1. A sum becomes Rs.1352 in 2 years at 4% per annum compound interest. The sum is =
a) Rs. 1270
b) Rs. 1225
c) Rs. 1245
d) Rs. 1250
Discussion
Explanation: Let the sum be Rs. x
$$\eqalign{ & \therefore 1352 = x{\left( {1 + \frac{4}{{100}}} \right)^2} \cr & \Rightarrow 1352 = x{\left( {1 + \frac{1}{{25}}} \right)^2} \cr & \Rightarrow 1352 = x{\left( {\frac{{26}}{{25}}} \right)^2} \cr & \Rightarrow x = \frac{{1352 \times 25 \times 25}}{{26 \times 26}} \cr & \Rightarrow x = {\text{Rs}}{\text{.}}\,1250 \cr} $$
2. What will be the compound interest accrued on an amount of Rs.10000 @ 20 p.c.p.a in 2 years if the interest is compounded half-yearly?
a) Rs. 4600
b) Rs. 4641
c) Rs. 4400
d) Rs. 4680
Discussion
Explanation:
$$\eqalign{ & {\text{P = Rs}}.10000, \cr & {\text{R}} = 20\% \,p.a. \cr & \,\,\,\,\,\,\, = 10\% \,{\text{per}}\,{\text{half year}} \cr & T = 2\,{\text{years}} = 4\,{\text{half}}\,{\text{years}} \cr & {\text{Amount}} \cr & {\text{ = Rs}}.\left[ {10000 \times {{\left( {1 + \frac{{10}}{{100}}} \right)}^4}} \right] \cr & = {\text{Rs}}.\left( {10000 \times \frac{{11}}{{10}} \times \frac{{11}}{{10}} \times \frac{{11}}{{10}} \times \frac{{11}}{{10}}} \right) \cr & = {\text{Rs}}.14641 \cr & \therefore {\text{C}}{\text{.I}}{\text{. = Rs}}.\left( {14641 - 10000} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}.\,4641 \cr} $$
3. A sum of money on compound interest amounts to Rs. 10648 in 3 years and Rs. 9680 in 2 years. The rate of interest per annum is = ?
a) 15%
b) 10%
c) 20%
d) 5%
Discussion
Explanation: 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) :
$$\eqalign{ & 1 + \frac{R}{{100}} = \frac{{10648}}{{9680}} \cr & \Rightarrow \frac{R}{{100}} = \frac{{10648}}{{9680}} - 1 \cr & \Rightarrow \frac{R}{{100}} = \frac{{10648 - 9680}}{{9680}} \cr & \Rightarrow \frac{R}{{100}} = \frac{{968}}{{9680}} \cr & \Rightarrow \frac{R}{{100}} = \frac{1}{{10}} \cr & \Rightarrow R = \frac{1}{{10}} \times 100 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = 10\% \cr} $$
4. The principal which will amounts to Rs. 270.40 in 2 years at the rate of 4% per annum compound interest, is = ?
a) Rs. 250
b) Rs. 225
c) Rs. 200
d) Rs. 220
Discussion
Explanation:
$$\eqalign{ & 4\% = \frac{1}{{25}} \cr & \,\,\,\,\,\,\,\,\, = \frac{{26 \to {\text{Amount}}}}{{25 \to {\text{Principal}}}} \cr & {\text{Time = 2 years}} \cr & {\text{Principal}}\,\,\,\,\,{\text{Amount}} \cr & \,\,\,\,\,\,\,\,\,{\text{25}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{26}} \cr & \,\,\,\,\,\,\,\,\,{\text{25}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{26}} \cr & \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \cr & \,\,\,\,\,\,\,\,625\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,676 \cr & \,\,\,\, \downarrow \times 0.4\,\,\,\,\,\, \downarrow \times 0.4 \cr & \,\,\,\,\,\,\,\,\,250\,\,\,\,\,\,\,\,\,\,\,\,270.40 \cr & {\text{Hence required principal}} \cr & {\text{ = Rs.250}} \cr} $$
5. The compound interest on a certain sum of money at a certain rate for 2 years is Rs. 40.80 and the simple interest on the same sum is Rs. 40 at the same rate and for the same time. The rate of interest is = ?
a) 5% per annum
b) 3% per annum
c) 4% per annum
d) 2% per annum
Discussion
Explanation: Difference in CI and SI for 2 years
$$\eqalign{ & = \left( {40.80 - 40} \right) \cr & = {\text{Rs 0}}{\text{.80}} \cr & {\text{SI for first year }} \cr & {\text{ = }}\frac{{40}}{2} = {\text{Rs}}{\text{.}}\,20 \cr & {\text{Required Rate }}\% \cr & {\text{ = }}\frac{{0.80}}{{20}} \times 100 = 4\% \cr} $$
6. In how many years will Rs. 2000 amounts to Rs. 2420 at 10% per annum compound interest?
a) 3 years
b) $$2\frac{1}{2}$$ years
c) $$1\frac{1}{2}$$ years
d) 2 years
Discussion
Explanation:
$$\eqalign{ & {\text{Principal = Rs. 2000}} \cr & {\text{Amount = Rs. 2420}} \cr & {\text{Rate = 10% }} \cr & {\text{By using formula,}} \cr & \Rightarrow 2420 = 2000{\left( {1 + \frac{{10}}{{100}}} \right)^n} \cr & \Rightarrow \frac{{2420}}{{2000}} = {\left( {1 + \frac{{10}}{{100}}} \right)^n} \cr & \Rightarrow \frac{{121}}{{100}} = {\left( {\frac{{11}}{{10}}} \right)^n} \cr & \Rightarrow {\left( {\frac{{11}}{{10}}} \right)^2} = {\left( {\frac{{11}}{{10}}} \right)^n} \cr & \Rightarrow n = 2 \cr & {\text{Hence,}} \cr & {\text{required time = 2 years}} \cr} $$
7. The compound interest on Rs.2800 for 18 months at 10% p.a is = ?
a) Rs. 441.35
b) Rs. 436.75
c) Rs. 434
d) Rs. 420
Discussion
Explanation:
$$\eqalign{ & {\text{Given,}}\,{\text{Principal}},\,P = Rs.\,2800 \cr & {\text{Compound}}\,{\text{rate}},\,R = 10\% \,{\text{per}}\,{\text{annum}} \cr & = \frac{{10}}{2} = 5\% \,{\text{half - yearly}} \cr & {\text{Amount}} \cr & = {\text{Rs}}{\text{.}}\left[ {2800 \times \left( {1 + \frac{{10}}{{100}}} \right)\left( {1 + \frac{{5}}{{100}}} \right)} \right] \cr & = {\text{Rs.}}\left( {2800 \times \frac{{11}}{{10}} \times \frac{{21}}{{20}}} \right) \cr & = {\text{Rs. }}3234 \cr & \therefore {\text{C}}{\text{.I}}{\text{. = Rs.}}\left( {3234 - 2800} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs. }}434 \cr} $$
8. If the rate of interest be 4% per annum for first year, 5% per annum for second year and 6% per annum for third year, then the compound interest of Rs.10000 for three years will be ?
a) Rs. 1575.20
b) Rs. 1625.80
c) Rs. 1600
d) Rs. 2000
Discussion
Explanation:
$$\eqalign{ & {\text{ = Rs}}.10000\left[ {\left( {1 + \frac{4}{{100}}} \right)\left( {1 + \frac{5}{{100}}} \right)\left( {1 + \frac{6}{{100}}} \right)} \right] \cr & = {\text{Rs}}.\left( {10000 \times \frac{{26}}{{25}} \times \frac{{21}}{{20}} \times \frac{{53}}{{50}}} \right) \cr & = {\text{Rs}}.\left( {\frac{{57876}}{5}} \right) = {\text{Rs}}.11575.20 \cr & {\text{C}}{\text{.I}}{\text{. = Rs}}{\text{.}}\left( {11575.20 - 10000} \right) \cr & \,\,\,\,\,\,\,\,\, = {\text{Rs}}.1575.20 \cr} $$
9. A bank offers 5% compound interest calculated on half yearly basis. A customer deposits Rs.1600 each on 1st January and 1st July of a year. At the end of the year, the amount he would have gained by way of interest is = ?
a) Rs. 120
b) Rs. 121
c) Rs. 122
d) Rs. 123
Discussion
Explanation:
$$\eqalign{ & {\text{ = Rs}}.\left[ {1600 \times {{\left( {1 + \frac{5}{{2 \times 100}}} \right)}^2} + 1600 \times \left( {1 + \frac{5}{{2 \times 100}}} \right)} \right] \cr & {\text{ = Rs}}.\left[ {1600 \times \frac{{41}}{{40}} \times \frac{{41}}{{40}} + 1600 \times \frac{{41}}{{40}}} \right] \cr & {\text{ = Rs}}.\left[ {1600 \times \frac{{41}}{{40}}\left( {\frac{{41}}{{40}} + 1} \right)} \right] \cr & {\text{ = Rs}}.\left( {\frac{{1600 \times 41 \times 81}}{{40 \times 40}}} \right) \cr & {\text{ = Rs}}.\,3321 \cr & \therefore {\text{C}}{\text{.I}}{\text{. = Rs}}.\left( {3321 - 3200} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}.\,121 \cr} $$
10. The compound interest on Rs 16000 for 9 months at 20% per annum, interest being compounded quarterly, is = ?
a) Rs. 2520
b) Rs. 2518
c) Rs. 2522
d) Rs. 2524
Discussion
Explanation:
The interest is compounded quarterly,
$$\therefore R = \frac{{20}}{4} = 5\% $$
Time = 3 quarters
$$\eqalign{ & \therefore C.I. = P\left[ {{{\left( {1 + \frac{R}{{100}}} \right)}^T} - 1} \right] \cr & = 16000\left[ {{{\left( {1 + \frac{5}{{100}}} \right)}^3} - 1} \right] \cr & = 16000\left[ {{{\left( {\frac{{21}}{{20}}} \right)}^3} - 1} \right] \cr & = 16000\left( {\frac{{9261 - 8000}}{{8000}}} \right) \cr & = 16000 \times \frac{{1261}}{{8000}} \cr & = {\text{Rs}}{\text{.}}\,\,2522 \cr} $$
11. A sum of money invested at compound interest amounts to Rs. 650 at the end of first year and Rs. 676 at the end of second year. The sum of money is =
a) Rs. 540
b) Rs. 560
c) Rs. 625
d) Rs. 600
Discussion
Explanation: Interest on 650 for one year = 676 - 650 = 26
$$\eqalign{ & 26 = \frac{{650 \times r \times 1}}{{100}} \cr & r = 4\% \cr & 650 = P\left[ {1 + \frac{4}{{100}}} \right] \cr & \Rightarrow 650 = P \times \frac{{26}}{{25}} \cr & \Rightarrow p = \frac{{650 \times 25}}{{26}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\,625 \cr} $$
12. On a certain sum of money the compound interest for 2 years is Rs. 282.15 and the simple interest for the same period of time is Rs. 270. The rate of interest per annum is =
a) 6.07%
b) 9%
c) 10%
d) 12.15%
Discussion
Explanation:
$$\eqalign{ & {\text{CI for 2 years}}\,{\text{ = Rs. 282}}{\text{.15}} \cr & {\text{SI for 2 year}}\,{\text{ = Rs. 270}} \cr & {\text{SI for 1 year = }}\frac{{270}}{2} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs.}}\,135 \cr & {\text{Difference between CI and SI}} \cr & {\text{ = }}\left( {282.15 - 270} \right) \cr & {\text{ = Rs. 12}}{\text{.15}} \cr & {\text{Required rate % }} \cr & {\text{ = }}\frac{{12.15}}{{135}} \times 100 = 9\% \cr} $$
13. If the difference between the compound interest and simple interest on a sum of 5% rate of interest per annum for three years is Rs. 36.60, then the sum is = ?
a) Rs. 8000
b) Rs. 4400
c) Rs. 8400
d) Rs. 4800
Discussion
Explanation:
Rate % = 5%
Effective Rate of CI for 3 years = 15.7625%
Effective Rate of SI for 3 years = 5 × 3 = 15%
According to the question
$$\eqalign{ & \left( {15.7625 - 15} \right)\% \,{\text{of sum}} {\text{ = Rs. 36}}{\text{.60}} \cr & {\text{0}}{\text{.7625% of sum}} {\text{ = Rs. 36}}{\text{.60}} \cr & {\text{Sum = }}\frac{{36.60}}{{0.7625}} \times 100 \cr & \,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs. }}4800 \cr} $$
14. Compound interest on a sum of money for 2 years at 4% per annum is Rs. 2448. simple interest on the same sum of money at the same rate of interest for 2 years will be = ?
a) Rs. 2360
b) Rs. 2400
c) Rs. 2250
d) Rs. 2500
Discussion
Explanation:
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
$$\eqalign{ & {\text{8}}{\text{.16% of sum}} \cr & {\text{ = Rs. 2448}} \cr & {\text{1% of sum}} \cr & {\text{ = Rs. }}\frac{{2448}}{{8.16}} \cr & {\text{8% of sum}} \cr & {\text{ = }}\frac{{2448}}{{8.16}} \times {\text{8}} \cr & {\text{ = Rs. 2400 }} \cr} $$
15. A man gets a simple interest on Rs. 1000 on a certain principal at the rate of 5 p.c.p.a. in 4 years. What compound interest will the man get on twice the principal in 2 years at the same rate ?
a) Rs. 1005
b) Rs. 1000
c) Rs. 10125
d) None of the above
Discussion
Explanation:
$$\eqalign{ & {\text{Principal}} \cr & {\text{ = Rs}}{\text{.}}\left( {\frac{{100 \times 1000}}{{5 \times 4}}} \right) \cr & = {\text{Rs}}{\text{. 5}}000 \cr & {\text{Now, P = Rs}}{\text{.}}\,10000, \cr & {\text{T = 2 years,}} \cr & {\text{R = 5% }} \cr & {\text{Amount}} \cr & {\text{ = Rs}}{\text{.}}\left[ {10000 \times {{\left( {1 + \frac{5}{{100}}} \right)}^2}} \right] \cr & = {\text{Rs}}{\text{.}}\left( {10000 \times \frac{{21}}{{20}} \times \frac{{21}}{{20}}} \right) \cr & = {\text{Rs}}. 11025 \cr & \therefore {\text{C}}{\text{.I}}{\text{. = }}\left( {11025 - 10000} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}. 1025 \cr} $$
16. Mr. Duggal invested Rs. 20000 with rate of interest @ 20 p.c.p.a. The interest was compounded half-yearly for first one year and in the next year it was compounded yearly. What will be the total interest earned at the end of 2 year ?
a) Rs. 9040
b) Rs. 8800
c) Rs. 9800
d) Rs. 8040
Discussion
Explanation:
$$\eqalign{ & {\text{Amount}} \cr & {\text{ = Rs}}.\left[ {20000{{\left( {1 + \frac{{10}}{{100}}} \right)}^2}\left( {1 + \frac{{20}}{{100}}} \right)} \right] \cr & = {\text{Rs}}.\left( {20000 \times \frac{{11}}{{10}} \times \frac{{11}}{{10}} \times \frac{6}{5}} \right) \cr & = {\text{Rs}}.29040 \cr & {\text{C}}{\text{.I}}{\text{. = Rs}}.\left( {29040 - 20000} \right) \cr & \,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}. 9040 \cr} $$
17. A sum of money doubles itself in 4 years compound interest. It will amount to 8 times itself at the same rate of interest in = ?
a) 24 years
b) 16 years
c) 12 years
d) 18 years
Discussion
Explanation:
$$\eqalign{ & {\text{Let Principal = P}} \cr & {\text{Rate = R% }} \cr & {\text{T = 4 years}} \cr & \therefore {\text{Amount = 2P}} \cr & {\text{Case (I) 2P = P}}{\left( {1 + \frac{R}{{100}}} \right)^4} \cr & 2 = {\left( {1 + \frac{R}{{100}}} \right)^4}.....(i) \cr & {\text{Case (II) Let after t years it will be 8 times}} \cr & {\text{8P = P}}{\left( {1 + \frac{R}{{100}}} \right)^t} \cr & {\left( 2 \right)^3} = {\left( {1 + \frac{R}{{100}}} \right)^t}.....(ii) \cr & {\text{By using equation (i) & equation (ii)}} \cr & {\left( {1 + \frac{R}{{100}}} \right)^{12}} = {\left( {1 + \frac{R}{{100}}} \right)^t} \cr & {\text{By comparing both sides,}} \cr & {\text{t = 12 years}} \cr} $$
18. If the compound interest on a sum of money for 3 years at the rate of 5% per annum is Rs. 252.20, the simple interest on the same sum at the same rate and for the same time is ?
a) Rs. 240
b) Rs. 245
c) Rs. 220
d) Rs. 250
Discussion
Explanation:
$$\eqalign{ & {\text{Rate = 5% }} \cr & {\text{Time = 3 years}} \cr & {\text{Compound Interest Rs. 252}}{\text{.20}} \cr & {\text{Effective rate% of CI for 3 years}} \cr & {\text{ = 15}}{\text{.7625% }} \cr & {\text{Effective rate% of SI for 3 years}} \cr & {\text{ = 5}} \times {\text{3 = 15% }} \cr & {\text{Required SI }} \cr & {\text{ = }}\frac{{252.20}}{{15.7625}} \times 15 \cr & = 240 \cr} $$
19. The difference between simple interest and compound interest on Rs. P at R% p.a in 2 years is = ?
a) $${\text{Rs}}{\text{.}}\,\frac{{P{R^2}}}{{100}}$$
b) $${\text{Rs}}{\text{.}}\,\frac{{2PR}}{{100}}$$
c) $${\text{Rs}}{\text{.}}\,\frac{{PR}}{{100}}$$
d) $${\text{Rs}}{\text{.}}\,\frac{{P{R^2}}}{{{{\left( {100} \right)}^2}}}$$
Discussion
Explanation:
$$\eqalign{ & {\text{S}}{\text{.I}}{\text{. = Rs}}{\text{.}}\left( {\frac{{P \times R \times 2}}{{100}}} \right) \cr & \,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\left( {\frac{{2PR}}{{100}}} \right) \cr & {\text{C}}{\text{.I}}{\text{. = Rs}}{\text{.}}\left[ {P \times {{\left( {1 + \frac{R}{{100}}} \right)}^2} - P} \right] \cr & \,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.}}\left[ {\frac{{P{R^2}}}{{{{\left( {100} \right)}^2}}} + \frac{{2PR}}{{100}}} \right] \cr & \therefore {\text{Difference}} \cr & {\text{ = Rs}}{\text{.}}\left[ {\left\{ {\frac{{P{R^2}}}{{{{\left( {100} \right)}^2}}} + \frac{{2PR}}{{100}}} \right\} - \frac{{2PR}}{{100}}} \right] \cr & = {\text{Rs}}{\text{.}}\left[ {\frac{{P{R^2}}}{{{{\left( {100} \right)}^2}}}} \right] \cr} $$
20. What would be the compound interest accrued on an amount of Rs. 8400 @ 12.5 p.c.p.a at the end of 3 years ?
a) Rs. 2584.16
b) Rs. 3820.14
c) Rs. 3560.16
d) Rs. 4205.62
Discussion
Explanation:
$$\eqalign{ & {\text{Amount}} \cr & {\text{ = Rs}}{\text{.}}\left[ {8400 \times {{\left( {1 + \frac{{25}}{{2 \times 100}}} \right)}^3}} \right] \cr & = {\text{Rs}}{\text{.}}\left( {8400 \times \frac{9}{8} \times \frac{9}{8} \times \frac{9}{8}} \right) \cr & = {\text{Rs}}{\text{.}}\left( {\frac{{382725}}{{32}}} \right) \cr & = {\text{Rs}}.11960.156 \approx {\text{Rs}}.11960.16 \cr & {\text{C}}{\text{.I}}{\text{.}} = {\text{Rs}}{\text{.}}\left( {11960.16 - 8400} \right) \cr & = {\text{Rs}}{\text{.}}\,3560.16 \cr} $$
21. What does Rs. 250 amounts to in 2 years with compound interest at the rate of 4% in the 1st year and 8% in the second year ?
a) Rs. 280
b) Rs. 280.80
c) Rs. 468
d) Rs. 290.80
Discussion
Explanation:
$$\eqalign{ & {\text{Principal = Rs 250}} \cr & {{\text{R}}_1} = 4\% ,\,\,\,\,\,\,\,\,\,{{\text{R}}_2} = 8\% \cr & {\text{Amount}}\,{\text{after}}{1^{st}}\,{\text{year}} \cr & = 250\left( {1 + \frac{4}{{100}}} \right) = {\text{Rs}}{\text{. }}260 \cr & {\text{Amount after }}{{\text{2}}^{{\text{nd}}}}{\text{ year}} \cr & {\text{ = }}260\left( {1 + \frac{8}{{100}}} \right) \cr & = {\text{Rs}}{\text{. }}280.80 \cr} $$
22. The compound interest on a certain sum of money for 2 years at 5% is Rs. 328, then the sum is =
a) Rs. 3000
b) Rs. 3600
c) Rs. 3200
d) Rs. 3400
Discussion
Explanation: Go check with option one by one (Go with option (C) and check it.)
Principal is Rs. 3200
3200 of 5% for 1st year = 160
then, principal = 3200 + 160 = 3360
3360 of 5% for 2nd year = 168
Interest = 160 + 168 = 328
23. The compound interest on a certain sum of money for 2 years at 5% per annum is Rs 410. The simple interest on the same sum at the same rate and for the same time is =
a) Rs. 400
b) Rs. 300
c) Rs. 350
d) Rs. 405
Discussion
Explanation:
$$\eqalign{ & {\text{Rate of interest 5}}\% \cr & = \frac{1}{{20}} \cr & {\text{Let principal}} \cr & {\text{ = }}{\left( {20} \right)^2}{\text{ = 400 units}} \cr & \Rightarrow {\text{ Total compound interest }} \cr & {\text{41 Units }} \to {\text{Rs. 410 }} \cr & {\text{1 Units }} \to {\text{Rs. 10 }} \cr & {\text{400 Units }} \to {\text{Rs. 400 }} \cr & {\text{Total simple interest}} \cr & {\text{ = Rs. 400}} \cr} $$
24. A sum of money lent out at compound interest increases in value by 50% in 5 years. A person wants to lend three different sums x, y and z for 10, 15 and 20 years respectively at the above rate in such a way that he gets back equal sums at the end of their respective periods. The ratio x : y : z is =
a) 6 : 9 : 4
b) 9 : 4 : 6
c) 9 : 6 : 4
d) 6 : 4 : 9
Discussion
Explanation:
$$\eqalign{ & P{\left( {1 + \frac{R}{{100}}} \right)^5} = 150\% \,{\text{of }}P = \frac{3}{2}P \cr & \Rightarrow {\left( {1 + \frac{R}{{100}}} \right)^5} = \frac{3}{2} \cr} $$
$$x{\left( {1 + \frac{R}{{100}}} \right)^{10}} = y{\left( {1 + \frac{R}{{100}}} \right)^{15}} = $$ $$z{\left( {1 + \frac{R}{{100}}} \right)^{20}}$$
$$ \Rightarrow x{\left\{ {{{\left( {1 + \frac{R}{{100}}} \right)}^5}} \right\}^2} = $$ $$y{\left\{ {{{\left( {1 + \frac{R}{{100}}} \right)}^5}} \right\}^3} = $$ $$z{\left\{ {{{\left( {1 + \frac{R}{{100}}} \right)}^5}} \right\}^4}$$
$$\eqalign{ & \Rightarrow x \times {\left( {\frac{3}{2}} \right)^2} = y \times {\left( {\frac{3}{2}} \right)^3} = z \times {\left( {\frac{3}{2}} \right)^4} \cr & \Rightarrow \frac{{9x}}{4} = \frac{{27y}}{8} = \frac{{81z}}{{16}} = k({\text{say}}) \cr & \Rightarrow x = \frac{{4k}}{9},y = \frac{{8k}}{{27}},z = \frac{{16k}}{{81}} \cr & x:y:z = \frac{{4k}}{9}:\frac{{8k}}{{27}}:\frac{{16k}}{{81}} \cr & x:y:z = 36:24:16 \cr & x:y:z = 9:6:4 \cr} $$
25. Under the Rural Housing Scheme, the Delhi Development Authority (DDA) allotted a house to Kamal Raj for Rs. 126100. This payment is to be made in three equal annual instalments. If the money is reckoned at 5% per annum compound interest, then how much is to be paid by Kamal Raj in each instalment ?
a) Rs. 45205
b) Rs. 46305
c) Rs. 47405
d) Rs. 48505
Discussion
Explanation: Let the value of each instalment be Rs. x
Then, (P.W. of Rs. x due 1 year hence) + (P.W. of Rs. x due 2 year hence) + (P.W. of Rs. x due 3 year hence) = 126100
$$\eqalign{ & \frac{x}{{\left( {1 + \frac{5}{{100}}} \right)}} + \frac{x}{{{{\left( {1 + \frac{5}{{100}}} \right)}^2}}} + \frac{x}{{{{\left( {1 + \frac{5}{{100}}} \right)}^3}}} = 126100 \cr & \frac{{20x}}{{21}} + \frac{{400x}}{{441}} + \frac{{8000x}}{{9261}} = 126100 \cr & \frac{{8820x + 8400x + 8000x}}{{9261}} = 126100 \cr & \frac{{25220x}}{{9261}} = 126100 \cr & x = \left( {\frac{{126100 \times 9261}}{{25220}}} \right) \cr & x = 46305 \cr} $$
26. A sum of Rs 210 was taken as a loan. This is to be paid back in two equal installments. If the rate of interest be 10% compounded annually, then the value of each installment is = ?
a) Rs. 127
b) Rs. 121
c) Rs. 210
d) Rs. 225
Discussion
Explanation:
$$\eqalign{ & {\text{Rate of interest}} \Rightarrow {\text{ 10% = }}\frac{1}{{10}} \cr & {\text{Each installment of 2 years}} \cr & \frac{{10}}{{11}} \times \frac{{\left( {10 + 11} \right)}}{{11}} \times {\text{ Installment = P}}{\text{.A}} \cr & \frac{{10}}{{11}} \times \frac{{\left( {10 + 11} \right)}}{{11}} \times {\text{ Installment = 210}} \cr & {\text{Installment = 121}} \cr} $$
27. A certain sum will amount to Rs 12100 in 2 years at 10% per annum of compound interest, interest being compounded annually. The sum is = ?
a) Rs. 12000
b) Rs. 6000
c) Rs. 8000
d) Rs. 10000
Discussion
Explanation: Amount = 12,100; r = 10%, t = 2 yrs
$$\eqalign{ & {\text{Amount}} = P{\left[ {1 + \frac{r}{{100}}} \right]^t} \cr & 12100 = P{\left[ {1 + \frac{{10}}{{100}}} \right]^2} \cr & 12100 = P{\left[ {\frac{{11}}{{10}}} \right]^2} \cr & 12100 = P \times \frac{{11}}{{10}} \times \frac{{11}}{{10}} \cr & P = \frac{{12100 \times 10 \times 10}}{{11 \times 11}} \cr & P = 10000 \cr} $$
28. Find the rate percent per annum if Rs. 2000 amounts to Rs. 2315.25 in one and half years interest being compounded half yearly.
a) 10 %
b) 11.5 %
c) 5 %
d) 20 %
Discussion
Explanation:
$$\eqalign{ & {\text{compounded half yearly}} \cr & {\text{Rate = }}\frac{{\text{R}}}{2} \cr & {\text{Time = }}\frac{{{\text{2T}}}}{3} \cr & {\text{Amount = P}}{\left( {1 + \frac{{\text{R}}}{{2 \times 100}}} \right)^3} \cr & 2315.25 = 2000{\left( {1 + \frac{{\text{R}}}{{2 \times 100}}} \right)^3} \cr & \frac{{2315.25}}{{2000}} = {\left( {1 + \frac{{\text{R}}}{{2 \times 100}}} \right)^3} \cr & \frac{{231525}}{{200000}} = {\left( {1 + \frac{{\text{R}}}{{2 \times 100}}} \right)^3} \cr & \frac{{9261}}{{8000}} = {\left( {1 + \frac{{\text{R}}}{{2 \times 100}}} \right)^3} \cr & {\left( {\frac{{21}}{{20}}} \right)^3} = {\left( {1 + \frac{{\text{R}}}{{2 \times 100}}} \right)^3} \cr & 1 + \frac{{\text{R}}}{{200}} = \frac{{21}}{{20}} \cr & {\text{R = 10}}\% \cr} $$
29. One can purchase a flat from a house building society for Rs. 55000 cash or on the terms that he should pay Rs. 4275 as cash down payment and get the rest in three equal installments. The society charges interest at the rate of 16% per annum compounded half-yearly. If the flat is purchased under installment plan, find the value of each installment ?
a) Rs. 18756
b) Rs. 19292
c) Rs. 19683
d) Rs. 20285
Discussion
Explanation: Total cost of the flat = Rs. 55000
Down payment = Rs. 4275
Balance = Rs. (55000 - 4275) = Rs. 50725
Rate of interest = 8% per half year
Let the value of each instalment be Rs. x
P.W. of Rs. x due 6 months hence + P.W. of Rs. x due 1 year hence + P.W. of Rs. x due $$1\frac{1}{2}$$ years hence = 50725
$$ \frac{x}{{\left( {1 + \frac{8}{{100}}} \right)}} + $$ $$\frac{x}{{{{\left( {1 + \frac{8}{{100}}} \right)}^2}}} + $$ $$\frac{x}{{{{\left( {1 + \frac{8}{{100}}} \right)}^3}}} = $$ $$50725$$
$$\eqalign{ & \frac{{25x}}{{27}} + \frac{{625x}}{{729}} + \frac{{15625x}}{{19683}} = 50725 \cr & \frac{{50725x}}{{19683}} = 50725 \cr & x = \left( {\frac{{50725 \times 19683}}{{50725}}} \right) = 19683 \cr} $$
30. The sum of money which when given on compound interest at 18% per annum would fetch Rs 960 more when the interest is payable half-yearly then when it was payable annually for 2 years is =
a) Rs. 60000
b) Rs. 30000
c) Rs. 40000
d) Rs. 50000
Discussion
Explanation: Rate of interest = 18%
Time = 2 year
When the interest is payable half yearly
Then, rate of interest = 9%
Time = 4 half - years
Let the principal be Rs. x
$$\eqalign{ & {\text{C}}{\text{.I}}{\text{. = }}x\left[ {{{\left( {1 + \frac{R}{{100}}} \right)}^T} - 1} \right]{\text{ }} \cr & = x\left[ {{{\left( {1 + \frac{9}{{100}}} \right)}^4} - 1} \right] \cr & = x\left[ {{{\left( {\frac{{109}}{{100}}} \right)}^4} - 1} \right] \cr & = x\left[ {1.4116 - 1} \right] \cr & = Rs.\,0.4116x \cr & {\text{According to question}} \cr & = x\left[ {{{\left( {1 + \frac{{18}}{{100}}} \right)}^2} - 1} \right] \cr & = x\left[ {{{\left( {\frac{{118}}{{100}}} \right)}^2} - 1} \right] \cr & = x\left[ {{{\left( {1.18} \right)}^2} - 1} \right] \cr & = x\left[ {1.3924 - 1} \right] \cr & = Rs.\,0.3924x \cr & {\text{According to question,}} \cr & 0.4116x - 0.3924x = 960 \cr & x = \frac{{960}}{{0.0192}} \cr & x = \frac{{960 \times 10000}}{{192}} \cr & x = 50000 \cr} $$
31. The difference between compound interest and simple interest on a certain sum of money for 2 years at 5% per annum is Rs. 41. What is the sum of money ?
a) Rs. 7200
b) Rs. 9600
c) Rs. 16400
d) Rs. 8400
Discussion
Explanation:
$$\eqalign{ & {\text{C}}{\text{.I}}{\text{.}} - {\text{S}}{\text{.I}}{\text{.}} = 41 \cr & {\text{C}}{\text{.I}}{\text{.}} - {\text{S}}{\text{.I}}{\text{.}} = P{\left( {\frac{r}{{100}}} \right)^2} \cr & 41 = P\left( {\frac{{25}}{{10000}}} \right) \cr & P = 16400 \cr} $$
32. The compound interest on a certain sum for 2 years at 10% per annum is Rs. 525. The simple interest on the same sum for double the time at half the rate percent per annum is = ?
a) Rs. 520
b) Rs. 550
c) Rs. 500
d) Rs. 515
Discussion
Explanation:
$$\eqalign{ & 10\% = \frac{1}{{10}} \cr & {\text{Let P}} = {\text{ }}{\left( {10} \right)^2} = 100 \cr & {\text{Total CI = 21 unit = 525}} \cr & {\text{1 unit = 25}} \cr & {\text{P}} = {\text{ 100 unit }} \cr & {\text{ = 100}} \times {\text{25}} = {\text{2500}} \cr & {\text{New Time = 4 years}} \cr & {\text{and new rate = 5}}\% \cr & {\text{SI = }}\frac{{2500 \times 4 \times 5}}{{100}}{\text{ }} \cr & {\text{SI = Rs. 500}} \cr} $$
33. Shashi had a certain amount of money. He invested $$\frac{2}{3}$$ of the total money in scheme A for 6 years and rest of the money he invested in scheme B for 2 years. Scheme A offers simple interest at a rate of 12% p.a. and scheme B offers compound interest ( compound annually) at a rate of 10% p.a. If the total interest obtained from both the schemes is Rs. 2750. What was the total amount invested by him in scheme A and scheme B together ? (Approximate value)
a) Rs. 4500
b) Rs. 4200
c) Rs. 4050
d) Rs. 5000
Discussion
Explanation: Let the total sum of money invested by Shashi be Rs. x
In scheme A money invested at simple interest for 6 years at a rate of 12% p.a.
$$ \frac{2}{3}{\text{of }}x \times \frac{{12 \times 6}}{{100}} = \frac{{48x}}{{100}}....(i)$$
In scheme B money at compound interest for 2 year at a rate of 10% p.a.
$$\eqalign{ & \frac{x}{3}{\left( {1 + \frac{{10}}{{100}}} \right)^2} - \frac{x}{3} \cr & \Rightarrow \frac{x}{3}{\left( {1 + \frac{{10}}{{100}}} \right)^2} - \frac{x}{3} = \frac{{7x}}{{100}} \cr} $$
According to given information,
$$\eqalign{ & \frac{{48x}}{{100}} + \frac{{7x}}{{100}} = 2750 \cr & 55x = 2750 \times 100 \cr & x = \frac{{2750 \times 100}}{{55}} \cr & x = Rs.\,5000 \cr} $$
34. The difference between CI and SI on a certain sum of money for 3 years at 5% p.a. is Rs. 122. Find the sum invested ?
a) Rs. 10000
b) Rs. 12000
c) Rs. 16000
d) Rs. 20000
Discussion
Explanation:
$$\eqalign{ & {\text{Rate of interest = 5}}\% {\text{ p}}{\text{.c}}{\text{.p}}{\text{.a}}{\text{.}} \cr & {\text{If time 3 years than CI}} - {\text{SI}} \cr & {\text{ = }}P\left[ {{{\left( {\frac{R}{{100}}} \right)}^3} + 3{{\left( {\frac{R}{{100}}} \right)}^2}} \right] \cr & \Rightarrow 122 = P\left[ {{{\left( {\frac{5}{{100}}} \right)}^3} + 3{{\left( {\frac{5}{{100}}} \right)}^2}} \right] \cr & \Rightarrow 122 = P\left( {\frac{{125}}{{1000000}} + \frac{{75}}{{10000}}} \right) \cr & 122 = P\left[ {\frac{{125 + 7500}}{{1000000}}} \right] \cr & 122 = P\left[ {\frac{{7525}}{{1000000}}} \right] \cr & P = \frac{{122 \times 1000000}}{{7625}} \cr & P = {\text{Rs}}{\text{. 16000}} \cr} $$
35. A man invested a sum of money at compound interest. It amounted to Rs. 2420 in 2 years and to Rs. 2662 in 3 years. Find the sum ?
a) Rs. 1000
b) Rs. 2000
c) Rs. 5082
d) Rs. 3000
Discussion
Explanation:
$$\eqalign{ & {\text{R}}\% {\text{ = }}\frac{{2662 - 2420}}{{2420}} \times 100 \cr & = \frac{{242}}{{2420}} \times 100 \cr & = 10\% \cr & {\text{2 years CI}}\% \cr & {\text{ = 10 + 10 + }}\frac{{10 \times 10}}{{100}} \cr & = 21\% \cr & {\text{So, 121}}\% {\text{ = 2420}} \cr & \Rightarrow {\text{100}}\% {\text{ = 2000}} \cr} $$
36. A sum of Rs. 8000 will amount to Rs. 8820 in 2 years if the interest is calculated every year. The rate of compound interest is = ?
a) 6%
b) 7%
c) 3%
d) 5%
Discussion
Explanation:
$$\eqalign{ & {\text{Principal = Rs 8000}} \cr & {\text{Amount = Rs 8820}} \cr & {\text{Let Rate = }}R \cr & {\text{Time = 2 years}} \cr & {\text{By using formula, }} \cr & 8820 = 8000{\left( {1 + \frac{R}{{100}}} \right)^2} \cr & \frac{{8820}}{{8000}} = {\left( {1 + \frac{R}{{100}}} \right)^2} \cr & \frac{{441}}{{400}} = {\left( {1 + \frac{R}{{100}}} \right)^2} \cr & {\text{Taking square root of both sides,}} \cr & \frac{{21}}{{20}} = \left( {1 + \frac{R}{{100}}} \right) \cr & R = 5\% \cr} $$
37. The compound interest on a certain some of money for 2 years at 10% per annum is Rs 420. The simple interest on the same sum at the same rate and for the same time will be ?
a) Rs. 350
b) Rs. 375
c) Rs. 380
d) Rs. 400
Discussion
Explanation:
$$\eqalign{ & {\text{Rate = 10}}\% \cr & {\text{Time = 2 years}} \cr & {\text{Effective rate of CI for 2 years}} \cr & {\text{ = 10 + 10 + }}\frac{{10 \times 10}}{{100}} = 21\% \cr & {\text{Effective rate of SI for 2 years}} \cr & {\text{ = 2}} \times {\text{10 = 20}}\% \cr & {\text{Required SI}} \cr & {\text{ = }}\frac{{420}}{{21}} \times {\text{20 = Rs. 400}} \cr} $$
38. A sum of money at compound interest amounts to thrice of itself in 3 years. In how many years it will be 9 times of itself ?
a) 9 years
b) 27 years
c) 6 years
d) 3 years
Discussion
Explanation: x becomes 3x in 3 years
Therefore 3x also becomes 9x in 3 years
Required years = 3 + 3 = 6
39. A father left a will of Rs. 16400 for his two sons aged 17 and 18 years. They must get equal amount when they are 20 years, at 5% compound interest. Find the present share of the younger son = ?
a) Rs. 8000
b) Rs. 8200
c) Rs. 8400
d) Rs. 8800
Discussion
Explanation:
Let the share of the younger and elder sons be Rs. x and Rs. (16400 - x)
Then, amount of Rs. x after 3 years = Amount of Rs. (16400 - x) after 2 years
$$\eqalign{ & x{\left( {1 + \frac{5}{{100}}} \right)^3} = \left( {16400 - x} \right){\left( {1 + \frac{5}{{100}}} \right)^2} \cr & x\left( {1 + \frac{5}{{100}}} \right) = \left( {16400 - x} \right) \cr & \frac{{21x}}{{20}} + x = 16400 \cr & \frac{{41x}}{{20}} = 16400 \cr & x = \left( {\frac{{16400 \times 20}}{{41}}} \right) \cr & x = 8000 \cr} $$
40. A sum of money put at compound interest amounts in 2 years to Rs. 672 and in 3 years Rs. 714. The rate of interest per annum is = ?
a) 5.5%
b) 6.0%
c) 6.25%
d) 6.75%
Discussion
Explanation:
$$\eqalign{ & {\text{S}}{\text{.I}}{\text{. on Rs}}{\text{. 672 for 1 year}} \cr & {\text{ = Rs}}{\text{. }}\left( {714 - 672} \right) \cr & {\text{ = Rs}}{\text{. 42}} \cr & {\text{Rate = }}\left( {\frac{{100 \times 42}}{{672 \times 1}}} \right){\text{% }} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{ = 6}}{\text{.25% }} \cr} $$
41. In what time will Rs 64000 amounts to Rs 68921 at 5% per annum interest being compounded half yearly ?
a) $$1\frac{1}{2}$$ years
b) 2 years
c) 3 years
d) $$2\frac{1}{2}$$ years
Discussion
Explanation:
$$\eqalign{ & {\text{Amount}} = {\text{ }}{\left( {1 + \frac{{\text{R}}}{{2 \times 100}}} \right)^{2 \times {\text{t}}}} \cr & 68921 = 64000{\left( {1 + \frac{5}{{2 \times 100}}} \right)^{2 \times {\text{t}}}} \cr & \frac{{68921}}{{64000}} = {\left( {1 + \frac{1}{{40}}} \right)^{2 \times {\text{t}}}} \cr & {\left( {\frac{{41}}{{40}}} \right)^3} = {\left( {\frac{{41}}{{40}}} \right)^{2 \times {\text{t}}}} \cr & 2{\text{t = 3}} \cr & {\text{t = }}\frac{3}{2} \cr & {\text{t = 1}}\frac{1}{2}{\text{ years}} \cr} $$
42. When principal = Rs. S, rate of interest = 2r % p.a., then a person will get after 3 years at compound interest = ?
a) $${\text{Rs}}{\text{. }}\frac{{6{\text{Sr}}}}{{100}}$$
b) $${\text{Rs}}{\text{. S}}{\left( {1 + \frac{{\text{r}}}{{50}}} \right)^3}$$
c) $${\text{Rs}}{\text{. S}}{\left( {1 + \frac{{\text{r}}}{{100}}} \right)^3}$$
d) $${\text{Rs}}{\text{. 3S}}{\left( {1 + \frac{{\text{r}}}{{100}}} \right)^3}$$
Discussion
Explanation:
$$\eqalign{ & {\text{Principal = Rs S}} \cr & {\text{Rate }}\% {\text{ = 2r}}\,\% {\text{ p}}{\text{.a}}{\text{.}} \cr & {\text{Time = 3 years}} \cr & {\text{A = P}}{\left( {1 + \frac{{\text{r}}}{{100}}} \right)^T} \cr & {\text{A = S}}{\left( {1 + \frac{{{\text{2r}}}}{{100}}} \right)^3} \cr & {\text{A = S}}{\left( {1 + \frac{{\text{r}}}{{50}}} \right)^3} \cr} $$
43. At what rate of compound interest per annum will a sum of Rs. 1200 become Rs. 1348.32 in 2 years ?
a) 6.5%
b) 4.5%
c) 6%
d) 7.5%
Discussion
Explanation:
$$\eqalign{ & {\text{A = P }}{\left( {1 + \frac{{\text{R}}}{{100}}} \right)^n} \cr & 1348.32 = 1200{\left( {1 + \frac{{\text{R}}}{{100}}} \right)^2} \cr & \frac{{134832}}{{120000}} = {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^2} \cr & \frac{{231525}}{{200000}} = {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^2} \cr & \frac{{2809}}{{2500}} = {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^2} \cr & {\left( {\frac{{53}}{{50}}} \right)^2} = {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^2} \cr & \frac{{53}}{{50}} = 1 + \frac{{\text{R}}}{{100}} \cr & {\text{R}} = {\text{ 6% }} \cr} $$
44. On what sum of money will the difference between simple interest and compound interest for 2 years at 5% per annum be equal to Rs. 63 ?
a) Rs. 24600
b) Rs. 24800
c) Rs. 25200
d) Rs. 25500
Discussion
Explanation:
$$\eqalign{ & {\text{Rate of interest = 5}}\% {\text{ per annum}} \cr & {\text{Time = 2 year}} \cr & P\left[ {{{\left( {1 + \frac{r}{{100}}} \right)}^n} - 1} \right] - \frac{{P \times r \times t}}{{100}}{\text{ = 63}} \cr & P\left[ {{{\left( {1 + \frac{5}{{100}}} \right)}^2} - 1} \right] - \frac{{P \times 5 \times 2}}{{100}}{\text{ = 63}} \cr & P\left[ {{{\left( {1 + \frac{5}{{100}}} \right)}^2} - 1} \right] - \frac{{10P}}{{100}}{\text{ = 63}} \cr & P\left[ {{{\left( {\frac{{105}}{{100}}} \right)}^2} - 1} \right] - \frac{{10P}}{{100}}{\text{ = 63}} \cr & P\left( {\frac{{11025 - 10000}}{{10000}}} \right) - \frac{{10P}}{{100}} = 63 \cr & \frac{{1025P}}{{10000}} - \frac{{10P}}{{100}} = 63 \cr & \frac{{1025P - 1000P}}{{10000}} = 63 \cr & 25P = Rs.630000 \cr & P = \frac{{630000}}{{25}} \cr & P = Rs. 25200 \cr} $$
45. The sum for 2 years given a compound interest of Rs. 3225 at 15% rate. Then the sum is =
a) Rs. 10000
b) Rs. 20000
c) Rs. 15000
d) Rs. 32250
Discussion
Explanation: Interest for 2 years at the rate of 15%
$$\eqalign{ & {\text{ = 15 + 15 + }}\frac{{15 \times 15}}{{100}} = 32.25\% \cr & {\text{According to question,}} \cr & {\text{32}}{\text{.25}}\% {\text{ = 3225}} \cr & {\text{100}}\% {\text{ = }}\frac{{3225}}{{32.25}} \times 100 \cr & = 100 \times 100 = 10000 \cr} $$
46. The compound interest on Rs. 5000 for 3 years at 10% p.a. will amount to = ?
a) Rs. 1654
b) Rs. 1655
c) Rs. 1600
d) Rs. 1565
Discussion
Explanation:
$$\eqalign{ & {\text{Principal = Rs. 5000}} \cr & {\text{Time = 3 years}} \cr & {\text{Rate = 10}}\% {\text{ = }}\frac{1}{{10}} \cr & {\text{Principal}}\,\,\,\,\,\,\,{\text{Amount}} \cr & \,\,\,\,\,\,\,\,\,\,{\text{10}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{11}} \cr & \,\,\,\,\,\,\,\,\,\,{\text{10}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{11}} \cr & \,\,\,\,\,\,\,\,\,\,{\text{10}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{11}} \cr & \underbrace {\overline {\,\,\,\,\,\,1000\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{1331}}\,\,\,\,\,} }_{331{\text{ units}}} \cr & 1000{\text{ units = Rs 5000}} \cr & 1{\text{ units = Rs 5}} \cr & 331{\text{ units = 331}} \times {\text{5}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{ = Rs. 1655}} \cr} $$
47. If the compound interest on a certain sum for two years at 12% per annum is Rs. 2544, the simple interest on it at the same rate for 2 years will be = ?
a) Rs. 2400
b) Rs. 2500
c) Rs. 2480
d) Rs. 2440
Discussion
Explanation:
$$\eqalign{ & {\text{Rate = 12}}\% \cr & {\text{Time = 2 years}} \cr & {\text{Effective rate of CI for 2 years}} \cr & {\text{ = 12 + 12 + }}\frac{{12 \times 12}}{{100}} \cr & = 25.44\,\% \cr & {\text{Effective rate of SI for 2 years}} \cr & {\text{ = 12}} \times 2{\text{ = 24}}\,\% \cr & {\text{Required SI}} \cr & {\text{ = }}\frac{{2544}}{{25.44}} \times {\text{24}} = {\text{ Rs. 2400}} \cr} $$
48. A sum becomes Rs. 2916 in 2 years at 8% per annum compound interest. The simple interest at 9% per annum for 3 years on the same amount will be = ?
a) Rs. 600
b) Rs. 675
c) Rs. 650
d) Rs. 625
Discussion
Explanation:
$$\eqalign{ & {\text{Amount = Rs}}{\text{. 2916}} \cr & {\text{Time = 2 years }} \cr & {\text{Rate = 8}}\% \cr & {\text{Effective rate }}\% {\text{ CI for 2 years}} \cr & {\text{ = 8 + 8 + }}\frac{{8 \times 8}}{{100}} = 16.64\% \cr & {\text{Required sum}} \cr & {\text{ = }}\frac{{2916}}{{\left( {100 + 16.64} \right)}} \times 100 = {\text{Rs}}{\text{. }}2500 \cr & {\text{Required simple interest}} \cr & {\text{ = }}\frac{{2500 \times 9 \times 3}}{{100}} \cr & = {\text{Rs}}{\text{. }}675 \cr} $$
49. A sum of money is compound interest became doubles itself in 15 years. It will become eight times of itself in =
a) 45 years
b) 48 years
c) 54 years
d) 60 years
Discussion
Explanation:
$$\eqalign{ & P{\left( {1 + \frac{{\text{R}}}{{100}}} \right)^{15}} = 2P \cr & \Rightarrow {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^{15}} = 2 \cr & {\text{Let }}P{\left( {1 + \frac{{\text{R}}}{{100}}} \right)^n} = 8P \cr & \Rightarrow {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^n} = 8 = {2^3} = {\left\{ {{{\left( {1 + \frac{{\text{R}}}{{100}}} \right)}^{15}}} \right\}^3} \cr & \Rightarrow {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^n} = {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^{45}} \cr & \Rightarrow n = 45 \cr} $$
50. A finance company declares that, at a certain compound interest rate, a sum of money deposited by anyone will become 8 times in 3 years. If the same amount is deposited at the same compound rate of interest, then in how many years will it become 16 times ?
a) 4 years
b) 5 years
c) 6 years
d) 7 years
Discussion
Explanation:
$$\eqalign{ & P{\left( {1 + \frac{{\text{R}}}{{100}}} \right)^3} = 8P \cr & \Rightarrow {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^3} = 8 \cr & {\text{Let }}P{\left( {1 + \frac{{\text{R}}}{{100}}} \right)^n} = 16P \cr & \Rightarrow {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^n} = 16 = {2^4} = {\left( {{2^3}} \right)^{\frac{4}{3}}} \cr & \Rightarrow {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^n} = {\left( 8 \right)^{\frac{4}{3}}} \cr & \Rightarrow {\left\{ {{{\left( {1 + \frac{{\text{R}}}{{100}}} \right)}^3}} \right\}^{\frac{4}{3}}} = {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^4} \cr & \Rightarrow n = 4 \cr} $$
51. A certain amount money at R% compounded annually after two and three years becomes Rs. 1440 and Rs. 1728 respectively, R% is ?
a) 5%
b) 10%
c) 15%
d) 20%
Discussion
Explanation: b – a = 3 – 2 = 1
B = Rs. 1728, A = Rs.1440
$$\eqalign{ & R\% = \left( {\frac{B}{A} - 1 \times 100} \right)\% \cr & \,\,\,\,\,\,\,\,\,\,\,\, = \left( {\frac{{1728}}{{1440}} - 1 \times 100} \right)\% \cr & \,\,\,\,\,\,\,\,\,\,\,\, = \left( {\frac{{288}}{{1440}} \times 100} \right)\% \cr & \,\,\,\,\,\,\,\,\,\,\,\, = 20\% \cr} $$
52. The compound interest on a certain sum for 2 successive years are Rs. 225 and Rs. 238.50. The rate of interest per annum is = ?
a) $$7\frac{1}{2}$$%
b) 5%
c) 10%
d) 6%
Discussion
Explanation:
$$\eqalign{ & {\text{Required rate }}\% \cr & {\text{ = }}\frac{{\left( {238.50 - 225} \right)}}{{225}} \times 100 \cr & = 6\,\% \cr} $$
53. A man, borrow Rs 21000 at 10% compound interest. How much he has to pay annually at the end of each year, to settle his loan in two years ?
a) Rs. 12000
b) Rs. 12100
c) Rs. 12200
d) Rs. 12300
Discussion
Explanation:
$$\eqalign{ & {\text{Rate }} \Rightarrow {\text{ 10% = }}\frac{1}{{10}} \cr & {\text{Each installment of 2 years}} \cr & \Rightarrow \frac{{10}}{{11}} \times \frac{{\left( {10 + 11} \right)}}{{11}} \times {\text{ Installment = P}}{\text{.A}} \cr & {\text{P}}{\text{.A = 21000}} \cr & {\text{Each installment = 12100}} \cr} $$
54. A sum of money invested at compound interest amounts to Rs. 4624 in 2 years and Rs. 4913 in 3 years. The sum of money is = ?
a) Rs. 4096
b) Rs. 4260
c) Rs. 4335
d) Rs. 4360
Discussion
Explanation: S.I. on Rs. 4624 for 1 year
$$\eqalign{ & {\text{ = Rs. }}\left( {4913 - 4624} \right) \cr & {\text{ = Rs. 289}} \cr & {\text{Rate}} = \left( {\frac{{100 \times 289}}{{4624 \times 1}}} \right)\% \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 6\frac{1}{4}\% \cr & {\text{Now, }} x{\left( {1 + \frac{{25}}{{400}}} \right)^2} = 4624 \cr & x \times \frac{{17}}{{16}} \times \frac{{17}}{{16}} = 4624 \cr & x = \left( {4624 \times \frac{{16}}{{17}} \times \frac{{16}}{{17}}} \right) \cr & x = 4096 \cr} $$
55. A sum of Rs. 12000 deposited at compound interest become double after 5 years. After 20 years it will become ?
a) Rs. 96000
b) Rs. 120000
c) Rs. 124000
d) Rs. 192000
Discussion
Explanation:
$$\eqalign{ & 12000 \times {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^5} = 24000 \cr & \Rightarrow {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^5} = 2 \cr & {\left[ {{{\left( {1 + \frac{{\text{R}}}{{100}}} \right)}^5}} \right]^4} = {2^4} = 16 \cr & \Rightarrow {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^{20}} = 16 \cr & \Rightarrow {\text{P}}{\left( {1 + \frac{{\text{R}}}{{100}}} \right)^{20}}{\text{ = 16P}} \cr & \Rightarrow 12000{\left( {1 + \frac{{\text{R}}}{{100}}} \right)^{20}} = 16 \times 12000 \cr & \Rightarrow 12000{\left( {1 + \frac{{\text{R}}}{{100}}} \right)^{20}} = 192000 \cr} $$
56. The compound interest on Rs. 4000 for 4 years at 10% per annum will be =
a) Rs. 1856.40
b) Rs. 1600
c) Rs. 1856
d) Rs. 1756.60
Discussion
Explanation:
$$\eqalign{ & {\text{10}}\% {\text{ = }}\frac{1}{{10}} \cr & {\text{Principal}}\,\,\,\,\,\,\,{\text{Amount}} \cr & \,\,\,\,\,\,\,{\text{10}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,11 \cr & \,\,\,\,\,\,\,{\text{10}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,11 \cr & \,\,\,\,\,\,\,{\text{10}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,11 \cr & \,\,\,\,\,\,\,{\text{10}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,11 \cr & \underbrace {\overline {\,\,\,10000\,\,\,\,\,\,{\text{:}}\,\,\,\,\,{\text{14641}}\,\,\,} }_{{\text{CI = 4641}}} \cr & {\text{Principal = 10000 units}} \cr & {\text{ = Rs}}{\text{. 4000 (given)}} \cr & {\text{1 unit = }}\frac{2}{5} \cr & {\text{CI = 4641 unit}} \cr & {\text{ = Rs}}{\text{. }}\left( {\frac{2}{5} \times 4641} \right) \cr & = {\text{Rs}}{\text{. }}1856.40 \cr} $$
57. What will be the difference between S.I. and C.I. on a sum of Rs. 15000 for 2 years at the same rate of interest of $$12\frac{1}{2}$$ % per annum ?
a) Rs. 230.550
b) Rs. 234.375
c) Rs. 250.129
d) Rs. 324.357
Discussion
Explanation:
$$\eqalign{ & {\text{S}}{\text{.I}}{\text{. = Rs}}{\text{. }}\left( {15000 \times \frac{{25}}{2} \times 2 \times \frac{1}{{100}}} \right) \cr & \,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. }}3750 \cr & {\text{C}}{\text{.I}}{\text{. = Rs}}{\text{. }}\left[ {15000{{\left( {1 + \frac{{25}}{{2 \times 100}}} \right)}^2} - 15000} \right] \cr & = {\text{Rs}}{\text{. }}\left( {15000 \times \frac{9}{8} \times \frac{9}{8} - 15000} \right) \cr & = {\text{Rs}}{\text{. }}\left( {18948.375 - 15000} \right) \cr & = {\text{Rs}}{\text{. }}3984.375 \cr & {\text{Difference }}{\text{ = Rs}}{\text{. }}\left( {3984.375 - 3750} \right) \cr & = {\text{Rs}}{\text{. }}234.375 \cr} $$
58. The compound interest on a certain sum of money for 2 years at 10% per annum is Rs. 525.The simple interest on the same sum of money for double the time at half the rate percent per annum is ?
a) Rs. 1000
b) Rs. 500
c) Rs. 200
d) Rs. 800
Discussion
Explanation: Let the sum of money be rs. P
$$\eqalign{ & \left[ {P{{\left( {1 + \frac{R}{{100}}} \right)}^t} - P} \right] = {\text{C}}{\text{.I}}{\text{.}} \cr & \left[ {P{{\left( {1 + \frac{{10}}{{100}}} \right)}^2} - P} \right] = 525 \cr & P{\left( {\frac{{11}}{{10}}} \right)^2} - 1 = 525 \cr & P\left( {\frac{{121}}{{100}} - 1} \right) = 525 \cr & P\left( {\frac{{21}}{{100}}} \right) = 525 \cr & P = \frac{{525 \times 100}}{{21}} \cr & P = {\text{Rs}}{\text{.}}\,2500 \cr} $$
Simple interest on the same sum Rs. 2500 for 4 (double the time) years at 5% (half the rate of percent per annum) is
$$\eqalign{ & {\text{S}}{\text{.I}}{\text{. = Rs}}{\text{.}}\left( {\frac{{2500 \times 5 \times 4}}{{100}}} \right) \cr & \,\,\,\,\,\,\,\,\,{\text{ = Rs}}{\text{. 500}} \cr} $$
59. A certain sum, invested at 4% per annum compound interest, compounded half yearly, amounts to Rs. 7803 at the end of one year. The sum is ?
a) Rs. 7000
b) Rs. 7200
c) Rs. 7500
d) Rs. 7700
Discussion
Explanation: 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
$${\text{ = 2}} + {\text{2}} + \frac{{2 \times 2}}{{100}} = 4.04\% $$
(100 + 4.04)% of sum = Rs. 7803
$$\eqalign{ & {\text{Sum = }}\frac{{7803}}{{104.04}} \times 100 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{. }}7500 \cr} $$
60. The difference between CI and SI for 3 years Rs. 992. If rate of interest is 10%. Find the Principal ?
a) Rs. 22000
b) Rs. 30000
c) Rs. 28000
d) Rs. 32000
Discussion
Explanation:
$$\eqalign{ & {\text{Rate}} = 10\% ,\, \cr & {\text{Let}}\,{\text{Principal}} = P \cr & {\text{S}}{\text{.I}}{\text{.}} = \frac{{P \times 10 \times 3}}{{100}} = \frac{{3P}}{{10}} \cr & {\text{C}}{\text{.I}}{\text{.}} = P\left\{ {{{\left( {1 + \frac{1}{{10}}} \right)}^3} - 1} \right\} \cr & {\text{C}}{\text{.I}}{\text{.}}\,\, - \,\,{\text{S}}{\text{.I}}{\text{.}} = 992 \cr & P\left\{ {{{\left( {1 + \frac{1}{{10}}} \right)}^3} - 1} \right\} - \frac{{3P}}{{10}} = 992 \cr & P\left\{ {{{\left( {\frac{{11}}{{10}}} \right)}^3} - 1 - \frac{3}{{10}}} \right\} = 992 \cr & P\left\{ {\frac{{\left( {1331 - 1000 - 300} \right)}}{{1000}}} \right\} = 992 \cr & P\left( {\frac{{31}}{{1000}}} \right) = 992 \cr & P = 32000 \cr} $$
61. A certain some of money and Rs. 2420 in 2 years and Rs. 2662 in 3 years at same rate of compound interest, compounded annually. The rate of interest per annum is =
a) 6%
b) 8%
c) 9%
d) 10%
Discussion
Explanation:
$$\eqalign{ & {\text{Amount after three years}} {\text{ = Rs. 2662}} \cr & {\text{Amount after two years}} {\text{ = Rs. 2420}} \cr & {\text{Net interest earned in the }}{{\text{3}}^{{\text{rd}}}}{\text{ year}} \cr & {\text{ = }}\,{\text{2662}} - {\text{2420}} \cr & {\text{ = Rs}}{\text{. 242}} \cr & {\text{Rate of interest (r)}} \cr & {\text{ = }}\frac{{242}}{{2420}} \times {\text{100 = 10% }} \cr} $$
(2nd year's amount is principal for 3rd year)
62. Kamal took Rs. 6800 as a loan which along with interest is to be repaid in two equal annual installment. If the rate of interest is $$12\frac{1}{2}$$ % compounded annually, then the value of each installment is =
a) Rs. 8100
b) Rs. 4150
c) Rs. 4050
d) Rs. 4000
Discussion
Explanation:
$$\eqalign{ & {\text{Rate of interest}} \cr & {\text{r}} = {\text{12}}\frac{1}{2}\% = \frac{1}{8} \cr} $$
| Year | Principal | Installment | |
| I | 8×9 → | 9×9 | ......(i) |
| II | 64 → | 81 | ......(ii) |
Total principal = 72 + 64 = 136 units
136 units → 6800
1 units → 50
81 units → 4050
Each installment = Rs. 4050
63. A man invests Rs 4000 for 3 years at compound interest. After one year the money amounts to Rs. 4320. What will be the amount (to the nearest rupee) due at the end of 3 years ?
a) Rs. 4939
b) Rs. 5039
c) Rs. 5789
d) Rs. 6129
Discussion
Explanation:
$$\eqalign{ & {\text{Le the rate be R }}\% {\text{ p}}{\text{.a}}{\text{.}} \cr & {\text{4000}}\left( {1 + \frac{{{\text{R }}}}{{100}}} \right) = 4320 \cr & 1 + \frac{{{\text{R }}}}{{100}} = \frac{{4320}}{{4000}} = \frac{{108}}{{100}} \cr & \frac{{{\text{R }}}}{{100}} = \frac{8}{{100}} \cr & {\text{R }} = 8 \cr & {\text{Amount after 3 yeras}} \cr & {\text{ = Rs}}{\text{. }}\left[ {4000 + {{\left( {1 + \frac{8}{{100}}} \right)}^3}} \right] \cr & {\text{ = Rs}}{\text{. }}\left( {4000 \times \frac{{27}}{{25}} \times \frac{{27}}{{25}} \times \frac{{27}}{{25}}} \right) \cr & {\text{ = Rs}}{\text{. }}\left( {\frac{{629856}}{{125}}} \right) \cr & {\text{ = Rs}}{\text{. }}5038.848 \approx 5039 \cr} $$
64. A sum of Rs. 13360 was borrowed at $${\text{8}}\frac{3}{4}$$ % per annum compound interest and paid back in two years in two equal annual installments. What was the amount of each installment ?
a) Rs. 5769
b) Rs. 7569
c) Rs. 7009
d) Rs. 7500
Discussion
Explanation:
$$\eqalign{ & {\text{Rate of interest (r)}} \cr & {\text{ = 8}}\frac{3}{4}\% = \frac{7}{{80}} = \frac{{87 \to {\text{ Installment}}}}{{80 \to {\text{Principal}}}} \cr} $$
| ⇒ | I | 80×87 | → | 87×87 | ......(i) |
| ⇒ | II | 6400 | → | 7569 | ......(ii) |
Total principal = 6960 + 6400 = 13360
13360 units = Rs. 13360
1 units = Rs. 1
7569 units = Rs. 7569
Each installment = Rs. 7569
65. An amount of Rs. 10000 becomes Rs. 14641 in 2 years if the interest is compounded half yearly. What is the rate of compound interest p.c.p.a. ?
a) 10%
b) 12%
c) 16%
d) 20%
Discussion
Explanation:
$$\eqalign{ & {\text{Let the rate be R% p}}{\text{.a}}{\text{. }} \cr & {\text{10000}}{\left( {1 + \frac{{\text{R}}}{{2 \times 100}}} \right)^4} = 14641 \cr & \Rightarrow {\left( {1 + \frac{{\text{R}}}{{200}}} \right)^4} = \frac{{14641}}{{10000}} = {\left( {\frac{{11}}{{10}}} \right)^4} \cr & 1 + \frac{{\text{R}}}{{200}} = \frac{{11}}{{10}} \cr & \frac{{\text{R}}}{{200}} = \frac{1}{{10}} \cr & {\text{R}} = {\text{20% }} \cr} $$
66. What will be the compound interest on a sum of Rs. 25,000 after 3 years at the rate of 12 p.c.p.a.?
a) Rs. 9000.30
b) Rs. 9720
c) Rs. 10123.20
d) Rs. 10483.20
Discussion
Explanation:
$$\eqalign{ & {\text{Amount}} = Rs.\,\left[ {25000 \times {{\left( {1 + \frac{{12}}{{100}}} \right)}^3}} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,\left( {25000 \times \frac{{28}}{{25}} \times \frac{{28}}{{25}} \times \frac{{28}}{{25}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,35123.20 \cr & {\text{C}}{\text{.I}}{\text{.}} = Rs.\left( {35123.20 - 25000} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,10123.20 \cr} $$
67. At what rate of compound interest per annum will a sum of Rs. 1200 become Rs. 1348.32 in 2 years?
a) 6%
b) 6.5%
c) 7%
d) 7.5%
Discussion
Explanation:
$$\eqalign{ & {\text{Let}}\,{\text{the}}\,{\text{rate}}\,{\text{be}}\,R\% \,p.a. \cr & 1200 \times {\left( {1 + \frac{R}{{100}}} \right)^2} = 1348.32 \cr & \Rightarrow {\left( {1 + \frac{R}{{100}}} \right)^2} = \frac{{134832}}{{120000}} = \frac{{11236}}{{10000}} \cr & {\left( {1 + \frac{R}{{100}}} \right)^2} = {\left( {\frac{{106}}{{100}}} \right)^2} \cr & 1 + \frac{R}{{100}} = \frac{{106}}{{100}} \cr & R = 6\% \cr} $$
68. The least number of complete years in which a sum of money put out at 20% compound interest will be more than doubled is
a) 3
b) 4
c) 5
d) 6
Discussion
Explanation:
$$\eqalign{ & P{\left( {1 + \frac{{20}}{{100}}} \right)^n} > 2P\,\,\, \Rightarrow \,\,\,{\left( {\frac{6}{5}} \right)^n} > 2 \cr & \left( {\frac{6}{5} \times \frac{6}{5} \times \frac{6}{5} \times \frac{6}{5}} \right) > 2 \cr & n = 4\,{\text{years}} \cr} $$
69. Albert invested an amount of Rs. 8000 in a fixed deposit scheme for 2 years at compound interest rate 5 p.c.p.a. How much amount will Albert get on maturity of the fixed deposit?
a) Rs. 8600
b) Rs. 8620
c) Rs. 8820
d) None of these
Discussion
Explanation:
$$\eqalign{ & {\text{Amount}} = Rs.\left[ {8000 \times {{\left( {1 + \frac{5}{{100}}} \right)}^2}} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,\left( {8000 \times \frac{{21}}{{20}} \times \frac{{21}}{{20}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,8820 \cr} $$
70. The effective annual rate of interest corresponding to a nominal rate of 6% per annum payable half-yearly is:
a) 6.06%
b) 6.07%
c) 6.08%
d) 6.09%
Discussion
Explanation:
$$\eqalign{ & {\text{Amount}}\,{\text{of}}\,{\text{Rs}}{\text{.}}\,{\text{100}}\,{\text{for}}\,{\text{1}}\,{\text{year}}\,{\text{when}}\, {\text{compounded}}\,{\text{half - yearly}} \cr & = Rs.\,\left[ {100 \times {{\left( {1 + \frac{3}{{100}}} \right)}^2}} \right] \cr & = Rs.\,106.09 \cr & {\text{Effective}}\,{\text{rate}} = \left( {106.09 - 100} \right)\% \cr & = 6.09\% \cr} $$
71. A man borrow Rs. 4000 at 15%, compound rate of interest. At the end of each year he pays back Rs. 1500. How much amount should be pay at the end of the third year to clear all his dues ?
a) Rs. 874.75
b) Rs. 824.50
c) Rs. 924.25
d) Rs. 974.25
Discussion
Explanation:
$$\eqalign{ & {\text{Amount after }}{{\text{1}}^{{\text{st}}}}{\text{ year}} \cr & {\text{ = Rs}}{\text{. }}\left[ {4000\left( {1 + \frac{{15}}{{100}}} \right) - 1500} \right] \cr & = {\text{Rs}}{\text{. }}\left[ {\left( {4000 \times \frac{{23}}{{20}}} \right) - 1500} \right] \cr & = {\text{Rs}}{\text{. }}\left( {4600 - 1500} \right) \cr & = {\text{Rs}}{\text{. }}3100 \cr & {\text{Amount after }}{{\text{2}}^{{\text{nd}}}}{\text{ year}} \cr & {\text{ = Rs}}{\text{. }}\left[ {3100\left( {1 + \frac{{15}}{{100}}} \right) - 1500} \right] \cr & = {\text{Rs}}{\text{. }}\left[ {\left( {3100 \times \frac{{23}}{{20}}} \right) - 1500} \right] \cr & = {\text{Rs}}{\text{. }}\left( {3565 - 1500} \right) \cr & = {\text{Rs}}{\text{. }}2065 \cr & {\text{Amount after }}{{\text{3}}^{{\text{rd}}}}{\text{ year}} \cr & {\text{ = Rs}}{\text{. }}\left[ {2065\left( {1 + \frac{{15}}{{100}}} \right) - 1500} \right] \cr & = {\text{Rs}}{\text{. }}\left[ {\left( {2065 \times \frac{{23}}{{20}}} \right) - 1500} \right] \cr & = {\text{Rs}}{\text{. }}\left( {2374.75 - 1500} \right) \cr & = {\text{Rs}}{\text{. }}874.75 \cr} $$
72. A certain sum of amounts to Rs. 5832 in 2 years at 8% per annum compound interest, the sum is = ?
a) Rs. 5000
b) Rs. 5200
c) Rs. 5280
d) Rs. 5400
Discussion
Explanation: $${\text{Rate 8% = }}\frac{2}{{25}}$$
| Principal | Amount |
| 25 | 27 |
| 25 | 27 |
| 625 | 729 |
| ↓ × 8 | ↓ × 8 |
| 5000 | 5832 |
73. A person deposited a sum of of Rs 6000 in a bank at 5% per annum simple interest. Another person deposited Rs 5000 at 8% per annum compound interest. After two years, the difference of their interest will be =
a) Rs. 230
b) Rs. 232
c) Rs. 832
d) Rs. 600
Discussion
Explanation:
$$\eqalign{ & {\text{Principal (}}{{\text{P}}_1}{\text{) = Rs. 6000}} \cr & {\text{Time (t) = 2 years}} \cr & {\text{Rate % = 5% }} \cr & {\text{Simple interest}} {\text{ = }}\frac{{6000 \times 5 \times 2}}{{100}}{\text{ = Rs. 600}} \cr & {\text{Principal (}}{{\text{P}}_2}{\text{) = Rs. 5000}} \cr & {\text{Time (t) = 2 years}} \cr & {\text{Rate % = 8% }} \cr} $$
2 year effective rate for Compound interest
$$\eqalign{ & = 8 + 8 + \frac{{8 \times 8}}{{100}} = 16.64\% \cr & {\text{Compound}}\,{\text{Interest}} \cr & {\text{ = 5000}} \times \frac{{16.64}}{{100}} = {\text{Rs}}{\text{. 832}} \cr & {\text{Difference}} {\text{ = Rs}}{\text{. }}\left( {832 - 600} \right) \cr & = {\text{Rs}}{\text{.}}\,232 \cr & {\text{ }} \cr} $$
74. A man invests Rs. 5000 for 3 years at 5% p.a. compound interest reckoned yearly. Income tax at the rate of 20% on the interest earned is deducted at the end of each year. Find the amount at the end of the third year = ?
a) Rs. 5624.32
b) Rs. 5627.20
c) Rs. 5630.50
d) Rs. 5788.125
Discussion
Explanation:
$$\eqalign{ & {\text{C}}{\text{.I}}{\text{. earned during }}{{\text{1}}^{{\text{st}}}}{\text{ year}} \cr & {\text{= }}\,{\text{Rs}}{\text{.}}\left[ {5000\left( {1 + \frac{5}{{100}}} \right) - 5000} \right]{\text{ }} \cr & = {\text{Rs}}{\text{. }}\left( {5250 - 5000} \right) \cr & = {\text{Rs}}{\text{. 250}} \cr & {\text{Amount after }}{{\text{1}}^{{\text{st}}}}{\text{ year}} \cr & = {\text{Rs}}{\text{. }}\left( {5250 - 20\% {\text{ of }}250} \right) \cr & = {\text{Rs}}{\text{.}}\left( {5250 - 50} \right){\text{ }} \cr & {\text{= }}\,{\text{Rs}}{\text{.}}\,{\text{5200 }} \cr & {\text{C}}{\text{.I}}{\text{. earned during }}{{\text{2}}^{{\text{nd}}}}{\text{ year}} \cr & = {\text{Rs}}{\text{.}}\left[ {5200\left( {1 + \frac{5}{{100}}} \right) - 5200} \right]{\text{ }} \cr & = {\text{Rs}}{\text{. }}\left( {5460 - 5200} \right) \cr & = {\text{Rs}}{\text{.260 }} \cr & {\text{Amount after }}{{\text{2}}^{{\text{nd}}}}{\text{ year}} \cr & {\text{= Rs}}{\text{. }}\left( {5460 - 20\% {\text{ of }}260} \right) \cr & {\text{= Rs}}{\text{. }}\left( {5460 - 52} \right) \cr & = {\text{Rs}}{\text{.}}\,{\text{5408 }} \cr & {\text{C}}{\text{.I}}{\text{. earned during }}{{\text{3}}^{{\text{rd}}}}{\text{ year}} \cr & {\text{= Rs}}{\text{. }}\left[ {5408\left( {1 + \frac{5}{{100}}} \right) - 5408} \right] \cr & = {\text{Rs}}{\text{. }}\left( {5678.40 - 5408} \right) \cr & = {\text{Rs}}{\text{.}}\,{\text{270}}{\text{.40 }} \cr & {\text{Amount after }}{{\text{3}}^{{\text{rd}}}}{\text{ year}} \cr & = {\text{Rs}}{\text{. }}\left( {5678.40 - 20\% \,{\text{of }}270.40} \right) \cr & = {\text{Rs}}{\text{. }}\left( {5678.40 - 54.08} \right) \cr & = {\text{Rs}}{\text{. 5624}}{\text{.32}} \cr} $$
75. At a certain rate per annum, the simple interest on a sum of money for one year is Rs. 260 and the compound interest on the same sum for two years is Rs. 540.80. The rate of interest per annum is =
a) 4%
b) 6%
c) 8%
d) 10%
Discussion
Explanation:
$$\eqalign{ & {\text{SI for 1 year}} {\text{ = Rs 260}} \cr & {\text{SI for 2 year}} {\text{ = 260}} \times {\text{2}} \cr & {\text{ = Rs}}{\text{. 520 }} \cr & {\text{Difference in (CI}} - {\text{SI)}} \cr & \left( {540.80 - 520} \right){\text{ = Rs 20}}{\text{.8}} \cr & {\text{Required rate % }} \cr & {\text{ = }}\frac{{20.8}}{{260}} \times {\text{100}} \cr & {\text{ = 8% }} \cr} $$
76. In how many years will a sum of Rs. 800 at 10% per annum compounded semi annually become Rs. 926.10?
a) $$1\frac{1}{3}$$ years
b) $$1\frac{1}{2}$$ years
c) $$2\frac{1}{3}$$ years
d) $$2\frac{1}{2}$$ years
Discussion
Explanation:
$$\eqalign{ & {\text{Let the time be }}n{\text{ year}} \cr & {\text{800}} \times {\left( {1 + \frac{5}{{100}}} \right)^{2n}} = 926.10 \cr & {\left( {1 + \frac{5}{{100}}} \right)^{2n}} = \frac{{9261}}{{8000}} \cr & {\left( {\frac{{21}}{{20}}} \right)^{2n}} = {\left( {\frac{{21}}{{20}}} \right)^3} \cr & 2n = 3 \cr & n = \frac{3}{2} \cr & n = 1\frac{1}{2}{\text{years}} \cr} $$
77. A loan of Rs. 12300 at 5% per annum compound interest, is to be repaid in two equal annual installments at the end of every year. Find the amount of each installment ?
a) Rs. 6651
b) Rs. 6615
c) Rs. 6516
d) Rs. 6156
Discussion
Explanation: $$5\% = \frac{1}{{20}} = \frac{{21 \to {\text{ Installment}}}}{{20 \to {\text{ Principal}}}}$$
| Year | Principal | Installment | |
| ⇒ I | 20×21 → | 21×21 | ......(i) |
| ⇒ II | 400→ | 441 | .....(ii) |
Total principal = 420 + 400 = 820
820 units = Rs. 12300
1 units = Rs. 15
441 units = Rs. 6615
Each installment = Rs. 6615
78. An amount of Rs 6000 lent at 5% per annum compounded interest for 2 years will become =
a) Rs. 600
b) Rs. 6600
c) Rs. 6610
d) Rs. 6615
Discussion
Explanation:
$$\eqalign{ & {\text{Amount = 6000}}{\left( {1 + \frac{5}{{100}}} \right)^2} \cr & {\text{Amount = 6000}} \times \frac{{21}}{{20}} \times \frac{{21}}{{20}} \cr & {\text{Amount = Rs. 6615}} \cr} $$
79. The simple interest on a certain sum of money for 3 years at 8% per annum is half the compound interest on Rs. 4000 for 2 years at 10% per annum. The sum placed on simple interest is ?
a) Rs. 1550
b) Rs. 1650
c) Rs. 1750
d) Rs. 2000
Discussion
Explanation:
$$\eqalign{ & {\text{C}}{\text{.I}}{\text{.}} {\text{ = Rs}}{\text{.}}\left[ {4000 \times {{\left( {1 + \frac{{10}}{{100}}} \right)}^2} - 4000} \right] \cr & = {\text{ Rs}}{\text{.}}\left( {4000 \times \frac{{11}}{{10}} \times \frac{{11}}{{10}} - 4000} \right) \cr & = {\text{Rs}}{\text{. 840}} \cr & {\text{Sum = Rs}}{\text{.}}\left( {\frac{{420 \times 100}}{{3 \times 8}}} \right) \cr & = {\text{Rs}}{\text{. }}1750 \cr} $$
80. There is 60% increase in an amount in 6 years at simple interest. What will be the compound interest of Rs. 12000 after 3 years at the same rate ?
a) Rs. 2160
b) Rs. 3120
c) Rs. 3972
d) Rs. 6240
Discussion
Explanation:
$$\eqalign{ & {\text{Let P}} = {\text{Rs}}.100 \cr & {\text{S}}{\text{.I}}{\text{. = Rs}}.60{\text{ and}} \cr & {\text{T = 6 years}} \cr & {\text{R = }}\frac{{100 \times 60}}{{100 \times 6}}{\text{ = 10% p}}{\text{.a}}{\text{.}} \cr & {\text{P = Rs 12000,}} \cr & {\text{T = 3 years and}} \cr & {\text{R = 10% p}}{\text{.a}}{\text{.}} \cr & {\text{C}}{\text{.I}}{\text{. = Rs}}{\text{.}}\left[ {12000 \times \left\{ {{{\left( {1 + \frac{{10}}{{100}}} \right)}^3} - 1} \right\}} \right] \cr & = {\text{Rs}}{\text{.}}\left( {12000 \times \frac{{331}}{{1000}}} \right) \cr & = {\text{Rs}}{\text{. }}3972 \cr} $$
81. A sum of money becomes eight times in 3 years, If the rate is compounded annually. In how much time will the same amount at the same compound rate become sixteen times ?
a) 6 years
b) 4 years
c) 8 years
d) 5 years
Discussion
Explanation:
$$\eqalign{ & {\text{Let principal = P}} \cr & {{Case (I)}} \cr & {\text{Time = 3 years,}} \cr & {\text{Amount = 8P}} \cr & 8{\text{P = P}}{\left( {1 + \frac{{\text{R}}}{{100}}} \right)^3} \cr & {\left( 2 \right)^3} = {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^3} \cr & {\text{Taking cube root of both sides,}} \cr & {\text{2 = }}\left( {1 + \frac{{\text{R}}}{{100}}} \right) \cr & {\text{R = 100 }}\% \cr & {{Case (II)}} \cr & {\text{Let after t years it will be 16 times}} \cr & 16{\text{P = P}}{\left( {1 + \frac{{\text{R}}}{{100}}} \right)^{\text{t}}} \cr & 16 = {\left( 2 \right)^{\text{t}}} \cr & {\left( 2 \right)^4} = {\left( 2 \right)^{\text{t}}} \cr & {\text{t}} = 4 \cr & {\text{Required time}} {\text{(t) = 4 years}} \cr} $$
82. A sum of money placed at compound interest double itself in 4 years. In how many years will it amount to four times itself ?
a) 12 years
b) 13 years
c) 8 years
d) 16 years
Discussion
Explanation:
$$\eqalign{ & {\text{Let}}, {\text{ Principal}} = Rs.\,100\% \cr & {\text{Amount}} = Rs.\,200 \cr & {\text{Rate}} = r\% \cr & {\text{Time}} = 4\,{\text{years}} \cr & A = P \times {\left[ {1 + \left( {\frac{r}{{100}}} \right)} \right]^n} \cr & 200 = 100 \times {\left[ {1 + \left( {\frac{r}{{100}}} \right)} \right]^4} \cr & 2 = {\left[ {1 + \left( {\frac{r}{{100}}} \right)} \right]^4} - - - - \left( i \right) \cr & {\text{If}}\,{\text{sum}}\,{\text{become}}\,{\text{8}}\,{\text{times}}\,{\text{in}}\,{\text{the}}\,{\text{time}}\,n\,{\text{years}} \cr & 4 = {\left( {1 + \left( {\frac{r}{{100}}} \right)} \right)^n} \cr & {2^2} = {\left( {1 + \left( {\frac{r}{{100}}} \right)} \right)^n} - - - - \left( {ii} \right) \cr & {\text{Using}}\,{\text{eqn}}\,\left( i \right)in\left( {ii} \right),\,{\text{we}}\,{\text{get}} \cr & {\left( {{{\left[ {1 + \left( {\frac{r}{{100}}} \right)} \right]}^4}} \right)^2} = {\left( {1 + \left( {\frac{r}{{100}}} \right)} \right)^n} \cr & {\left[ {1 + \left( {\frac{r}{{100}}} \right)} \right]^{8}} = {\left( {1 + \left( {\frac{r}{{100}}} \right)} \right)^n} \cr & n = 8\,{\text{years}}. \cr} $$
83. The compound interest on Rs. 30000 at 7% per annum for a certain time is Rs. 4347. The times is = ?
a) 3 years
b) 4 years
c) 2 years
d) 2.5 years
Discussion
Explanation:
$$\eqalign{ & {\text{Principal = Rs}}{\text{. 30000}} \cr & {\text{CI = Rs 4347}} \cr & {\text{Rate = 7}}\% \cr & {\text{By using formula, }} \cr & \left( {30000 + 4347} \right) = 30000{\left( {1 + \frac{7}{{100}}} \right)^{\text{t}}} \cr & 34347 = 30000{\left( {1 + \frac{7}{{100}}} \right)^{\text{t}}} \cr & \frac{{34347}}{{30000}} = {\left( {\frac{{107}}{{100}}} \right)^{\text{t}}} \cr & \left( {\frac{{11449}}{{10000}}} \right) = {\left( {\frac{{107}}{{100}}} \right)^{\text{t}}} \cr & {\left( {\frac{{107}}{{100}}} \right)^2} = {\left( {\frac{{107}}{{100}}} \right)^{\text{t}}} \cr & {\text{t}} = 2\,{\text{years}} \cr} $$
84. A money lender borrows money at 4% per annum and pays the interest at the end of the year. He lends it at 6% per annum compound interest compounded half yearly and receives the interest at the end of the year. In this way, he gains Rs. 104.50, a year. The amount of money be borrows, is ?
a) Rs. 4500
b) Rs. 5000
c) Rs. 5500
d) Rs. 6000
Discussion
Explanation:
$$\eqalign{ & {\text{Let the sum Rs}}{\text{. }}x{\text{ }} \cr & {\text{C}}{\text{.I}}{\text{. when compounded half yearly}} {\text{ = Rs}}{\text{.}}\left[ {x \times {{\left( {1 + \frac{3}{{100}}} \right)}^2} - x} \right] \cr & = {\text{Rs}}{\text{.}}\left( {\frac{{10609}}{{10000}}x - x} \right) \cr & = {\text{Rs}}{\text{.}}\left( {\frac{{609x}}{{10000}}} \right) \cr & {\text{C}}{\text{.I}}{\text{. when compounded yearly}} {\text{ = Rs}}{\text{.}}\left[ {x \times \left( {1 + \frac{4}{{100}}} \right) - x} \right] \cr & = {\text{Rs}}{\text{.}}\left( {\frac{{26x}}{{25}} - x} \right) \cr & = {\text{Rs}}{\text{.}}\frac{x}{{25}} \cr & \therefore \frac{{609x}}{{10000}} - \frac{x}{{25}} = 104.50 \cr & \frac{{209x}}{{10000}} = 104.50 \cr & x = \left( {\frac{{104.50 \times 10000}}{{209}}} \right) \cr & x = 5000 \cr} $$
85. The effective annual rate of interest corresponding to a nominal rate of 6% per annum payable half yearly is = ?
a) 6.06%
b) 6.07%
c) 6.08%
d) 6.09%
Discussion
Explanation: Amount of Rs. 100 for 1 year when compounded half yearly
$$\eqalign{ & {\text{ = Rs}}{\text{.}}\left[ {100 \times {{\left( {1 + \frac{3}{{100}}} \right)}^2}} \right] \cr & = {\text{Rs}}.106.09 \cr & {\text{Effective rate}} \cr & {\text{ = }}\left( {106.09 - 100} \right)\% \cr & = 6.09\,\% \cr} $$
86. A bank offers 5% compound interest calculated on half-yearly basis. A customer deposits Rs. 1600 each on 1st January and 1st July of a year. At the end of the year, the amount he would have gained by way of interest is:
a) Rs. 120
b) Rs. 121
c) Rs. 122
d) Rs. 123
Discussion
Explanation:
$$\eqalign{ & {\text{Amount}} = {1600 \times {{\left( {1 + \frac{5}{{2 \times 100}}} \right)}^2} + 1600 \times \left( {1 + \frac{5}{{2 \times 100}}} \right)} \cr & = {1600 \times \frac{{41}}{{40}} \times \frac{{41}}{{40}} + 1600 \times \frac{{41}}{{40}}} \cr & = {1600 \times \frac{{41}}{{40}}\left( {\frac{{41}}{{40}} + 1} \right)} \cr & = {\frac{{1600 \times 41 \times 81}}{{40 \times 40}}} \cr & = Rs.\,3321 \cr & C.I. = Rs.\,\left( {3321 - 3200} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,121 \cr} $$
87. The difference between simple and compound interests compounded annually on a certain sum of money for 2 years at 4% per annum is Rs. 1. The sum (in Rs.) is:
a) 625
b) 630
c) 640
d) 650
Discussion
Explanation:
$$\eqalign{ & {\text{Let}}\,{\text{the}}\,{\text{sum}}\,{\text{be}}\,Rs.\,x.\,{\text{Then}}, \cr & {\text{C}}{\text{.I}}{\text{.}} = {x{{\left( {1 + \frac{4}{{100}}} \right)}^2} - x} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\frac{{676}}{{625}}x - x} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{51}}{{625}}x \cr & {\text{S}}{\text{.I}}{\text{.}} = {\frac{{x \times 4 \times 2}}{{100}}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{2x}}{{25}} \cr & \therefore \frac{{51x}}{{625}} - \frac{{2x}}{{25}} = 1 \cr & x = 625 \cr} $$
88. There is 60% increase in an amount in 6 years at simple interest. What will be the compound interest of Rs. 12,000 after 3 years at the same rate?
a) Rs. 2160
b) Rs. 3120
c) Rs. 3972
d) Rs. 6240
Discussion
Explanation:
$$\eqalign{ & {\text{Let}}\,{\text{P = Rs}}{\text{.}}\,{\text{100}}\,{\text{Then}},\, \cr & \,\,\,\,\,{\text{S}}{\text{.I}}{\text{. = }}\,{\text{Rs}}{\text{.}}\,{\text{60}}\,{\text{and}} \cr & \,\,\,\,\,\,\,\,{\text{T = 6}}\,{\text{years}} \cr & R = {\frac{{100 \times 60}}{{100 \times 6}}} = 10\% \,p.a. \cr & {\text{Now}},\,P = Rs.\,12000 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,T = 3\,{\text{year}}\,{\text{and}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,R = \,10\% \,p.a. \cr & {\text{C}}{\text{.I}}{\text{.}} = Rs.\,\left[ {12000 \times \left\{ {{{\left( {1 + \frac{{10}}{{100}}} \right)}^3} - 1} \right\}} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,\left( {12000 \times \frac{{331}}{{1000}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,3972 \cr} $$
89. What is the difference between the compound interests on Rs. 5000 for $$1\frac{1}{2}$$ years at 4% per annum compounded yearly and half-yearly?
a) Rs. 2.04
b) Rs. 3.06
c) Rs. 4.80
d) Rs. 8.30
Discussion
Explanation:
$$\eqalign{ & {\text{C}}{\text{.I}}{\text{.}}\,{\text{when}}\,{\text{interest}}\,{\text{compounded}}\,{\text{yearly}} = Rs.\left[ {5000 \times \left( {1 + \frac{4}{{100}}} \right) \times \left( {1 + \frac{{\frac{1}{2} \times 4}}{{100}}} \right)} \right] \cr & = Rs.\left( {5000 \times \frac{{26}}{{25}} \times \frac{{51}}{{50}}} \right) \cr & = Rs.5304 \cr & {\text{C}}{\text{.I}}{\text{.}}\,{\text{when}}\,{\text{interest}}\,{\text{in}}\,{\text{compounded}}\,{\text{half - yearly}} = Rs.\,\left[ {5000 \times {{\left( {1 + \frac{2}{{100}}} \right)}^3}} \right] \cr & = Rs.\,\left( {5000 \times \frac{{51}}{{50}} \times \frac{{51}}{{50}} \times \frac{{51}}{{50}}} \right) \cr & = Rs.\,5306.04 \cr & {\text{Difference}} = Rs.\,\left( {5306.04 - 5304} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,2.04 \cr} $$
90. The compound interest on Rs. 30,000 at 7% per annum is Rs. 4347. The period (in years) is:
a) 2
b) $$2\frac{1}{2}$$
c) 3
d) 4
Discussion
Explanation:
$$\eqalign{ & {\text{Amount}} = Rs.\,\left( {30000 + 4347} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Rs.\,34347 \cr & {\text{Let}}\,{\text{the}}\,{\text{time}}\,{\text{be}}\,n\,{\text{years}} \cr & 30000\,{\left( {1 + \frac{7}{{100}}} \right)^n} = 34347 \cr & \Rightarrow {\left( {\frac{{107}}{{100}}} \right)^n} = \frac{{34347}}{{30000}} = \frac{{11449}}{{10000}} = {\left( {\frac{{107}}{{100}}} \right)^2} \cr & n = 2\,{\text{years}} \cr} $$
91. 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%
Discussion
Explanation:
$$\eqalign{ & {\text{Principal}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Amount}} \cr & \,\,\,\,\,\,\,\,\,{\text{1}}\,\,\,\,\,\,\,\,\,\,\,\, \to \,\,\,\,\,\,\,\,\,\,\,\,{\text{4}} \cr & 4 = 1{\left( {1 + \frac{r}{{100}}} \right)^2} \cr & 4 = {\left( {1 + \frac{r}{{100}}} \right)^2} \cr & r = 100\% \cr} $$
92. 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
Discussion
Explanation:
$$\eqalign{ & {\text{S}}{\text{.I}}{\text{. = Rs}}{\text{.}}\left( {\frac{{19200 \times 12 \times 3}}{{100}}} \right) \cr & \,\,\,\,\,\,\,\,\, = {\text{Rs}}{\text{.6912}} \cr & {\text{C}}{\text{.I}}{\text{. = Rs}}{\text{.}}\left[ {19200 \times {{\left( {1 + \frac{{12}}{{100}}} \right)}^3} - 19200} \right] \cr & = {\text{Rs}}{\text{.}}\left[ {\left( {19200 \times \frac{{28}}{{25}} \times \frac{{28}}{{25}} \times \frac{{28}}{{25}}} \right) - 19200} \right] \cr & = {\text{Rs}}{\text{. }}\left( {\frac{{16859136}}{{625}} - 19200} \right) \cr & = {\text{Rs}}{\text{. }}\left( {26974.6176 - 19200} \right) \cr & = {\text{Rs}}{\text{. 7774}}{\text{.6176}} \cr & {\text{Difference }} {\text{ = Rs}}{\text{.}}\left( {7774.6176 - 6912} \right) \cr & = {\text{Rs}}{\text{. 862}}{\text{.6176}} \cr} $$
93. 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
Discussion
Explanation: Rate % = 10%,
Time = 1 year
Case (I) : When interest is calculated yearly, Rate = 10%
Case (II) : When interest is calculated half yearly
$$\eqalign{ & \Rightarrow {\text{New rate }}\% = \frac{{10}}{2} = 5\% \cr & {\text{Time = 1}} \times {\text{2}} = {\text{2 years}} \cr & {\text{Effective rate}}\% \cr & {\text{ = 5 + 5 + }}\frac{{5 \times 5}}{{100}} = 10.25\% \cr & {\text{Difference in rates}} \cr & {\text{ = }}\left( {10.25 - 10} \right)\% = 0.25\% \cr & {\text{0}}{\text{.25% of sum = Rs 180}} \cr & {\text{Sum = }}\frac{{180}}{{0.25}} \times 100 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\text{Rs. }}72000 \cr} $$
94. 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
Discussion
Explanation:
$$\eqalign{ & {\text{Let the rate be R}}\% {\text{ p}}{\text{.a}}{\text{. }} \cr & {\text{25500}}{\left( {1 + \frac{{\text{R}}}{{100}}} \right)^3} \cr & = 25500 + 8440.50 \cr & = 33940.50 \cr} $$
$$ \Rightarrow {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^3} = \frac{{33940.50}}{{25500}} = $$ $$\frac{{1331}}{{1000}} = $$ $${\left( {\frac{{11}}{{10}}} \right)^3}$$
$$\eqalign{ & 1 + \frac{{\text{R}}}{{100}} = \frac{{11}}{{10}} \cr & \frac{{\text{R}}}{{100}} = \frac{1}{{10}} \cr & {\text{R}} = 10\,\% \cr & S.I. = {\text{R}}s.\left( {\frac{{25500 \times 10 \times 3}}{{100}}} \right) \cr & = {\text{Rs}}{\text{.}}\,7650 \cr} $$
95. 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
Discussion
Explanation:
Let the R% p.a.
$$\left[ {26000 \times {{\left( {1 + \frac{{\text{R}}}{{100}}} \right)}^3} - 26000} \right] - $$ $$\left( {\frac{{26000 \times {\text{R}} \times 3}}{{100}}} \right) = $$ $$2994.134$$
$$ 26000\left[ {{{\left( {1 + \frac{{\text{R}}}{{100}}} \right)}^3} - 1 - \frac{{3{\text{R}}}}{{100}}} \right] = $$ $$2994.134$$
$$ 26000$$ $$\left[ {\frac{{{{\left( {100 - {\text{R}}} \right)}^3} - 1000000 - 30000{\text{R}}}}{{1000000}}} \right] = $$ $$2994.134$$
$$ 26\left[ {\left\{ {1000000 + {{\text{R}}^3} + 300{\text{R}}\left( {100 + {\text{R}}} \right) - 1000000 - 30000{\text{R}}} \right\}} \right] = 2994134$$
$$ {{\text{R}}^3} + 300{{\text{R}}^2} = \frac{{2994134}}{{26}} = $$ $$115159$$
$$ {{\text{R}}^2}\left( {{\text{R}} + 300} \right) = 115159$$
$$ {\text{R = 19}}\% $$
96. 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
Discussion
Explanation:
$$\eqalign{ & {\text{Rate }}\% {\text{ = 4}}\% \cr & {\text{Time (}}{{\text{t}}_1}) = 2\,{\text{years}} \cr & {\text{SI for 2 years}} \cr & {\text{ = 4}} \times {\text{2 = 8}}\% \cr & {\text{CI for 2 years}} \cr & {\text{ = 4 + 4 + }}\frac{{4 \times 4}}{{100}} \cr & = 8.16\% \cr & \operatorname{Required} \,CI = \frac{{80}}{8} \times 8.16 \cr & = Rs.\,81.60 \cr} $$
97. 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
Discussion
Explanation:
$$\eqalign{ & {\text{Amount = Rs}}{\text{. }}\left( {30000 - 4347} \right) \cr & {\text{Amount = Rs}}{\text{. }} 34347 \cr & {\text{Let the time be }}n{\text{ years}} \cr & {\text{30000}}{\left( {1 + \frac{7}{{100}}} \right)^n} = 34347 \cr & {\left( {\frac{{107}}{{100}}} \right)^n} = \frac{{34347}}{{30000}} \cr & {\left( {\frac{{107}}{{100}}} \right)^n} = \frac{{11449}}{{10000}} = {\left( {\frac{{107}}{{100}}} \right)^2} \cr & n = {\text{ 2 years}} \cr} $$
98. 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
Discussion
Explanation:
$$\eqalign{ & {\text{Effective rate of CI for 2 years}} \cr & {\text{= 5 + 5 + }}\frac{{5 \times 5}}{{100}} \cr & = 10.25\% \cr & {\text{Effective rate of SI for 3 years}} \cr & {\text{= 6}} \times {\text{3 = 18% }} \cr & {\text{Required SI}} {\text{= }}\frac{{246}}{{10.25}} \times 18 \cr & = {\text{Rs. 432}} \cr} $$
99. 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
Discussion
Explanation:
$$\eqalign{ & P\left[ {{{\left( {\frac{{21}}{{20}}} \right)}^3} - 1 - \frac{3}{{20}}} \right] = 122 \cr & P\left[ {\frac{{{{21}^3} - {{20}^3} - 3 \times {{20}^2}}}{{{{20}^3}}}} \right] = 122 \cr & P\left[ {\frac{{9261 - 8000 - 1200}}{{8000}}} \right] = 122 \cr & P \times \frac{{61}}{{8000}} = 122 \cr & P = \frac{{8000 \times 122}}{{61}} \cr & P = {\text{Rs}}{\text{.}}\,16000 \cr} $$
100. 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%
Discussion
Explanation:
$$\eqalign{ & {\text{Let the rate be R}}\% {\text{ p}}{\text{.a}}{\text{.}} \cr & {\text{2000}}{\left( {1 + \frac{{\text{R}}}{{100}}} \right)^2} = 2226.05 \cr & {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^2} = \frac{{222605}}{{200000}} \cr & {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^2} = \frac{{44521}}{{40000}} \cr & {\left( {1 + \frac{{\text{R}}}{{100}}} \right)^2} = {\left( {\frac{{221}}{{200}}} \right)^2} \cr & 1 + \frac{{\text{R}}}{{100}} = \frac{{211}}{{200}} \cr & \frac{{\text{R}}}{{100}} = \frac{{11}}{{200}} \cr & {\text{R}} = \frac{{11}}{2}\% \cr & {\text{R}} = 5.5\% \cr} $$