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R 14 Veto follow us: facebook.com/vetotrivandrum 14 compound interest hnPbIpamÀ imkvXmwtIm« Important formulae Principal - Rs.P Rate - r% per annum Time period - t years Amount - Rs.A I. When interest is compouned annually, II. When interest is compounded half-yearly III. When interest is compounded quarterly IV. When interest is compouned monthly t r A= P ⎡ ⎢ 1+ ⎤ ⎣ 100 ⎥ ⎦ 2t r A= P ⎡ ⎢ 1+ ⎤ ⎣ 200 ⎥ ⎦ 4t r 1 400 A= P ⎡ ⎢ + ⎤ ⎣ ⎥ ⎦ 12t r 1 1200 A= P ⎡ ⎢ + ⎤ ⎣ ⎥ ⎦ Time → 1 Year 2 Year 3 Year 4 Year Rate ↓ 4% 5% 10% 20% ⎡ r ⎤ ⎢ 1+ ⎣ 100 ⎥ ⎦ 26 25 21 20 11 10 6 5 ⎡ r ⎤ ⎢ 1+ ⎣ 100 ⎥ ⎦ 676 625 441 400 ü ü 36 25 1. Find compound interest on Rs.2000 at 5% per annum compounded yearly, for 2 years? (a) 105 (b) 175 (c) 200 (d) 205 Ans: (d) 205 2 ⎡ r ⎤ ⎢ 1+ ⎣ 100 ⎥ ⎦ 17576 15625 9261 8000 1331 1000 216 125 3 ⎡ r ⎤ ⎢ 1+ ⎣ 100 ⎥ ⎦ 4 456976 390625 194481 160000 14641 10000 1296 625 Solution: CI = A - P t r P ⎡ ⎛ ⎞ = ⎢ 1 1 ⎤ ⎜ + ⎟ − ⎥ ⎢⎣⎝ 100 ⎠ ⎥⎦ Here P = 2000 r = 5% t = 2 years 2 ⎡⎛ 5 ⎞ ⎤ CI = 2000 ⎢⎜1+ ⎟ −1⎥ ⎢⎣⎝ 100 ⎠ ⎥⎦ ⎡441 ⎤ = 2000 ⎢ −1 ⎣400 ⎥ ⎦ 41 = 2000× 400 = 205 2. Find the compound interest on Rs.5000 for 2 years at 10% per annum, compounded half-yearly? (a) 6077.5 ( b ) 5 0 0 0 (c) 1077.5 (d) 1000 Ans: (c) 1077.5 Solution: CI = 6077.5 - 5000 = 1077.5 3. Find the compound interest on Rs.1000 at 40% per annum, compounded quarterly for 1 year? (a) 1464.1 ( b ) 1 4 0 0 (c) 14641 (d) 464.1 Ans: (d) 464.1 2t r 1 200 A= P ⎡ ⎢ + ⎤ ⎣ ⎥ ⎦ Here P = 5000 r = 10% t = 2 years 4 ⎡ 10 ⎤ A = 5000 ⎢ 1+ ⎣ 200 ⎥ ⎦ 4 ⎡ 5 ⎤ = 5000 ⎢ 1+ ⎣ 100 ⎥ ⎦ 194481 = 5000× 160000 194481 = 32 = 6077.5 Solution: 4t r A= P ⎡ ⎢ 1+ ⎤ ⎣ 100 ⎥ ⎦ Here P = 1000 r = 40% t = 1 years ⎡ 40 ⎤ A = 1000 ⎢ 1+ ⎣ 100 ⎥ ⎦ ⎡ 10 ⎤ = 1000 ⎢ 1+ ⎣ 100 ⎥ ⎦ 14641 = 1000× 10000 = 1464.1 CI = 1464.1 - 1000 = 464.1 4 4 4. Find the compound interest on Rs.4000 at 24% per annum for 3 months, compounded monthly? (a) 244.83 (b) 4244.83 (c) 4000 (d) None of these Ans: (a) 244.83 Solution: 12t r A= P ⎡ ⎢ 1+ ⎤ ⎣ 1200 ⎥ ⎦ Here P = 4000 r = 24% 1 t = 3 months = 4 years ⎡ ⎢ ⎣ 24 A = 4000 1 + 1200 ⎡ ⎢ ⎣ 1 = 4000 1 + 50 CI = 4244.83 - 4000 = 244.83 5. Find the present worth of Rs.9261 due 3 years hence at 5% per annum compounded yearly? (a) 9261 (b) 8000 (c) 9000 (d) 10000 Ans: (b) 8000 ⎤ ⎥ ⎦ 12× 1 4 ⎤ ⎥ ⎦ Short-cut Present worth of Rs. A due n years hence is given by: A n Present worth = ⎛ R ⎞ ⎜1+ ⎟ ⎝ ü ⎠ Solution: A P = n ⎛ R ⎞ ⎜1+ ⎟ ⎝ 100 ⎠ Here A = 9261 R = 5% n = 3 years 9261 P = 5 3 ⎛ ⎞ ⎜1+ ⎟ ⎝ 100 ⎠ 9261 = 9261 8000 = 8000 3 51 51 51 = 4000 × × × 50 50 50 530604 = 125 = 4244.83
R 14 Veto follow us: facebook.com/vetotrivandrum 15 6. A certain sum of money doubles itself in 5 years. In how many years will it become 8 times at the same rate of compound interest? (a) 15 years (c) 5 years Ans: (a) 15 years (b) 10 years (d) 3 years Short-cut If a sum becomes x times in y years at CI then it will be x (n) times in ny years Solution: 2 times in 5 years 8 times 2 (3) in 3 x 5 = 15 years 7. A sum of money tribles in 4 years. In how many years will it amount to 9 times at the same rate of compound interest? (a) 2 years (c) 8 years Ans: (c) 8 years Solution: 3 times in 4 years (b) 4 years (d) 12 years 9 times 3 (2) in 4 x 2 = 8 years 8. At what rate percent compound interest does a sum of money become nine-fold in 2 years? (a) 50% (b) 100% (c) 15% (d) 200% Ans: (d) 200% Short-cut If a certain sum becomes ‘m’ times in ‘t’ years, the rate of compound interest ‘r’ is equal to 100 ⎡( ) 1 t m −1⎤ ⎣ ⎦ Solution: Rate % = ( ) 1 2 100 ⎡ 9 −1⎤ ⎣ ⎦ = 100[3-1] = 100 x 2 = 200% 9. At what rate percentage (compound interest) will a sum of money become 8 times in 3 years? (a) 100% (b) 200% (c) 50% (d) 150% Ans: (a) 100% Solution: Rate % = ( ) 1 3 100 ⎡ 8 −1⎤ ⎣ ⎦ = 100[2-1] = 100% 10. Find the difference between the compound interest and the simple interest for the sum of Rs.1500 at 10% per annum for 2 years? (a) 1000 (b) 100 (c) 50 (d) 15 Ans: (d) 15 Short-cut The difference between SI and CI on a certain sum of money for 2 2 years is PR 2 100 Solution: P=1500 R = 10% 2 PR 1500× 10× 10 = = 2 15 100 100× 100 11. The difference between the simple and the compound interest on a certain sum of money for 2 years at 4% per annum is Rs.1. Find the sum? (a) 625 (b) 125 (c) 525 (d) None of these Ans: (a) 625 Solution: 2 PR = 1 2 100 P× 4× 4 = 1 100× 100 100× 100 P = = 625 4× 4 12. Find the difference between CI and SI on Rs.8000 for 3 years at 2.5% per annum? (a) 12.31 (b) 10.25 (c) 15.125 (d) 20.38 Ans: (c) 15.125 Short-cut Difference between CI and SI for 2 3 year is PR ⎡ R ⎤ 3 2 100 ⎢ + ⎣ 100⎥ ⎦ Solution: Here P = 8000 R = 2.5% Difference between SI and CI 2 PR ⎡ R ⎤ = 3 2 100 ⎢ + ⎣ 100⎥ ⎦ 8000× 2.5× 2.5 ⎡ 2.5 ⎤ = 3+ 100× 100 ⎢ ⎣ 100⎥ ⎦ 8000× 2.5× 2.5× 302.5 = 100× 100× 100 8× 25× 25× 3025 = 100× 100× 100 121 = 8 = 15.125 13. A certain sum of money invested at compound interest, compounded annually, becomes Rs.8820 in 2 years and Rs.9261 in 3 years. Find the rate of interest and the sum? (a) 5%, 4000 (b) 5%, 8000 (c) 10%, 4000 (d) 10%, 8000 Ans: (b) 5%, 8000 Short-cut A certain sum of money invested at compound interest, compounded yearly, becomes A1 in n years and Rs.A2 in (n+1) years then 100 Rate % = ⎛ A ⎞ 1 P = A1 ⎜ ⎟ ⎝ A 2 ⎠ Solution: n ( A − A ) A 2 1 1 % ( ) 100 9261 8820 Rate% = P = − 8820 × 8820 n ⎛ A ⎞ 1 A1 ⎜ ⎟ ⎝ A 2 ⎠ 2 100 441 = = 5% ⎛8820 ⎞ = 8820⎜ ⎟ ⎝ 9261 ⎠ 2 ⎛20 ⎞ = 8820× ⎜ ⎟ ⎝ 21 ⎠ = 8000 POINTS TO REMEMBER 1. Average of first ‘n’ natural numbers ⎛n+ 1⎞ = ⎜ ⎟ ⎝ 2 ⎠ 2. Average of first ‘n’ even numbers = n+1 3. Average of first ‘n’ odd numbers = n 4. Average of cons-ecutive numbers Revision Laws of exponents a xa = a m n m+n m a n a = a m n (a ) = a 1 m-n mn -n a = a n 0 a = 1 (ab) = a xb ⎛a⎞ ⎜ ⎟ ⎝b⎠ ⎛1 ⎞ ⎜ ⎟ ⎝a ⎠ n n n n -m = NOTE a b n n = a m POWERS OF 10 10 0 = One 10 1 = Ten 10 2 = Hundred 10 3 = Thousand 10 4 = Myriad 10 6 = Million 10 9 = Billion 10 12 = Trillion 10 15 = Quadrillion 10 18 = Quintillion 10 21 = Sextillion First No. + Last No. = 2
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