Interest - Aptitude MCQ Questions and Solutions with Explanations

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

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

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

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
$$\eqalign{ & P{\left( {\frac{6}{5}} \right)^3}\left( {\frac{1}{5}} \right) - \frac{P}{5} = 7280 \cr & \frac{P}{5}\left[ {{{\left( {\frac{6}{5}} \right)}^3} - 1} \right] = 7280 \cr & \frac{P}{5}\left[ {\frac{{216}}{{125}} - 1} \right] = 7280 \cr & \frac{P}{5} \times \frac{{91}}{{125}} = 7280 \cr & \therefore P = 50000 \cr} $$