Compound Interest - Aptitude MCQ Questions and Solutions with Explanations

Let the sum be Rs. P and rate of interest be R% per annum. Then,
$$\eqalign{ & P{\left( {1 + \frac{R}{{100}}} \right)^2} = 9680\,.....\,\left( 1 \right) \cr & P{\left( {1 + \frac{R}{{100}}} \right)^3} = 10648\,.....\,\left( 2 \right) \cr} $$
On dividing equation (2) by (1) :
$$\eqalign{ & 1 + \frac{R}{{100}} = \frac{{10648}}{{9680}} \cr & \Rightarrow \frac{R}{{100}} = \frac{{10648}}{{9680}} - 1 \cr & \Rightarrow \frac{R}{{100}} = \frac{{10648 - 9680}}{{9680}} \cr & \Rightarrow \frac{R}{{100}} = \frac{{968}}{{9680}} \cr & \Rightarrow \frac{R}{{100}} = \frac{1}{{10}} \cr & \Rightarrow R = \frac{1}{{10}} \times 100 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = 10\% \cr} $$

The interest is compounded quarterly,
$$\therefore R = \frac{{20}}{4} = 5\% $$
Time = 3 quarters
$$\eqalign{ & \therefore C.I. = P\left[ {{{\left( {1 + \frac{R}{{100}}} \right)}^T} - 1} \right] \cr & = 16000\left[ {{{\left( {1 + \frac{5}{{100}}} \right)}^3} - 1} \right] \cr & = 16000\left[ {{{\left( {\frac{{21}}{{20}}} \right)}^3} - 1} \right] \cr & = 16000\left( {\frac{{9261 - 8000}}{{8000}}} \right) \cr & = 16000 \times \frac{{1261}}{{8000}} \cr & = {\text{Rs}}{\text{.}}\,\,2522 \cr} $$

$$\eqalign{ & {\text{Principal}} \cr & {\text{ = Rs}}{\text{.}}\left( {\frac{{100 \times 1000}}{{5 \times 4}}} \right) \cr & = {\text{Rs}}{\text{. 5}}000 \cr & {\text{Now, P = Rs}}{\text{.}}\,10000, \cr & {\text{T = 2 years,}} \cr & {\text{R = 5% }} \cr & {\text{Amount}} \cr & {\text{ = Rs}}{\text{.}}\left[ {10000 \times {{\left( {1 + \frac{5}{{100}}} \right)}^2}} \right] \cr & = {\text{Rs}}{\text{.}}\left( {10000 \times \frac{{21}}{{20}} \times \frac{{21}}{{20}}} \right) \cr & = {\text{Rs}}. 11025 \cr & \therefore {\text{C}}{\text{.I}}{\text{. = }}\left( {11025 - 10000} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}. 1025 \cr} $$

---

Details Explanation

Understanding Simple Interest:
Simple interest is calculated only on the principal amount (the initial amount of money).
The formula for simple interest is: Simple Interest = (Principal × Rate × Time) / 100
Solving for the Principal:
In this problem, we know the simple interest (Rs. 1000), the rate (5% per year), and the time (4 years). We can use the simple interest formula to find the principal:
1000 = (Principal × 5 × 4) / 100
Solving for Principal, we get: Principal = Rs. 5000
Understanding Compound Interest:
Compound interest is calculated on the principal amount plus accumulated interest from previous periods. It grows faster than simple interest.
The formula for compound interest is a bit more complex, but we can break it down step-by-step.
Calculating Compound Interest:
The question asks for the compound interest on twice the principal (2 × Rs. 5000 = Rs. 10000) for 2 years at 5% interest.
Year 1: Interest = (10000 × 5 × 1) / 100 = Rs. 500
Amount at the end of Year 1 = 10000 + 500 = Rs. 10500
Year 2: Interest = (10500 × 5 × 1) / 100 = Rs. 525
Total compound interest over 2 years = 500 + 525 = Rs. 1025
Therefore, the correct answer is B: Rs. 1025 (Note: There's a discrepancy in the provided options. The calculation leads to Rs. 1025, which is not listed.)

$$\eqalign{ & {\text{S}}{\text{.I}}{\text{. = Rs}}{\text{.}}\left( {\frac{{P \times R \times 2}}{{100}}} \right) \cr & \,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\left( {\frac{{2PR}}{{100}}} \right) \cr & {\text{C}}{\text{.I}}{\text{. = Rs}}{\text{.}}\left[ {P \times {{\left( {1 + \frac{R}{{100}}} \right)}^2} - P} \right] \cr & \,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\left[ {\frac{{P{R^2}}}{{{{\left( {100} \right)}^2}}} + \frac{{2PR}}{{100}}} \right] \cr & \therefore {\text{Difference}} \cr & {\text{ = Rs}}{\text{.}}\left[ {\left\{ {\frac{{P{R^2}}}{{{{\left( {100} \right)}^2}}} + \frac{{2PR}}{{100}}} \right\} - \frac{{2PR}}{{100}}} \right] \cr & = {\text{Rs}}{\text{.}}\left[ {\frac{{P{R^2}}}{{{{\left( {100} \right)}^2}}}} \right] \cr} $$

Time (t) = 2 years
Rate % = 4%
Effective rate of CI of 2 years
$$\eqalign{ & {\text{ = 4 + 4 + }}\frac{{4 \times 4}}{{100}} \cr & = 8.16\% \cr} $$
Effective Rate of SI for 2 years = 8%
According to the question
$$\eqalign{ & {\text{8}}{\text{.16% of sum}} \cr & {\text{ = Rs. 2448}} \cr & {\text{1% of sum}} \cr & {\text{ = Rs. }}\frac{{2448}}{{8.16}} \cr & {\text{8% of sum}} \cr & {\text{ = }}\frac{{2448}}{{8.16}} \times {\text{8}} \cr & {\text{ = Rs. 2400 }} \cr} $$

$$\eqalign{ & {\text{Principal (}}{{\text{P}}_1}{\text{) = Rs. 6000}} \cr & {\text{Time (t) = 2 years}} \cr & {\text{Rate % = 5% }} \cr & {\text{Simple interest}} \cr & {\text{ = }}\frac{{6000 \times 5 \times 2}}{{100}}{\text{ = Rs. 600}} \cr & {\text{Principal (}}{{\text{P}}_2}{\text{) = Rs. 5000}} \cr & {\text{Time (t) = 2 years}} \cr & {\text{Rate % = 8% }} \cr} $$
2 year effective rate for Compound interest
$$\eqalign{ & = 8 + 8 + \frac{{8 \times 8}}{{100}} = 16.64\% \cr & {\text{Compound}}\,{\text{Interest}} \cr & {\text{ = 5000}} \times \frac{{16.64}}{{100}} = {\text{Rs}}{\text{. 832}} \cr & {\text{Difference}} \cr & {\text{ = Rs}}{\text{. }}\left( {832 - 600} \right) \cr & = {\text{Rs}}{\text{.}}\,232 \cr & {\text{ }} \cr} $$

$$\eqalign{ & {\text{S}}{\text{.I}}{\text{. = Rs}}{\text{.}}\left( {\frac{{19200 \times 12 \times 3}}{{100}}} \right) \cr & \,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.6912}} \cr & {\text{C}}{\text{.I}}{\text{. = Rs}}{\text{.}}\left[ {19200 \times {{\left( {1 + \frac{{12}}{{100}}} \right)}^3} - 19200} \right] \cr & = {\text{Rs}}{\text{.}}\left[ {\left( {19200 \times \frac{{28}}{{25}} \times \frac{{28}}{{25}} \times \frac{{28}}{{25}}} \right) - 19200} \right] \cr & = {\text{Rs}}{\text{. }}\left( {\frac{{16859136}}{{625}} - 19200} \right) \cr & = {\text{Rs}}{\text{. }}\left( {26974.6176 - 19200} \right) \cr & = {\text{Rs}}{\text{. 7774}}{\text{.6176}} \cr & \therefore {\text{Difference }} \cr & {\text{ = Rs}}{\text{.}}\left( {7774.6176 - 6912} \right) \cr & = {\text{Rs}}{\text{. 862}}{\text{.6176}} \cr} $$

$$\eqalign{ & {\text{S}}{\text{.I}}{\text{. = Rs}}{\text{. }}\left( {15000 \times \frac{{25}}{2} \times 2 \times \frac{1}{{100}}} \right) \cr & \,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. }}3750 \cr & {\text{C}}{\text{.I}}{\text{. = Rs}}{\text{. }}\left[ {15000{{\left( {1 + \frac{{25}}{{2 \times 100}}} \right)}^2} - 15000} \right] \cr & = {\text{Rs}}{\text{. }}\left( {15000 \times \frac{9}{8} \times \frac{9}{8} - 15000} \right) \cr & = {\text{Rs}}{\text{. }}\left( {18948.375 - 15000} \right) \cr & = {\text{Rs}}{\text{. }}3984.375 \cr & {\text{Difference }} \cr & {\text{ = Rs}}{\text{. }}\left( {3984.375 - 3750} \right) \cr & = {\text{Rs}}{\text{. }}234.375 \cr} $$

Let the principal be x Rs. Now,
$$\eqalign{ & C.I. = P\left[ {{{\left( {1 + \frac{R}{{100}}} \right)}^T} - 1} \right] \cr & \Rightarrow 1261 = x\left[ {{{\left( {1 + \frac{5}{{100}}} \right)}^3} - 1} \right] \cr & \Rightarrow 1261 = x\left( {\frac{{9261}}{{8000}} - 1} \right) \cr & \Rightarrow 1261 = x\left( {\frac{{9261 - 8000}}{{8000}}} \right) \cr & \Rightarrow 1261 = \frac{{1261x}}{{8000}} \cr & \Rightarrow x = \frac{{1261 \times 8000}}{{1261}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,8000 \cr} $$

Time (t) = 1 years
Rate % = 4%
Amount = Rs. 7803
When interest is compounded half yearly
New Rate = $$\frac{4}{2}$$ = 2%
Time = 1 × 2 = 2 years
Required rate% for 2 years CI
$${\text{ = 2}} + {\text{2}} + \frac{{2 \times 2}}{{100}} = 4.04\% $$
According to question,
(100 + 4.04)% of sum = Rs. 7803
$$\eqalign{ & \therefore {\text{Sum = }}\frac{{7803}}{{104.04}} \times 100 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. }}7500 \cr} $$