Interest — Part 1 MCQs with Explanations

$$\eqalign{ & {\text{P = Principal}}\,{\text{Amount}} \cr & {\text{T = }}\,{\text{Time}}\,{\text{period}} \cr & {\text{R = Rate}}\,{\text{of}}\,{\text{Interest}} \cr & I = \frac{{P \times T \times R}}{{100}} \cr & I = \frac{{5200 \times 2 \times 6}}{{100}} \cr & I = 624. \cr} $$

$$\eqalign{ & A = P + I \cr & A = 1200 + {\frac{{PTR}}{{100}}} \cr & A = {1200 + {\frac{{1200 \times 5 \times 3}}{{100}}} } \cr & {\text{Amount}},\,A = Rs.\,1380 \cr} $$

$$\eqalign{ & {\text{Let rate is }}R\% \cr & {\text{We have}}, \cr & I = \frac{{PTR}}{{100}} \cr & {\text{Here}},1500 = \frac{{5000 \times 3 \times R}}{{100}} \cr & {\text{Thus}},R = 10\% \cr} $$

$$\eqalign{ & {\text{We}}\,{\text{can}}\,{\text{use}}\,{\text{formula}}\,{\text{of}}\,{\text{compound}}\,{\text{interest}} \cr & A = P \times {\left[ {1 + \left( {\frac{r}{{100}}} \right)} \right]^n} \cr & A = 2100 \times {\left[ {1 + \left( {\frac{5}{{100}}} \right)} \right]^2} \cr & A = 2100 \times {\left[ {\frac{{105}}{{100}}} \right]^2} \cr & A = \frac{{ {2100 \times 11025} }}{{10000}} \cr & {\text{Hence,}}\,{\text{Amount}}\,A = Rs.\,2315.25 \cr} $$

$$\eqalign{ & {\text{The}}\,{\text{difference}}\,{\text{between}}\,{\text{compound}}\,{\text{interest}}\,{\text{and}} \cr & {\text{simple}}\,{\text{interest}}\,{\text{over}}\,{\text{two}}\,{\text{years}}\,{\text{is}}\,{\text{given}}\,{\text{by}} \cr & \frac{{{{\Pr }^2}}}{{{{100}^2}}}\,or\,P{\left( {\frac{r}{{100}}} \right)^2} \cr & {\text{Here,}}\,{\text{Principal}}\,\left( P \right) = Rs.\,2000 \cr & {\text{Rate}}\,\left( r \right) = 5\% \cr & {\text{Now}}\,{\text{difference}}, \cr & D = \frac{{ {2000 \times 5 \times 5} }}{{ {100 \times 100} }} \cr & D = Rs.\,5 \cr} $$

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

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

$$\eqalign{ & {1^{{\text{st}}}}{\kern 1pt} {\text{case}}: \cr & {I_1} = \frac{{700 \times 3 \times 8}}{{100}} = {\text{Rs}}{\text{. }}168 \cr & {2^{{\text{nd}}}}{\kern 1pt} {\text{case}}: \cr & {I_2} = \frac{{700 \times 2 \times 7.5}}{{100}} = {\text{Rs}}{\text{. }}105 \cr} $$ Then total interest for five years $$ = \left( {{I_1} + {I_2}} \right) = {\text{Rs}}{\text{. }}273$$ Alternatively, As interest is calculated as simple interest So, we can add up rates for all given 5 years and calculate it easily i.e. For the five years rate = (8 × 3 + 7.5 × 2) = 39% Now, 700 = 39% $$ \uparrow $$ ⇒ 973 Interest = Rs. 273 The thought can go this way we internally calculated $$\eqalign{ & {\kern 1pt} 10\% \,{\text{of}}\,700 = {\frac{{700}}{{10}}} = 70 \cr & {\text{Then}}, \cr & 39\% \,{\text{of}}\,700 \cr & = \left( {40\% - 1\% } \right)\,{\text{of}}\,700 \cr & = \left( {280 - 7} \right) \cr & = 273 \cr} $$

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

$$\eqalign{ & {\text{Given}}, \cr & {\text{Compound}}\,{\text{rate}},\,R = 10\% \,{\text{per}}\,{\text{annum}} \cr & {\text{Time}} = 2\,{\text{years}} \cr & CI - Rs.\,420 \cr & {\text{Let}}\,P\,{\text{be}}\,{\text{the}}\,{\text{required}}\,{\text{principal}} \cr & A = \left( {P + CI} \right) \cr & {\text{Amount}},A = \left\{ {P \times {{\left[ {1 + \left( {\frac{R}{{100}}} \right)} \right]}^n}} \right\} \cr & \left( {P + CI} \right) = \left\{ {P \times {{\left[ {1 + \frac{{10}}{{100}}} \right]}^2}} \right\} \cr & \left( {P + 420} \right) = P \times {\left[ {\frac{{11}}{{10}}} \right]^2} \cr & P - 1.21P = - 420 \cr & 0.21P = 420 \cr & {\text{Hence}},P = \frac{{420}}{{0.21}} = Rs.\,2000 \cr} $$

$$\eqalign{ & P = Rs.\,3300 \cr & A = Rs.\,3399 \cr & R = 6\% \,{\text{per}}\,{\text{annum}} \cr & {\text{Let}}\,{\text{the}}\,{\text{time}}\,{\text{be}}\,{\text{n}}\,{\text{years}}{\text{.}} \cr & {\text{Compound}}\,{\text{interest}}\,{\text{is}}\,{\text{taken}}\,{\text{half - yearly}}. \cr & A = P \times {\left[ {1 + \left( {\frac{R}{2} \times 100} \right)} \right]^{2n}} \cr & 3399 = 3300{\left( {1 + \frac{3}{{100}}} \right)^{2n}} \cr & {\left( {1.03} \right)^{2n}} = \frac{{3399}}{{3300}} \cr & {\left( {1.03} \right)^{2n}} = {\left( {1.03} \right)^1} \cr & Thus,\,2n = 1\,year \cr & n = \frac{1}{2}{\text{year}} = 6\,{\text{months}} \cr} $$

$$\eqalign{ & {\text{Value}}\,{\text{of}}\,{\text{maruti}}\,{\text{Van}},\, \cr & {V_0} = Rs.\,196000 \cr & {\text{Rate}}\,{\text{of}}\,{\text{depreciation}},\, \cr & r = 14 {\frac{2}{7}} \% = \frac{{100}}{7}\% ; \cr & {\text{Time}},\,t = 2\,{\text{years}} \cr & {\text{Let}}\,{V_1}\,{\text{is}}\,{\text{the}}\,{\text{value}}\,{\text{after}}\,{\text{depreciation}}. \cr & {V_1} = {V_0} \times {\left[ {1 - \left( {\frac{r}{{100}}} \right)} \right]^t} \cr & {V_1} = 196000 \times {\left[ {1 - \left( {\frac{{\left( {\frac{{100}}{7}} \right)}}{{100}}} \right)} \right]^2} \cr & {V_1} = 196000 \times {\left( {1 - {\frac{1}{7}}} \right)^2} \cr & {V_1} = 196000 \times {\left( {\frac{7-1}{7}} \right)^2} \cr & {V_1} = 196000 \times {\left( {\frac{6}{7}} \right)^2} \cr & {V_1} = \frac{{\left( {196000 \times 36} \right)}}{{49}} \cr & {V_1} = Rs.\,144000 \cr} $$

$$\eqalign{ & {\text{Interest}}\,{\text{for}}\,{\text{the}}\,{\text{last}}\,{\text{5}}\,{\text{years}} \cr & = \frac{{PTR}}{{100}} \cr & = \frac{{360 \times 5 \times 6}}{{100}} = Rs.\,108 \cr & {\text{Interest}}\,{\text{for}}\,{\text{year}} = 540 - 360 = 180 \cr & {\text{So,}}\,{\text{interest}}\,{\text{for}}\,{\text{first}}\,{\text{four}}\,{\text{years}} \cr & = 180 - 108 = Rs.\,72; \cr & {\text{Now,}}\,{\text{rate}}\,{\text{for}}\,{\text{first}}\,{\text{four}}\,{\text{years}} \cr & = \frac{{ {72 \times 100} }}{{360 \times 4}} \cr & = 5\% \cr} $$

$$\eqalign{ & {\text{Here,}}\,{\text{time}}\,{\text{interval}}\,{\text{is}}\,{\text{given}}\,{\text{as}}\, \cr & {\text{February}}\,5,\,1994\,{\text{to}}\,{\text{April}}\,18,\,1994 \cr & = 73\,{\text{days}} = \frac{{73}}{{365}} = 0.2\,{\text{years}}. \cr & {\text{Now}}\,{\text{interest}} = \frac{{PTR}}{{100}} \cr & = \frac{{ {700 \times 9 \times 0.2} }}{{100}} \cr & = Rs.\,12.60 \cr} $$

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 $$\eqalign{ & \frac{{14x}}{{100}} + \frac{{ {18000 - 12x} }}{{100}} = 186 \cr & 14x + 18000 - 12x = 186 \times 100 \cr & 2x = 18600 - 18000 \cr & x = \frac{{600}}{2} = {\text{Rs}}{\text{. }}300 \cr & {\text{Hence, money lent}}{\kern 1pt} {\text{at }}12\% \cr & = 1500 - 300 \cr & = {\text{Rs}}{\text{.}}\,1200 \cr} $$

Let the rate be R at which Principal P is invested for 2 years. According to question, {Interest at Rate (R + 2)}% - (interest at rate R%) = Rs. 60 $$\frac{{\left( {P \times 2 \times \left( {R + 2} \right)} \right)}}{{100}} - $$ $$\frac{{\left( {P \times 2 \times R} \right)}}{{100}}$$ $$ = 60$$ $$\eqalign{ & \frac{{ {2PR + 4P - 2PR} }}{{100}} = 60 \cr & 4P = 60 \times 100 \cr & {\text{Or}},P = \frac{{60 \times 100}}{4} \cr & {\text{Hence}},P = {\text{Rs}}{\text{.}}\,1500 \cr} $$

$$\eqalign{ & {\text{Let}}\,{\text{sum}}\,{\text{is}}\,P. \cr & {\text{The}}\,{\text{difference}}\,{\text{between}}\,{\text{compound}}\,{\text{interest}}\,{\text{and}} \cr & \,{\text{simple}}\,{\text{interest}}\,{\text{over}}\,{\text{three}}\,{\text{years}}\,{\text{is}}\,{\text{given}}\,{\text{by}} \cr & = P\left( {\frac{r}{{100}}} \right)2 \times \left\{ {\left( {\frac{r}{{100}}} \right) + 3} \right\} \cr & 48 = P \times \left( {\frac{{20}}{{100}}} \right)2 \times \left\{ {\left( {\frac{{20}}{{100}}} \right) + 3} \right\} \cr & 48 = P \times \frac{4}{{100}} \times \frac{{16}}{5} \cr & 48 = P \times \frac{{64}}{{500}} \cr & {\text{Or}},\,64P = 48 \times 500 \cr & {\text{Hence}},\,P = Rs.\,375 \cr} $$

$$\eqalign{ & {\text{Let}}\,{\text{time}}\,{\text{is}}\,T\,{\text{years}}. \cr & {\text{According}}\,{\text{to}}\,{\text{questions}}, \cr & \frac{{1750 \times 9 \times T}}{{100}} = \frac{{\left( {2500 \times 10.5 \times 4} \right)}}{{100}} \cr & Or,\,T = \frac{{ {2500 \times 10.5 \times 4} }}{{1750 \times 9}} \cr & Or,\,T = 6.66 = 6\,{\text{years}}\,{\text{and}}\,8\,{\text{months}} \cr} $$

$$\eqalign{ & {\text{Let}}\,{\text{rate}}\,{\text{is}}\,R\% \cr & {\text{According}}\,{\text{to}}\,{\text{the}}\,{\text{question}}, \cr & \left[ {\frac{{400 \times 2 \times R}}{{100}}} \right] + \left[ {\frac{{100 \times 4 \times R}}{{100}}} \right] = 60 \cr & 8R + 4R = 60 \cr & {\text{Hence}},\,R = 5\% \cr} $$

$$\eqalign{ & {1^{{\text{st}}}}\,{\text{Method}}: \cr & {\text{Let rate is }}R\% \cr & {\text{Now}}, \cr & P = 100, \cr & A = 400, \cr & I = 400 - 100 = 300, \cr & {\text{Time}},\,T = 10\,{\text{years}} \cr & I = \frac{{PTR}}{{100}} \cr & {\text{Or}},R = \frac{{ {100 \times I} }}{{PT}} \cr & {\text{Or}},R = \frac{{ {100 \times 300} }}{{ {100 \times 10} }} \cr & {\text{Hence}},{\kern 1pt} R = 30\% \cr} $$ 2 nd Method : Here, the sum become 4 times that means 100, become 400. Rate of such question is given by $$R = \frac{{{\text{interest}}}}{{{\text{time}}}} = \frac{{300}}{{10}} = 30\% $$ 3 rd Method : Here, 300% of rise in the sum so $$\eqalign{ & 100 - - - 300\% \uparrow - - - {\kern 1pt} 400; \cr & R = {\frac{{{\text{total}}\,{\text{percentage rise}}}}{{{\text{given time}}}}} \cr & \,\,\,\,\,\,\,\, = \frac{{300\% }}{{10}} \cr & \,\,\,\,\,\,\,\, = 30\% \cr} $$

$$\eqalign{ & {\text{Let}}, \cr & {\text{Principal}} = Rs.\,100 \cr & {\text{Amount}} = Rs.\,200 \cr & {\text{Rate}} = r\% \cr & {\text{Time}} = 4\,{\text{years}} \cr & {\text{Now}}, \cr & A = P \times {\left[ {1 + \left( {\frac{r}{{100}}} \right)} \right]^n} \cr & 200 = 100 \times {\left[ {1 + \left( {\frac{r}{{100}}} \right)} \right]^4} \cr & 2 = {\left[ {1 + \left( {\frac{r}{{100}}} \right)} \right]^4} - - - - - - \left( i \right) \cr & {\text{If}}\,{\text{sum}}\,{\text{become}}\,{\text{8}}\,{\text{times}}\,{\text{in}}\,{\text{the}}\,{\text{time}}\,n\,{\text{years}} \cr & {\text{then,}} \cr & 8 = {\left( {1 + \left( {\frac{r}{{100}}} \right)} \right)^n} \cr & {2^3} = {\left( {1 + \left( {\frac{r}{{100}}} \right)} \right)^n} - - - - - - \left( {ii} \right) \cr & {\text{Using}}\,{\text{eqn}}\,\left( i \right)in\left( {ii} \right),\,{\text{we}}\,{\text{get}} \cr & {\left( {{{\left[ {1 + \left( {\frac{r}{{100}}} \right)} \right]}^4}} \right)^3} = {\left( {1 + \left( {\frac{r}{{100}}} \right)} \right)^n} \cr & {\left[ {1 + \left( {\frac{r}{{100}}} \right)} \right]^{12}} = {\left( {1 + \left( {\frac{r}{{100}}} \right)} \right)^n} \cr & {\text{Thus}},\,n = 12\,{\text{years}}. \cr} $$

$$\eqalign{ & {\text{Let}}\,{1^{st}}\,{\text{part}}\,{\text{is}}\,x\,{\text{and}}\,{2^{nd}}\,{\text{part}}\,{\text{is}}\,\left( {6000 - x} \right) \cr & {\text{According}}\,{\text{to}}\,{\text{question}}, \cr & {\frac{{x \times 2 \times 6}}{{100}}} = \frac{{ {\left( {6000 - x} \right) \times 3 \times 8} }}{{100}} \cr & 12x = 144000 - 24x \cr & Or,\,36x = 144000 \cr & Or,\,x = \frac{{144000}}{{36}} = Rs.\,4000 \cr & {1^{st}}\,{\text{part}} = Rs.\,4000 \cr & {2^{nd}}\,{\text{part}} = Rs.\,2000 \cr} $$

$$\eqalign{ & {\text{Let}}\,{\text{sum}}\,is\,{\text{Rs}}.\,100, \cr & {\text{Then}}\,{\text{it}}\,{\text{become}}\,\frac{7}{4}\,{\text{times}} \cr & i.e.\,{\text{Rs}}.\,\frac{{700}}{4}\,{\text{in}}\,6\,{\text{years}}. \cr & {\text{Interest}} = {\frac{{700}}{4}} - 100 = Rs.\,\frac{{300}}{4} \cr & {\text{Hence}}, \cr & {\text{Rate}} = \frac{{{\text{Total}}\,{\text{interest}}}}{{{\text{Given}}\,{\text{time}}}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{300}}{4} \times 6 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = 12 {\frac{1}{2}} \% \cr} $$

Let, Principal, P = Rs. 100; Amount, A = Rs. 200; Time = 12 years; Interest = Rs. 100; Rate of interest $$\eqalign{ & = \frac{{{\text{Total Interest}}}}{{{\text{Given Time}}}} \cr & = \frac{{100}}{{12}} \cr & = 8\frac{1}{3}\% \cr} $$

$$\eqalign{ & {\text{Let}}\,{\text{each}}\,{\text{installment}}\,{\text{be}}\,P \cr & {\text{Hence}}, \cr & {\frac{x}{{ {\left( {\frac{{100}}{{100 + r}}} \right) + {{\left( {\frac{{100}}{{100 + r}}} \right)}^2}} }}} \cr & Or,\,\frac{x}{{ {1 + {\frac{{15}}{{400}}} } }} + \frac{x}{{ {1 + {{\left( {\frac{{15}}{{400}}} \right)}^2}} }} = Rs.\,13040 \cr & {\text{On}}\,{\text{solving,}}\,{\text{it}}\,{\text{gives}}\,x = Rs.\,6889 \cr} $$

As short - cut of installment for 2 installments is given by Installment = $$ {\frac{{\text{P}}}{{ {\left( {\frac{{100}}{{100}} + {\text{R}}} \right) + {{\left( {\frac{{100}}{{100}} + {\text{R}}} \right)}^2}} }}} $$ There is the need of rate (R) which is unavailable in the question. so, we cannot determine the answer

$$\eqalign{ & {\text{Total}}\,{\text{cow}}\,{\text{milksold}}\,{\text{in}}\,{\text{five}}\,{\text{days}} \cr & = 5 \times 8 = 40\,{\text{litres}} \cr & {\text{Total}}\,{\text{buffalo}}\,{\text{milk}}\,{\text{sold}}\,{\text{in}}\,{\text{five}}\,{\text{days}} \cr & = 5 \times 10 = 50\,{\text{litres}} \cr & {\text{Hence}}, \cr & {\text{SP}}\,{\text{of}}\,{\text{40}}\,{\text{litres}}\,{\text{cow}}\,{\text{milk}} \cr & = 40 \times 55 = Rs.\,2200 \cr & {\text{Hence}}, \cr & {\text{Profit}}\,{\text{on}}\,{\text{cow}}\,{\text{milk}} \cr & = \frac{{ {2200 \times 12} }}{{112}} = Rs.\,235.71 \cr & {\text{Profit}}\,{\text{on}}\,{\text{buffalo}}\,{\text{milk}} \cr & = \frac{{ {50 \times 36 \times 20} }}{{120}} = Rs.\,300 \cr & {\text{Thus}}, \cr & {\text{Total}}\,{\text{Profit}} \cr & = 235.71 + 300 = Rs.\,535.71 \cr} $$

$$\eqalign{ & \frac{{{\text{Decrease in second year}}}}{{{\text{Decrease in third year}}}} \cr & = \frac{{100}}{{\left( {100 - r} \right)}} \cr & = \frac{{10}}{9} \to r = 10\% \cr} $$ Let the population of vultures 3 years ago be P, then $$\eqalign{ & P \times {\left[ {1 - \left( {\frac{{10}}{{100}}} \right)} \right]^3} = 29160 \cr & \to P = 40000 \cr} $$

1000 == 10%↑ ⇒ 1100 == Double ⇒ 2200 == 10%↑ ⇒ 2420 == Double ⇒ 4840 == 10%↑ ⇒ 5324 == Double ⇒ 10648

Difference between Compound interest and Simple interest for the fourth year is Rs. 7280 $$\eqalign{ & P{\left( {\frac{6}{5}} \right)^3}\left( {\frac{1}{5}} \right) - \frac{P}{5} = 7280 \cr & \frac{P}{5}\left[ {{{\left( {\frac{6}{5}} \right)}^3} - 1} \right] = 7280 \cr & \frac{P}{5}\left[ {\frac{{216}}{{125}} - 1} \right] = 7280 \cr & \frac{P}{5} \times \frac{{91}}{{125}} = 7280 \cr & \therefore P = 50000 \cr} $$

Let X = equal installment at the end of one year( rate% annually) . Now 1st year, P =210, Interest = $$\frac{{{\text{PTR}}}}{{100}}$$ = 210 * 0.1 = 21. Let X is to be paid as an equal installment. Now, at the beginning of 2nd year, P = 210 + 21 - X, Interest at the end of 2nd year, = (231 - X) * 0.1 = 23.1 - 0.1X. Hence,total installment, 2X = 210 + 21 + 23.1 - 0.1X, X = $$\frac{{{\text{254}}{\text{.1}}}}{{2.1}}$$ = 121.

$$A = P {\left( {1 - \frac{R}{{100}}} \right)^n}$$ Where A = Value of goods after n years P = Initial Price R = Rate of depriciation $$\eqalign{ & \therefore P = \frac{{5832}}{{{{\left( {1 - \frac{{10}}{{100}}} \right)}^3}}} \cr & \Rightarrow P = \frac{{5832}}{{{{\left( {1 - \frac{1}{{10}}} \right)}^3}}} \cr & \Rightarrow P = \frac{{5832}}{{{{\left( {\frac{9}{{10}}} \right)}^3}}} \cr & \Rightarrow P = 5832 \times \frac{{10}}{9} \times \frac{{10}}{9} \times \frac{{10}}{9} \cr & \Rightarrow P = 8000 \cr} $$

S.I. for 1 year = Rs. (854 - 815) = Rs. 39. S.I. for 3 years = Rs.(39 x 3) = Rs. 117. Therefore Principal = Rs. (815 - 117) = Rs. 698.

Let the sum invested in Scheme A be Rs. x and that in Scheme B be Rs. (13900 - x ) Then, $$ {\frac{{x \times 14 \times 2}}{{100}}} + $$ $$ {\frac{{\left( {13900 - x} \right) \times 11 \times 2}}{{100}}} $$ $$ = 3508$$ ⇒ 28x - 22x = 350800 - (13900 x 22) ⇒ 6x = 45000 ⇒ x = 7500 So, sum invested in Scheme B = Rs. (13900 - 7500) = Rs. 6400

$$\eqalign{ & {\text{Principal}} \cr & = Rs.\,\left( {\frac{{100 \times 4016.25}}{{9 \times 5}}} \right) \cr & = Rs.\,\left( {\frac{{401625}}{{45}}} \right) \cr & = Rs.\,8925 \cr} $$

$$\eqalign{ & {\text{Time}} = \left( {\frac{{100 \times 81}}{{450 \times 4.5}}} \right){\text{years}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = 4\,{\text{years}} \cr} $$

$$\eqalign{ & {\text{Let}}\,{\text{rate}}\, = R\% \,{\text{and}} \cr & {\text{Time}}\,{\text{ = }}\,{\text{R}}\,{\text{years}}{\text{}} \cr & {\text{Then,}}\, {\frac{{1200 \times R \times R}}{{100}}} = 432 \cr & \Rightarrow 12{R^2} = 432 \cr & \Rightarrow {R^2} = 36 \cr & \Rightarrow R = 6 \cr} $$

$$\eqalign{ & {\text{S}}{\text{.I}}{\text{.}} = Rs.\,\left( {15500 - 12500} \right) \cr & \,\,\,\,\,\,\,\,\,\, = Rs.\,3000 \cr & {\text{Rate}} = \left( {\frac{{100 \times 3000}}{{12500 \times 4}}} \right)\% \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = 6\% \cr} $$

$$\eqalign{ & {\text{Let}}\,{\text{the}}\,{\text{sum}}\,{\text{be}}\,{\text{Rs}}{\text{.}}\,{\text{100}}{\text{.}}\,{\text{Then,}} \cr & {\text{S}}{\text{.I}}{\text{.}}\,{\text{for}}\,{\text{first}}\,{\text{6}}\,{\text{months}} \cr & = Rs.\,\left( {\frac{{100 \times 10 \times 1}}{{100 \times 2}}} \right) \cr & = Rs.\,5 \cr & {\text{S}}{\text{.I}}{\text{.}}\,{\text{for}}\,{\text{last}}\,{\text{6}}\,{\text{months}} \cr & = Rs.\,\left( {\frac{{105 \times 10 \times 1}}{{100 \times 2}}} \right) \cr & = Rs.\,5.25 \cr} $$ So, amount at the end of 1 year = Rs. (100 + 5 + 5.25) = Rs. 110.25 ∴ Effective rate = (110.25 - 100) = 10.25%

$$\eqalign{ & {\text{Let}}\,{\text{the}}\,{\text{rate}}\,{\text{be}}\,R\% \,{\text{p}}{\text{.a}}{\text{.}}\,{\text{Then,}} \cr & {\frac{{5000 \times R \times 2}}{{100}}} + {\frac{{3000 \times R \times 4}}{{100}}} \cr & = 2200. \cr & \Rightarrow 100R + 120R = 2200 \cr & \Rightarrow R = {\frac{{2200}}{{220}}} = 10 \cr & \therefore {\text{Rate}} = 10\% \cr} $$