1. At what rate percent per annum of compound interest, will a sum of money become four times of itself in two years ?
a) 100%
b) 75%
c) 50%
d) 20%
Explanation:
$$\eqalign{ & {\text{Principal}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Amount}} \cr & \,\,\,\,\,\,\,\,\,{\text{1}}\,\,\,\,\,\,\,\,\,\,\,\, \to \,\,\,\,\,\,\,\,\,\,\,\,{\text{4}} \cr & 4 = 1{\left( {1 + \frac{r}{{100}}} \right)^2} \cr & 4 = {\left( {1 + \frac{r}{{100}}} \right)^2} \cr & r = 100\% \cr} $$
2. What will be the difference between the simple interest and compound interest accrued on an amount of Rs. 19200 of 3 years @ 12 p.c.p.a. ?
a) Rs. 722.6826
b) Rs. 798.1824
c) Rs. 802.5144
d) Rs. 862.6176
Explanation:
$$\eqalign{ & {\text{S}}{\text{.I}}{\text{. = Rs}}{\text{.}}\left( {\frac{{19200 \times 12 \times 3}}{{100}}} \right) \cr & \,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.6912}} \cr & {\text{C}}{\text{.I}}{\text{. = Rs}}{\text{.}}\left[ {19200 \times {{\left( {1 + \frac{{12}}{{100}}} \right)}^3} - 19200} \right] \cr & = {\text{Rs}}{\text{.}}\left[ {\left( {19200 \times \frac{{28}}{{25}} \times \frac{{28}}{{25}} \times \frac{{28}}{{25}}} \right) - 19200} \right] \cr & = {\text{Rs}}{\text{. }}\left( {\frac{{16859136}}{{625}} - 19200} \right) \cr & = {\text{Rs}}{\text{. }}\left( {26974.6176 - 19200} \right) \cr & = {\text{Rs}}{\text{. 7774}}{\text{.6176}} \cr & {\text{Difference }} {\text{ = Rs}}{\text{.}}\left( {7774.6176 - 6912} \right) \cr & = {\text{Rs}}{\text{. 862}}{\text{.6176}} \cr} $$
3. On a certain sum of money, the difference between the compound interest for a year payable half yearly, and the simple interest for a year is Rs. 180. If the rate of interest in both the cases is 10%, then the sum is = ?
a) Rs. 60000
b) Rs. 72000
c) Rs. 62000
d) Rs. 54000
Explanation: Rate % = 10%,
Time = 1 year
Case (I) : When interest is calculated yearly, Rate = 10%
Case (II) : When interest is calculated half yearly
$$\eqalign{ & \Rightarrow {\text{New rate }}\% = \frac{{10}}{2} = 5\% \cr & {\text{Time = 1}} \times {\text{2}} = {\text{2 years}} \cr & {\text{Effective rate}}\% \cr & {\text{ = 5 + 5 + }}\frac{{5 \times 5}}{{100}} = 10.25\% \cr & {\text{Difference in rates}} \cr & {\text{ = }}\left( {10.25 - 10} \right)\% = 0.25\% \cr & {\text{0}}{\text{.25% of sum = Rs 180}} \cr & {\text{Sum = }}\frac{{180}}{{0.25}} \times 100 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs. }}72000 \cr} $$
4. The compound interest accrued on an amount of Rs. 25500 at the end of 3 years is Rs. 8440.50. What would be the simple interest accrued on the same amount at the same rate in the same period ?
a) Rs. 4650
b) Rs. 5650
c) Rs. 6650
d) Rs. 7650
Explanation:
$$\eqalign{ & {\text{Let the rate be R}}\% {\text{ p}}{\text{.a}}{\text{. }} \cr & {\text{25500}}{\left( {1 + \frac{{\text{R}}}{{100}}} \right)^3} \cr & = 25500 + 8440.50 \cr & = 33940.50 \cr} $$
$$ \Rightarrow {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^3} = \frac{{33940.50}}{{25500}} = $$ $$\frac{{1331}}{{1000}} = $$ $${\left( {\frac{{11}}{{10}}} \right)^3}$$
$$\eqalign{ & 1 + \frac{{\text{R}}}{{100}} = \frac{{11}}{{10}} \cr & \frac{{\text{R}}}{{100}} = \frac{1}{{10}} \cr & {\text{R}} = 10\,\% \cr & S.I. = {\text{R}}s.\left( {\frac{{25500 \times 10 \times 3}}{{100}}} \right) \cr & = {\text{Rs}}{\text{.}}\,7650 \cr} $$
5. The difference between the amount of compound interest and simple interest accrued on an amount of Rs. 26000 at the end of 3 years is Rs. 2994.134. What is the rate of interest p.c.p.a ?
a) 17%
b) 19%
c) 22%
d) Cannot be determined
Explanation:
Let the R% p.a.
$$\left[ {26000 \times {{\left( {1 + \frac{{\text{R}}}{{100}}} \right)}^3} - 26000} \right] - $$ $$\left( {\frac{{26000 \times {\text{R}} \times 3}}{{100}}} \right) = $$ $$2994.134$$
$$ 26000\left[ {{{\left( {1 + \frac{{\text{R}}}{{100}}} \right)}^3} - 1 - \frac{{3{\text{R}}}}{{100}}} \right] = $$ $$2994.134$$
$$ 26000$$ $$\left[ {\frac{{{{\left( {100 - {\text{R}}} \right)}^3} - 1000000 - 30000{\text{R}}}}{{1000000}}} \right] = $$ $$2994.134$$
$$ 26\left[ {\left\{ {1000000 + {{\text{R}}^3} + 300{\text{R}}\left( {100 + {\text{R}}} \right) - 1000000 - 30000{\text{R}}} \right\}} \right] = 2994134$$
$$ {{\text{R}}^3} + 300{{\text{R}}^2} = \frac{{2994134}}{{26}} = $$ $$115159$$
$$ {{\text{R}}^2}\left( {{\text{R}} + 300} \right) = 115159$$
$$ {\text{R = 19}}\% $$
6. The simple interest on a sum of money at 4% per annum for 2 years is Rs 80. The compound interest on the same sum for the same period is = ?
a) Rs. 82.60
b) Rs. 82.20
c) Rs. 81.80
d) Rs. 81.60
Explanation:
$$\eqalign{ & {\text{Rate }}\% {\text{ = 4}}\% \cr & {\text{Time (}}{{\text{t}}_1}) = 2\,{\text{years}} \cr & {\text{SI for 2 years}} \cr & {\text{ = 4}} \times {\text{2 = 8}}\% \cr & {\text{CI for 2 years}} \cr & {\text{ = 4 + 4 + }}\frac{{4 \times 4}}{{100}} \cr & = 8.16\% \cr & \operatorname{Required} \,CI = \frac{{80}}{8} \times 8.16 \cr & = Rs.\,81.60 \cr} $$
7. The compound interest on Rs. 30000 at 7% per annum is Rs. 4347. The period (in years) is = ?
a) 2 years
b) $${\text{2}}\frac{1}{2}$$ years
c) 3 years
d) 4 years
Explanation:
$$\eqalign{ & {\text{Amount = Rs}}{\text{. }}\left( {30000 - 4347} \right) \cr & {\text{Amount = Rs}}{\text{. }} 34347 \cr & {\text{Let the time be }}n{\text{ years}} \cr & {\text{30000}}{\left( {1 + \frac{7}{{100}}} \right)^n} = 34347 \cr & {\left( {\frac{{107}}{{100}}} \right)^n} = \frac{{34347}}{{30000}} \cr & {\left( {\frac{{107}}{{100}}} \right)^n} = \frac{{11449}}{{10000}} = {\left( {\frac{{107}}{{100}}} \right)^2} \cr & n = {\text{ 2 years}} \cr} $$
8. The compound interest on a certain sum of money at 5% per annum for 2 years is Rs 246. The simple interest on the same sum for 3 years at 6% per annum is = ?
a) Rs. 435
b) Rs. 450
c) Rs. 430
d) Rs. 432
Explanation:
$$\eqalign{ & {\text{Effective rate of CI for 2 years}} \cr & {\text{= 5 + 5 + }}\frac{{5 \times 5}}{{100}} \cr & = 10.25\% \cr & {\text{Effective rate of SI for 3 years}} \cr & {\text{= 6}} \times {\text{3 = 18% }} \cr & {\text{Required SI}} {\text{= }}\frac{{246}}{{10.25}} \times 18 \cr & = {\text{Rs. 432}} \cr} $$
9. The difference between compound and simple interest on a certain sum for 3 years at 5% per annum is Rs. 122. The sum is = ?
a) Rs. 16000
b) Rs. 15000
c) Rs. 12000
d) Rs. 10000
Explanation:
$$\eqalign{ & P\left[ {{{\left( {\frac{{21}}{{20}}} \right)}^3} - 1 - \frac{3}{{20}}} \right] = 122 \cr & P\left[ {\frac{{{{21}^3} - {{20}^3} - 3 \times {{20}^2}}}{{{{20}^3}}}} \right] = 122 \cr & P\left[ {\frac{{9261 - 8000 - 1200}}{{8000}}} \right] = 122 \cr & P \times \frac{{61}}{{8000}} = 122 \cr & P = \frac{{8000 \times 122}}{{61}} \cr & P = {\text{Rs}}{\text{.}}\,16000 \cr} $$
10. Rs.2000 amounts to Rs. 2226.05 in 2 years at compound interest. What will be the rate of interest ?
a) 5%
b) 5.25%
c) 5.5%
d) 6%
Explanation:
$$\eqalign{ & {\text{Let the rate be R}}\% {\text{ p}}{\text{.a}}{\text{.}} \cr & {\text{2000}}{\left( {1 + \frac{{\text{R}}}{{100}}} \right)^2} = 2226.05 \cr & {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^2} = \frac{{222605}}{{200000}} \cr & {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^2} = \frac{{44521}}{{40000}} \cr & {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^2} = {\left( {\frac{{221}}{{200}}} \right)^2} \cr & 1 + \frac{{\text{R}}}{{100}} = \frac{{211}}{{200}} \cr & \frac{{\text{R}}}{{100}} = \frac{{11}}{{200}} \cr & {\text{R}} = \frac{{11}}{2}\% \cr & {\text{R}} = 5.5\% \cr} $$