Interest - Aptitude MCQ Questions and Solutions 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 r₁ and r₂ 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%

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

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