 \times 2 \times 100}} = 32.50 \cr & \Rightarrow \frac{{225{\text{P}}}}{{2400}} - \frac{{200{\text{P}}}}{{2400}} = 32.50 \cr & \Rightarrow \frac{{25{\text{P}}}}{{2400}} = 32.50 \cr & \Rightarrow {\text{P = Rs 3120}} \cr & {\text{Hence required principal}} \cr & {\text{ = Rs}}{\text{. 3120}} \cr} $$

$$\eqalign{ & {\text{P}} = {\text{Rs}}{\text{.}}\,{\text{21250}} \cr & {\text{S}}{\text{.I}}{\text{.}} = {\text{Rs}}{\text{.}}\,\left( {{\text{26350}} - {\text{21250}}} \right) \cr & \,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,5100 \cr & {\text{T}} = {\text{6 years}} \cr & \therefore {\text{Rate}} = \left( {\frac{{100 \times 5100}}{{{\text{21250}} \times {\text{6}}}}} \right)\% \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 4\% \cr} $$

$$\eqalign{ & {\text{P}} = {\text{Rs.}}{\text{ 8630}} \cr & {\text{S}}{\text{.I}}{\text{.}} = {\text{Rs.}}{\text{ 1553}}{\text{.40}} \cr & {\text{T}} = {\text{3 years}} \cr & \therefore {\text{Rate}} \cr & = \left( {\frac{{100 \times 1553.40}}{{8630 \times 3}}} \right)\% \cr & = 6\% \cr} $$

$$\eqalign{ & {\text{Let the rate of be R}}\% {\text{p}}{\text{.a}}{\text{.}} \cr & Then, \cr & {\frac{{600 \times R \times 4}}{{100}}} + {\frac{{600 \times R \times 2}}{{100}}} = 180 \cr & \Rightarrow 2400{\text{R}} + 1200{\text{R}} = 18000 \cr & \Rightarrow 3600{\text{R}} = 18000 \cr & \Rightarrow {\text{R}} = 5\% \cr} $$

$$\eqalign{ & {\text{Let principal = P}} \cr & \therefore {\text{amount = 3P}} \cr & {\text{Interest = 3P}} - {\text{P}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{ = 2P}} \cr & {\text{According to the question,}} \cr & {\text{2P = }}\frac{{{\text{P}} \times {\text{R}} \times {\text{20}}}}{{100}} \cr & \Rightarrow {\text{R = 10% }} \cr} $$ Let after t years it will become double $$\eqalign{ & {\text{Hence,}} \cr & {\text{Interest = 2P}} - {\text{P = P}} \cr & \Rightarrow {\text{P = }}\frac{{{\text{P}} \times 10 \times {\text{t}}}}{{100}} \cr & \Rightarrow {\text{t = 10 years}} \cr} $$

$$\eqalign{ & {\text{Loan Amount}}:{\text{Total Interest}} \cr & {\text{5}}:{\text{2}} \cr & {\text{Principal}} \to {\text{5}} \cr & {\text{Interest for 1 year}} \Rightarrow {\text{2}} \cr & {\text{Interest for 5 year}} \Rightarrow {\text{10}} \cr & {\text{10}} = \frac{{5 \times {\text{R}} \times 5}}{{100}} \cr & {\text{R}}\% = 40\% \cr & = \frac{{40}}{{100}} = \frac{2}{5} \cr & {\text{P}}:{\text{R}} = 5:\frac{2}{5} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = 25:2 \cr} $$

$$\eqalign{ & {\text{Half yearly rate}} \cr & {\text{ = }}\frac{6}{2} = 3\% \cr & {\text{Effective rate % }} \cr & {\text{ = 3}} + {\text{3}} + \frac{{3 \times 3}}{{100}} \cr & = 6.09\% \cr} $$

$$\eqalign{ & {\text{S}}{\text{.I}}{\text{.}} = {\text{Rs}}{\text{.}}\,{\text{8376}} \cr & {\text{R}} = 8\% \cr & {\text{T}} = {\text{6}}{\kern 1pt} {\text{years}} \cr & \therefore {\text{Sum}} = {\text{Rs}}{\text{.}}\left( {\frac{{100 \times 8376}}{{8 \times 6}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. }}17450 \cr} $$

$$\eqalign{ & {\text{S}}{\text{.I}}{\text{.}} = {\text{Rs}}{\text{. 210}} \cr & {\text{R}} = 3\frac{3}{4}\% = \frac{{15}}{4}\% \cr & {\text{T}} = {\text{2}}\frac{{\text{1}}}{{\text{3}}}{\text{years}} = \frac{7}{3}{\text{years}} \cr & \therefore {\text{Sum}} = {\text{Rs}}{\text{.}}\left( {\frac{{100 \times 210}}{{\frac{{15}}{4} \times \frac{7}{3}}}} \right) \cr & = {\text{Rs}}{\text{.}}\left( {\frac{{100 \times 210 \times 4 \times 3}}{{15 \times 7}}} \right) \cr & = {\text{Rs}}{\text{. }}2400 \cr} $$

$$\eqalign{ & {\text{Let principal = 10P}} \cr & {\text{Interest = 10P}} \times \frac{{30}}{{100}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{ = 3P}} \cr & {\text{According to the question,}} \cr & {\text{Case (I)}} \cr & \Rightarrow 3{\text{P = }}\frac{{{\text{10P}} \times {\text{R}} \times {\text{6}}}}{{100}} \cr & \Rightarrow {\text{R = 5}}\% \cr & {\text{Case (II)}} \cr & {\text{Interest = Principal = 10P}} \cr & \Rightarrow {\text{10P = }}\frac{{{\text{10P}} \times {\text{5}} \times {\text{t}}}}{{100}} \cr & \Rightarrow {\text{t = 20 years}} \cr} $$

Let the principal in each case = 100 units According to the question, 1 st part 2 nd part 3 rd part Principal → 100 x6 100 x3 100 x2 Rate % → 10 12 15 Time → 6 10 12 Interest → 60 x6 120 x3 180 x2 Interest Interest is same in each, so equal the interest. Hence required ratio = 600 : 300 : 200 of sum = 6 : 3 : 2

$$\eqalign{ & {\text{Principal = Rs}}{\text{. 1000 }} \cr & {\text{Rate = 5}}\% \cr & {\text{Interest for first 10 years}} \cr & = \frac{{1000 \times 5 \times 10}}{{100}} \cr & = {\text{Rs}}{\text{. 500}} \cr & {\text{After 10 years principal}} \cr & = {\text{(1000}} + {\text{500)}} \cr & {\text{ = Rs}}{\text{. 1500}} \cr & {\text{Remaining interest}} \cr & {\text{ = Rs}}{\text{. (2000}} - {\text{1500)}} \cr & {\text{ = Rs}}{\text{. 500}} \cr & {\text{Required time }} \cr & {\text{ = }}\frac{{500}}{{1500}} \times \frac{{100}}{5} \cr & = \frac{{100}}{15} \cr & = \frac{{20}}{3} \cr & = 6\frac{2}{3}{\text{ years}} \cr & {\text{Total time}} \cr & = \left( {10 + 6\frac{2}{3}} \right){\text{years}} \cr & {\text{ = 16}}\frac{2}{3}{\text{ years}} \cr} $$

$$\eqalign{ & {\text{Sum}} = {\frac{{100 \times S.I.}}{{R \times T}}} \cr & = {\text{Rs}}{\text{.}}\,\, {\frac{{100 \times x}}{{x \times x}}} \cr & = {\text{Rs}}{\text{.}}\,\, {\frac{{100}}{x}} \cr} $$

$$\eqalign{ & {\text{P}} = {\text{Rs}}{\text{. 6200}} \cr & {\text{S}}{\text{.I}}{\text{.}} \cr & = {\text{Rs}}{\text{.}}\left( {{\text{9176}} - {\text{6200}}} \right) \cr & = {\text{Rs}}{\text{. }}2976 \cr & {\text{T}} = 4\,{\text{years}} \cr & \therefore {\text{Rate}} \cr & = \left( {\frac{{100 \times 2976}}{{6200 \times 4}}} \right)\% \cr & = 12\% \cr & {\text{New }}{\text{rate}} \cr & = \left( {12 + 3} \right)\% \cr & = 15\% \cr & {\text{New}}\,{\text{S}}{\text{.I}}{\text{.}} \cr & = {\text{Rs}}{\text{.}}\left( {\frac{{{\text{6200}} \times 15 \times 4}}{{100}}} \right) \cr & = \text{Rs.} \,3720 \cr & {\text{New}}\,{\text{amount}} \cr & = {\text{Rs}}{\text{.}}\left( {{\text{6200}} + 3720} \right) \cr & = {\text{Rs}}{\text{.}}\,9920 \cr} $$

$$\eqalign{ & {\text{Sum}} = {\text{Rs}}{\text{.}}\left( {\frac{{100 \times 1536}}{{12 \times 5}}} \right) \cr & = {\text{Rs}}{\text{. }}2560 \cr & Now, \cr & P = {\text{Rs}}{\text{.}}\left( {2560 + 1000} \right) \cr & \,\,\,\,\,\, = {\text{Rs}}{\text{. }}3560 \cr & {\text{T}} = {\text{2years}} \cr & {\text{R}} = {\text{12}}\% \cr & \therefore S.I. \cr & = {\text{Rs}}{\text{.}}\left( {\frac{{3560 \times 12 \times 2}}{{100}}} \right) \cr & = {\text{Rs}}{\text{.}}\,854.40 \cr} $$

$$\eqalign{ & {\text{S}}{\text{.I}}{\text{.}} = {\text{Rs}}{\text{. 840}} \cr & {\text{R}} = {\text{5}}\% \cr & {\text{T}} = 8\,{\text{years}} \cr & {\text{Principal}} \cr & = {\text{Rs}}{\text{. }}\left( {\frac{{100 \times 840}}{{5 \times 8}}} \right) \cr & = {\text{Rs}}{\text{.}}\,2100 \cr & {\text{Now,}} \cr & {\text{P}} = {\text{Rs}}{\text{.}}\,2100 \cr & {\text{S}}{\text{.I}}{\text{.}} = {\text{Rs}}{\text{.}}\,{\text{840}} \cr & {\text{T}} = {\text{5 years}}{\text{}} \cr & \therefore {\text{Rate}} \cr & = \left( {\frac{{100 \times 840}}{{2100 \times 5}}} \right)\% \cr & = {\text{8}}\% \cr} $$

$$\eqalign{ & {\text{Principal}}\,\,\,\,\,\,\,{\text{Amount}} \cr & \underbrace {\,\,\,\,\,{\text{6000}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{7200}}\,\,\,\,\,}_{ + 1200} \cr & {\text{By using formula,}} \cr & {\text{Rate }}\% \cr & = \frac{{1200}}{{6000}} \times \frac{{100}}{4} \cr & {\text{ = 5}}\% \cr & {\text{New rate}}\% \cr & = {\text{5}} \times \frac{3}{2} = {\text{7}}{\text{.5}}\% \cr & {\text{Interest after 5 years}} \cr & = \frac{{6000 \times 7.5 \times 5}}{{100}} \cr & {\text{ = Rs}}{\text{. 2250}} \cr & {\text{Hence,}} \cr & {\text{amount }} \cr & {\text{ = Rs}}{\text{. }}\left( {6000 + 2250} \right) \cr & {\text{ = Rs 8250}} \cr} $$

$$\eqalign{ & {\text{Let the required time = t years }} \cr & {\text{According to the question,}} \cr & \Leftrightarrow \frac{{500 \times 4 \times 6.25}}{{100}} = \frac{{400 \times 5 \times {\text{t}}}}{{100}} \cr & \Leftrightarrow 5 \times 4 \times 625 = 400 \times 5 \times {\text{t}} \cr & \Leftrightarrow {\text{t = }}\frac{{625}}{{100}} = \frac{{25}}{4} \cr & \Leftrightarrow {\text{t}} = 6\frac{1}{4}{\text{years}} \cr} $$