EXAMPLES.-1. Required the simple interest of 7651. 10s, for 4 years, at 5 per cent. per annum? 5 Here p (7651. 10s.=) 765.5, t=4. r=(=).05. 100 Then i ptr (theor. 1.)=765.5 × 4×.05=153.1=153l. 2s. Answer. 2. What is the amount of 751. 10s. 6d. for 8 years, at 4 per cent. per annum ? 4 100 Here p=(75l. 10s. 6d.=) 75.525, t=(8+=) 8.5, r=(==) .0475: whence (theor. 2.) ptr+p=75.525 × 8.5 x .0475+75.525 =106.01821875= 106l. Os. 4d4.49≈a, the amount. 3. What sum of money being put out at 3 per cent. simple interest, will amount to 4021. 10s. in 5 years? 3 =(4021. 10s.=) 402.5, t=5, r=(- =).03: where Here a= 100 4. In what time will 350l. amount to 4021. 10s. at 3 per cent. 5. At what rate per cent. will 751. amount to 771. Ss. 14d. in 1+ year? Here p=75, a=(771. 88. 14d.=) 77.40625, t=(14) 1.5. a-p 77.40625-75 Then (theor. 5.) r= =.02138 21' per cent. nearly,r, the answer. 6. What is the interest of 2541. 17s. 3d. for 2 years, at 4 per cent. per annum? Ans. 251. 98. 84d. 7. What is the amount of 250l. in 7 years, at 3 per cent. per annum ? Ans. 3021, 10s. Od. 8. What sum being lent for of a year, will amount to 158. 64d. at 5 per cent? Ans. 15 shillings. 9. In what time will 257. amount to 25l. 11s. 3d. at 44 per cent. per annum? Ans. half a year. 10. At what rate per cent. per annum will 7967. 15s. amount to 976l. Os. 44d. in 5 years? Ans. 4 per cent. 11. Required the interest of 140l. 10s. 6d. for 24 years, at 5 per cent. per annum ? 12. To find the amount of 2001. in 8 years, at 4 per cent. per annum? 13. Suppose a sum, which has been lent for 120 days at 4 per cent. per annum, amounts to 2431. 3s. 14d. what is the sum? 14. In what time will 7257. 15s. amount to 7311. 2s. 84d. at cent. per annum ? 4 per 15. At what rate per cent. per annum will 559l. 4s. Od. amount to 735l. 7s. Od. in 7 years? 23. To investigate the rules of discount. Def. 1. When a debt which by agreement between debtor and creditor should be paid some time hence, is paid immediately, it is usual and just to make an allowance for the early payment; this allowance is called the discount. 2. The sum actually paid (that is, the remainder, after the discount has been subtracted from the debt,) is called the present worth. 3. The debt is considered as the amount of the present worth, put out at simple interest, at the given rate, and for the given time 2. Let p the given debt, r=the interest of 1 pound for a year, t=the time the debt is paid before it is due, in years or parts of a year; then will 1+tr=the amount of 1 pound at the rate r, and for the time t: (Art. 22. theor. 2.) then also will the amount of 1 pound be to 1 pound, (or its present worth,) as the given debt, to its present worth; also the amount of 1 pound, is to the interest of 1 pound, as the given debt, to the discount; that is, 1+tr: 1 :: p: Ρ =the present worth of p pounds paid t time before due, at r 1+tr per cent. interest; also 1+tr: tr :: p: -- the discount al ptr 1+ tr lowed on p pounds, at the said rate, and for the said time. EXAMPLES.-1. What is the discount, and present worth of 250l. paid 2 years and 75 days before it falls due, at 5 per cent. per annum simple interest? 2 In Smart's Tables of Interest, there is inserted a table of discounts, by which the discount of any sum of money may be calculated with ease and expedition. Here p=250, r=.05, t=(2 y.75 d=) 2.20548 years. 2251. 38. 44d.=the present worth. 2. Required the present worth, and discount, of 4871. 12s. due 6 months hence, at 3 per cent. per annum? Ans. pr. worth 480l. 7s. 10 d. disc. 71. 4s. 1d. 3. Sold goods for 8751. 5s. 6d. to be paid for 5 months hence; what are the present worth and discount at 4 per cent. per annum ? Ans. pr. worth 8591. 3s. 34d. disc. 161. 2s. 24d. 4. What is the present worth of 150l. payable as follows; viz. one third at 4 months, one third at 8 months, and one third at 12 months; at 5 per cent. per annum discount? 5. How much present money can I have for a note of 351. 15s. 8d. due 13 months hence, at 4 per cent. per annum discount? OF RATIOS. 24. Ratio is the relation which one quantity bears to another in magnitude, the comparison being made by considering how often one of the quantities contains, or is contained in, the other. Thus, if 12 be compared with 3, we observe that it has a certain relative magnitude with respect to 3, it is 4 times as great as 3, or contains 3 four times; but in comparing it with 6, we discover that it has a different relative magnitude with respect to 6, for it contains 6 but twice. a Ratio is a Latin word implying comparison. The student must be careful not to confound the idea of ratio with that of proportion, as some through inattention have done: he must bear in mind, that ratio is simply the comparison of one quantity to another, both being quantities of the same kind; whereas proportion is the equality of two ratios: the former requires two quantities of the same kind to express it, the latter requires at least three quantities, which must be all of the same kind; or four quantities, whereof the two first must be of a kind, and the two last likewise of a kind. See the note on Art. 53, and the note on Art. 127. Part 1. Vol. I. 25. The ratio of two quantities is usually expressed by interposing two dots, placed vertically, between them. Thus the ratios of a to b, and of 5 to 4, are written, a : b, and 5: 4. 26. The former quantity is called the antecedent, and the latter the consequent. Thus in the above ratios, a and 5 are the antecedents, and b and 4 the consequents. The antecedent and consequent are called terms of the ratio. 27. To determine what multiple, part, or parts the antecedent is of the consequent, (that is, to find how often it contains or is contained in the consequent,) the former must be divided by the latter; and this division is expressed by placing the consequent below the antecedent like a fraction. Thus the ratio of a to b, or a b, is likewise properly expressed thus and 5: 4 thus a b' 5 4 28. Hence, two ratios are equal, when the antecedent of the first ratio is the same multiple, part, or parts of its consequent, that the antecedent of the other ratio is of its consequent; or in other words, when the fraction made by the terms of the former ratio (Art. 27.) is equal to the fraction made by the terms of the latter. 6 Thus the ratio of 68 is equal to the ratio of 3 : 4, for 3 29. Hence, if both terms of any ratio be multiplied or divided by the same quantity, the ratio is not altered. 4x6 24' dently equal to the given fraction ; that is, 3: 4 is the same as 18:24; in like manner, if the terms of the ratio a : b, or α — be both multiplied by any quantity n, the resulting ratio an : b 30. Hence, one ratio is greater than another, when the antecedent of the former ratio is a greater multiple, part, or parts of its consequent, than the antecedent of the latter ratio is of its consequent; or, when the fraction constituted by the terms of the first ratio, is greater than that constituted by the terms of the latter. Thus 6 : 2 is greater than 8: 4, for 6 contains 2 three times, whereas 8 contains 4 but twice, or 6 2 8 is greater than 4 31. Having two or more ratios given, to determine which is the greater. RULE. Having expressed the given ratios in the form of fractions, (Art. 27.) reduce these fractions to other equivalent ones having a common denominator, (Vol. I. P. 1. Art. 180.) The latter will express the given ratios having a common consequent, wherefore the numerators will express the relative magnitudes of the ratios respectively. EXAMPLES.-1. Which is the greater ratio, 7: 4, or 8: 5? 7 8 These ratios expressed in form of fractions, are and 4 5' whence 7x5=35, and 8x4=32, these are the new numerators; also 4×5=20, the common denominator. being the greater, shews that the ratio of 7: 4, is greater than the ratio of 8:5. 2. Which is the greater ratio, that of 8: 11, or that of 23:32 ? 8 11 23 These ratios expressed like fractions, are and which 32' reduced to other equivalent fractions with a common denominator, become 256 253 and respectively; the former of these being the 352 greater, shews that the ratio 8: 11, is greater than the ratio 23:32. 3. Which is greatest, the ratio of 18: 25, or that of 19: 27 ? Ans, the former. 4. Which is the greatest, and which the least, of the ratios 9: 10, 37: 41, and 75: 83 ? |