254. To find the principal, when the interest, rate per cent., and time, are given. Make the interest of $1 for the given time and rate, the numerator, and 100 the denominator of a common fraction ; then divide the given interest by this fraction ; (Art. 141 ;) and the quotient will be the principal required. Or, simply divide the given interest by the interest of $1 for the given time and rate expressed in decimals ; and the quotient will be the principal. 12. What sum put at interest will produce $13.30 in 6 months, at 7 per cent. ? Operation. 'The int. of $1 for 6 mo. at $.035)$13.300 7 per cent. is $.035. (Art. 380. Ans. $380.239.) PROOF.-$380 x .035=$13.30. (Art. 247.) 13. A father bequeaths his son $500 a year: what sum must be invested, at 5 per cent., to produce it? 14. What sum must be put at 6 per cent. interest, to give $350 interest semi-annually? 15. A gentleman retiring from business, loaned his money at 7 per cent., and received an income of $1200 a year : how much was he worth? PROBLEM IV. 16. A man loaned $80 at 5 per cent., and received $10 interest: how long was it loaned ? Note. In this example it will be observed that the principal, the interest, and the rate per cent., are given, to find the time. Analysis.—The interest of $80 at 5 per cent. for 1 year is $4. (Art. 237.) Now, since $4 interest requires the given principal 1 year at the given per cent., $10 inter Quest.--254. When the interest, rate per cent., and time, are given, how is the principal found ? est will require the same principal of 1 year, which is equal to 25 years. (Art. 121.) Or, we may reason thus : If $4 interest requires the use of the given principal 1 year, $10 interest will require the same principal as many years as $4 is contained times in $10. And $10-$4=2.5. Ans. 2.5 years. ProOF.-$80 x .05= $4, the interest for 1 year; $4 x 24=$10, the given interest. Hence, 255. To find the time, when the principal, interest, and rate per cent., are given. Make the given interest the numerator, and the interest of the given principal for 1 year at the given rate the denominator of a common fraction, which being reduced to a whole or mixed number, will give the time required. Or, simply divide the given interest by the interest of the given principal at the given rate for 1 year, and the quotient will be the time. Obs. If the quotient contains a decimal of a year, it should be reduced to months and days. (Art. 166.) 17. How long will it take $100, at 5 per cent., to double itself; that is, to gain $100 interest ? Operation. The interest of $100 for 1 year, at 5 per $5)$100 cent., is $5. (Art. 237.) 20 Ans. 20 years. Proof.—$100 x.05 x 20=$100. (Art. 238.) 18. In what time will $500, at 6 per cent., produce $100 interest? 19. How long will it take $100, at 6 per cent., to double itself? 20. How long will it take $100, at 7 per cent., to double itself? Quest.--255. When the principal, interest, and rate per cent., are given, how is the time found? Obs. When the quotient contains a decimal of a year, what should be done with it? TABLE, Showing in what time any given principal will double itself at any rate, from 1 to 20 per cent. Simple Interest. 256. Compound Interest is the interest arising not only from the principal, but also from the interest itself, after it becomes due. OBs. 1. Compound Interest is often called interest upon interest. 2. When the interest is paid on the principal only, it is called Simple Interest. Ex. 1. What is the compound interest of $500 for 3 years, at 6 per cent. ? Operation. $500 principal. 530 Amt. for 1 year. 561.80 Amt. for 2 years. $561.80 x .06= 33.70 Int. for 3d year. $595.50 Amt. for 3 years. 500.00 Prin. deducted. Ans. $95.50 compound Int. for 3 years. QUEST.--256. From what does compound interest arise ? Obs. What is compound interest often called ? What is Interest? 257, Hence, to calculate compound interest. Cast the interest on the. given principal for 1 year, or the specified time, and add it to the principal; then cast the interest on this amount for the next year, or specified time, and add it to the principal as before. Proceed in this manner with each successive year of the proposed time. Finally, subtract the given principal from the last amount, and the remainder will be the compound interest. 2. What is the compound interest of $350 for 4 years, at 6 per cent. ? 3. What is the compound interest of $865 for 5 years, at 7 per cent. ? 4. What is the amount of $250 for 6 years, at 5 per cent. compound interest ? 5. What is the amount of $1000 for 3 years, at 4 per cent. compound interest, payable semi-annually ? 6. What is the amount of $1200 for 2 years, at 6 per cent. compound interest, payable quarterly ? 7. What is the amount of $800 for 3 years, at 5 per cent. compound interest, payable semi-annually ? 8. What is the amount of $1500 for 5 years, at 7 per cent. compound interest ? 9. What is the amount of $2000 for 2 years, at 3 per cent. compound interest, payable quarterly? 10. What is the amount of $3500 for 6 years, at 6 per cent. compound interest ? Note.—This and the next two examples may be solved either by the rule, or by the Table below. 11. What is the amount of $1860 for 8 years, at 7 per cent. compound interest ? 12. What is the amount of $20000 for 10 years, at 8 per cent. compound interest ? QUEST.--257. How is compound interest calculated ? TABLE, Showing the Amount of $1, or £1, at 3, 4, 5, 6, and 7 per cent., compound interest, for any number of years, from 1 to 35. 13. Yrs. (3 per cent. I 4 per cent. I 5 per cent. 6 per cent. | 7 per cent. 1. 1.030,000 1.040,000 / 1.050,000 1.060,000 1.07,000 2. 1.060,900 | 1.081,600 1.102,500 | 1.123,600 1.14,490 3. 1.092,727 | 1.124,864 | 1.157,625 | 1.191,016 1.22,504 4. 1.125,509 1.169,859 1.215,506 1.262,477 1.31,079 5. 1.159,274 1.216,653 1.276,282 1.338,226 | 1.40,255 6. 1.194,052 1.265,319 1.340,096 | 1.418,519 | 1.50,073 7. 1.229,874 | 1.315,932 1.407,100 1.503,630 1.60,578 8. 1.266,770 1.368,569 1.477,455 1.593,848 1.71,818 9. 1.304,773 1.423,312 1.551,328 1.689,479 1.83,845 10. 1.343,916 1.480,244 1.628,895 1.790,848 | 1.96,715 11. 1.384,234 1.539,454 1.710,339 1.898,299 2.10,485 12. 1.425,761 1.601,032 ) 1.795,856 2.012,196 2.25,219 1.468,534 1.665,074 1.885,649 2.132,928 | 2.40,984 14. 1.512,590 1.731,676 | 1.979,932 2.260,904 | 2.57,853 15. 1.557,967 1.800,944 | 2.078,928 2.396,558 | 2.75,903 16. / 1.604,706 1.872,981 2.182,875 2.540,352 2.95,216 17. 1.652,848 1947,900 2.292,018 2.692,773 3.15,881 18. 1.702,433 2.025,817 | 2.406,619 2.854,339 3.37,293 1.753,506 | 2.106,849 2.526,950 3.025,600 3.61,652 20. 1.806,111 2.191,123 2.653,298 3.207,135 3.86,968 21. | 1.860,295 2.278,768 | 2.785,963 3.399,564 4.14,056 22. 1.916,103 2.369,919 2.925,261 3.603,537 4.43,040 23. 1.973,587 2.464,716 3.071,524 | 3.819,750 4.74,052 24. 2.032,794 2.563,304 | 3.225,100 | 4.048,935 5.07,236 25. 2.093,778 2.665,836 3.386,355 | 4.291,871 5.42,743 26. 2.156,592 2.772,470 | 3.555,673 | 4.549,383 5.80,735 27. | 2.221,289 2.883,369 | 3.733,456 | 4.822,346 | 6.21,386 28. 2.287,928 2.998,703 3.920,129 5.111,687 6.64,883 29. 2.356,566 3.118,651 4.116,136 5.418,388 7.11,425 30. 2.427,262 3.243,398 4.321.942 5.743,491 7.61,225 31. 2.500,080 3.373,133 4.538,039 6.088,101 8.14,571 32. 2.575,083 3.508,059 4.764,941 6.453,386 8.71,527 33. 2.652,335 3.648,381 5.003,189 6.840,590 | 9.32,533 34. 2.731,905 3.794,316 5.253,348 7.251,025 9.97,811 35. | 2.813,862 | 3.946,089 5.516,015 7.686,087 10.6,765 19. |