281. 4th case. Ex. 1. What principal will produce £121. 15s. 5d. in 2 yrs. I mo., the interest at 53 per cent.? .5 interest is gained in 12 mo. by 100, 100 x 12 Ex. 2. What principal, lent from the 24th of March, 1852, till the 17th November, 1853, at 4 per cent. per annum, will gain £1200? From March 24th, 1852, to November 17th, 1853, are 1338 yrs., or 603 days. 4 is gained in 365 days by 100, ..1 is gained in 1 day by 100 × 365 4/ 400 £1200 is gained in 603 days by 1200×100 × 365 = £16141. 10s. 3 d. nearly. 603×41 201 282. EXERCISES. 1. What is the interest of £862. 14s. 2d. for 5 years, at 4 per cent.? 2. Find the interest of £535 for 117 days, at 43 per cent. 3. What principal will amount to £1292. 6s. 41d. in 93 years, at 53 per cent. per annum? 4. In what time will £350 amount to £402. 10s., at 3 per cent. per annum? 5. At what rate per cent. will £1500 amount to £1850 in 41 years? 6. Find the amount of £1440. 15s. for 1 year 73 days, at 5 per cent. per annum. 7. In what time will any principal double itself, at 4 per cent. per annum ? 8. If the interest on £261. 11s. 8d. for 15 days be 8s. 8d., what is the rate per annum ? 9. At whatrate per cent. per annum will any principal triple itself in 20 years? 10. Find the amount of £4000 for 3 years 10 weeks, at 3 per cent. per annum? 11. Find the interest of £3996. 15s. for 4 years 225 days, at 3 per cent. per annum. 12. What principal will bring £1312. 13s. interest in 27 months, at 1 per cent. per month? 13. Borrowed, on March 12th, 1849, the sum of £2456. 12s. 6d., at 5 per cent.; on February the 2nd, 1851, I paid on account £932. 15s. 8d.; on August 13th, 1852, I paid £442. 18s. 6d. ; and on May 18th, 1854, I shall pay £925. 10s. 9d. What will be the balance for me to pay on the 31st of December, 1855 ? 14. At what rate per cent. will £827. 10s. amount to £986. 15s. 10 d. in 5 years? 15. What principal will amount to £1383. 15s. 11d. in 5 years, at 33 per cent. per annum ? 16. Find the interest and amount of £1000 for 2 yrs. 9 mo. 25 days, at 4 per cent. per annum. 17. The interest of £319. 6s. for 5 years was £68. 14s. 91d. Find the rate per cent. per annum ? 18. Required, the simple interest on £1064. 18s. 6d., from May 1st to October 8th, at 3 per cent. per annum ? 19. How long will £670 be in amounting to £840. 10s., at 61 per cent. per annum ? 20. A's income is derived from two equal principals, one being lent at 4, and the other at 5 per cent. per annum. If his annual income be £650, what is the value of each principal ? ་ of the principal is at at 4 per cent. per 21. A persons daily income is £3; 3 per cent., at 3 per cent., annum. How much is each principal, and also the whole fortune? COMPOUND INTEREST. 283. When the interest is added yearly to the principal, and we consider the sum accruing as a new principal, the interest is compound. Ex. Required, the compound interest of £1200, at 5 per cent. per annum, for 4 years 6 months. There are several methods of solving questions of this kind. 1st method: Find the interest, year after year, adding it each time to the last principal. For instance, the interest of 1st year's principal, or £1200, is £1200 × 5 100 = £60. = .. the interest of 2nd year's principal, or £1260, is = £63. .. the interest of 3rd year's principal, or £1323, is = £66.15 £1260 20 £1323 20 .. the interest of 4th year's principal, or £1389.15, is the interest of next year's principal, or £1458.6075 is £1458.6075 = £36.4652. 20 × 2 ..the amount=£1458.6075+36.4652+£1495.0727. Hence C.I.= £1495.0727 - £1200 £295.0727= £295. 1s. 51d. nearly. 2nd method: Find the amount of £1 for the given time, and multiply it by the given principal. · 100 bring 5, .. 1 brings .05. Then, at the end of 1st year, £1 amounts to £1.05 ; 2nd year, £1 amounts to 1.05 ×.05+1.05=1.1025; the next year £1 amounts to 1.245894. And .£1 is worth, at the end of 4 years, £1.245894, .. £1200 is worth, at the end of 4 years 1200 × £1.245894, or £1495.0728. To avoid the tediousness of this process, tables have been constructed on this principle, giving the amount of £1 for different numbers of years, at different rates of interest. The following concise table may be found useful: TABLE OF THE AMOUNT OF £1 FOR ANY NUMBER OF YEARS 1.092025 1.1025 1.1411661 1.157625 3 1.092727 1.1087179 1.124864 4 1.1255088 1.1475230 1.1698586 1.1925186 1.2155062 5 1.1592741 1.1876863 1.2166529 1.24618191.2762816 61.1940523 1.2292553 1.2653190 1.3022601 1.3400956 7 1.2298739 1.2722793 1.3159318 1.3608618 1.4071004 81.2667701 1.3168090 1.3685690 1.4221006 1.4774554 91.3047732 1.3628974 1.4233118 1.4860951 1.5513282 10 1.3439164 1.4105988 1.4802443 1.5529694 1.6288946 11 1.38423391.45996971.5394541 1.6228530 1.7103394 12 1.4257609 1.5110687 1.6010322 1.6958814 1.7958563 13 1.4685337 1.5639561 1.6650735 1.7721961 1.8856491 14 1.5125897 1.6186945 1.7316764 1.8519449 1.9799316 15 1.5579674 1.6753488 1.8009435 1.9352824 2.0789282 16 1.6047064 1.7339860 1.8729812 2.0223702 2.1828746 17 1.65284761.79467561.94790052.1133768 2.2920183 18 1.7024331 1.8574892 2.0258165 2.2084788 2.4066192 19 1.7535060 1.9225013 2.1068492 2.3078603 2.5269502 20 1.8061112 1.9897889 2.1911231 2.4117140 2.6532977 3rd method. 100 are worth in 1 year 105, ..£1200 are worth in 1 year at the end of 1st year. 1200 × £105 100 amount of £1200 =amount of £1200 at the end of 3rd year. 1200 × 105 × 105 × £105 Likewise, .. 100 x 100 x 100 1200 x 105 x 105 x 105 x £105 100 x 100 x 100 x 100 4th year. And.. 21 100 x 100 x 100 are worth in 1 year = amount of £1200 at the end of 1200 × 105 x 105 × 105 × £105 are worth in 1 year 100 x 100 x 100 × 100 21 21 21 4.1 20.3 1200 × 103 × 103 × 193 × 103 × £107.5 = amount of £1200 at the 100 × 100 × 100 × 100 × 100 20 1 20 1 end of the next year. Performing the work, we find as before, £1495.0727. Hence, we must multiply the principal by the amount of 100 for 1 year as many times as there are years in the question, and divide that product by 100, multiplied by itself as many times as there are years. The quotient will be the amount. 4th method depends upon the raising of quantities to different powers, and for that reason we postpone the explanation of it to a later part of the work. Ex. 2. Determine the amount of £725 in 4 years, at 5 per cent. C.I. |