4. The principal and interest being added together, the sum is called the amount. Let p the principal lent, r=the interest of 1 pound for a year, t=the time during which the principal has been lent, i= the interest of p pounds for t years, a=the amount; then will 1 (pound): r (interest) :: p (pounds): pr=the interest of p pounds for a year: and 1 (year): pr (interest) :: t (years): ptr=i (THEOR. 1.) the interest of p pounds for t years, or t parts of a i i i year: hence pr t= and r= If to this interest the pr pt principal be added, we shall have ptr+p=a (THEOR. 2.) hence a by transposition, &c. p= [r+1 (THEOR. 3.) t= and r tr a-p (THEOR. 5.) The following is a synopsis of the whole pt money, at different periods, from 5 to 50 per cent. but the law at present is, that not more than 5 per cent. per annum can be taken here, although the legal rate of interest is much higher in some of our colonies. The interest of money is computed as follows; In the courts of law ...... in years, quarters, and days. On South Sea and India bonds ...... calendar months and days. .... quarters of a year and days. On Exchequer bills Brokerage, or commission, is an allowance made to brokers and agents in foreign, or other distant places, for buying and selling goods, and performing other money transactions, on my account; it is reckoned at so much per cent. on the money which passes through their hands, and is calculated by the rules of simple interest, the time being always considered as 1. The same rules serve for finding the value of any quantity of stock to be bought or sold, and likewise for finding the price of insurance on houses, ships, goods, &c. 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 x 4 x .05=153.1=1531. 2s. Answer. 2. What is the amount of 75l. 10s. 6d. for 8 years, at 44 per cent. per annum ? Here p (751. 10s. 6d.=) 75.525, t=(84) 8.5, r==( 43/ 100 .0475: whence (theor. 2.) ptr+p=75.525 × 8.5×.0475+75.525 =106.01821875= 106l. Os. 4d4.49a, 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 Here a=(4021. 10s.=) 402.5, t=5, r=(; =).03: where 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. 8s. 14d. in 1+ year? Here p=75, a=(771. 8s. 1+d.=) 77.40625, t=(1+=) 1.5. 6. What is the interest of 254l. 17s. 3d. for 24 years, at 4 per cent. per annum? Ans. 251. 9s. 8d. 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 15s. 64d. at 5 per cent? Ans. 15 shillings. 9. In what time will 251. amount to 251. 11s. 3d. at 41 per cent. per annum ? Ans. half a year. 10. At what rate per cent. per annum will 7961. 15s. amount to 9761. 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 7251. 15s, amount to 731l. 2s, 8÷d, at 4 per cent. per annum ? 15. At what rate per cent. per annum will 559l. 4s. Od. amount to 735l. 7s. Od. in 7 years 's? 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: =the present worth of p pounds paid t time before due, at r 1+tr per cent. interest; also 1+tr : tr : :P : ptr 1 + tr =the discount al 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? 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. Then 1+tr ptr 250 x 2.20548 x .05 = = 1+2.20548 x .05 225.16964= 24l. 16s. 7d4=the discount. 1+tr 1+2.20548 x .05 1.110274 2251. 3s. 44d.=the present worth. 2. Required the present worth, and discount, of 4871. 12s. due 6 months hence, at 3 3. Sold goods for 875l. 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. 3 d. disc. 16l. 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? a 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. • 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 ex 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 6: 8 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. 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 "a Ъ be both multiplied by any quantity n, the resulting ratio an : |
