EXAMPLES. 1. A had in company 50l for 4 months, and B had 60l for 5 months; at the end of which time they find 24l gained: how must it be divided between them? Then as 500: 24 :: 200 : 2. C and D hold a piece of ground in common, for which they are to pay 541. C put in 23 horses for 27 days, and D 21 horses for 39 days; how much ought each man to pay for the rent ? Ans. C must pay 231 5s 9d. D........ 30 14 3. 3. Three persons, E, F, G, hold a pasture in common, for which they are to pay 30l per annum; into which E put 7 oxen for 3 months, F put 9 oxen for 5 months, and G put in 4 oxen for 12 months; how much must each person pay of the rent ? Ans. E must pay 51 10s 6d 19. 11 16 10 0%. G........ 12 12 7 2%. F ........ 4. A ship's company take a prize of 1000l, which they agree to divide among them according to their pay and the time they have been on board : now the officers and midshipmen have been on board 6 months, and the sailors 3 months; the officers have 40s a month, the midshipmen 30s, and the sailors 22s a month; moreover, there are 4 officers, 12 midshipmen, and 110 sailors: what will each man's share be? 5. H, with a capital of 1000l, began trade the first of January, and, meeting with success in business, took in I as a partner, with a capital of 1500l, on the first of March following. Three months after that they admit K as a third partner, who brought into stock 2800l. After trading together till the end of the year, they find there has been gained 17761 10s; how must this be divided among the partners? Ans. H must have 4571 9s 4dq. 571 16 84 87 747 3114 I 233123 6. X, Y, and Z made a joint-stock for 12 months; X at first put in 20l, and 4 months after 20l more; Y put in at first 30l, at the end of 3 months he put in 20l more, and 2 months after he put in 40l more; Z put in at first 60l, and 5 months after he put in 10l more, 1 month after which he took out 30l; during the 12 months they gained 50l; how much of it must each have? INTEREST is the premium or sum allowed for the loan or forbearance of money. The money lent or forborn is called the Principal; and the sum of the principal and its interest is called the Amount. Interest is allowed at so much per cent. per annum, or interest of 100l for a year, is called the rate of interest. Thus: when interest is at 3 per cent. the rate is 3; But, by law, interest ought not to be taken higher than at the rate of 5 per cent. Interest is of two sorts; Simple and Compound. Simple Interest is that which is allowed for the principal lent or forborn only, for the whole time of forbearance. As the interest of any sum, for any time, is directly proportional to the principal sum, and also to the time of continuance; hence arises the following general rule of calculation. As 100l is to the rate of interest, so is any given principal to its interest for one year. And again, As 1 year is to any given time, so is the interest for a year, just found, to the interest of the given sum for that time. OTHERWISE. Take the interest of 1 pound for a year, which multiply by the given principal, and this product again by the time of loan or forbearance, in years and parts, for the interest of the proposed sum for that time. Note. When there are certain parts of years in the time, as quarters, or months, or days; they may be worked for, either by taking the aliquot or like arts of the interest of a year, or by the Rule of Three in the usual way. Also, the division by 100 is done by pointing off two figures for decimals. EXAMPLES. 1. To find the interest of 230l 10s, for 1 year, at the rate of 4 per cent. 2. To find the interest of 5471 15s, for 3 years, at 5 per cent. per annum. As 100: 5:547-75 Or 20:1:: 547.75: 27-3875 interest for 1 year. 3 182 1625 ditto for 3 years. 20 s 3.2500 12 d 3.00 Ans. 821 3s 3d. 3. To find the interest of 200 guineas, for 4 years, 7 months and 25 days, at 4 per cent. per annum. 4. To find the interest of 4507, for a year, at 5 per cent. per annum. Ans. 221 10s. 5. To find the interest of 7151 12s 6d, for a year, at 41⁄2 per cent. per annum. Ans. 321 4s Oid. 6. To find the interest of 720l, for 3 years, at 5 per cent. per annum. annum. Ans. 1081. 7. To find the interest of 3551 15s, for 4 years, at 4 per cent. per annum. Ans. 56l 18s 4åd. 8. To find the interest of 321 5s 8d, for 7 years, at 44 per cent. per annum. £ 170 Ans. 91 12s 1d. 9. To find the interest of for 11⁄2 year, at 5 per cent. per annum. Ans. 121 15s. 10. To find the insurance on 2051 15s, for of a year, at 4 per cent. per Ans. 21 1s 1d. 11. To find the interest on 319l 6d, for 5 years, at 3 per cent. per annum. Ans. 681 15s 9 d. per cent per annum. Ans. 11 12s 7d. 13. To find the interest of 171 5s, for 117 days, at 4 per cent. per annum. Ans. 5s 3d. 14. To find the insurance on 7121 6s for 8 months, at 7 per cent. per annum. Ans. 351 128 3d. Note. The rules for Simple Interest serve also to calculate Insurances, or the Purchase of Stocks, or any thing else that is rated at so much per cent. See also more on the subject of Interest with the algebraical expression and investigation of the rules, at the end of the Algebra. 12. To find the insurance on 1071, for 117 days, at 4 COMPOUND INTEREST. COMPOUND INTEREST, called also interest upon interest, is that which arises from the principal and interest, taken together, as it becomes due, at the end of each stated time of payment. Though it be not lawful to lend money at compound interest, yet in purchasing annuities, pensions, or leases in reversion, it is usual to allow compound interest to the purchaser for his ready money. RULE 1. Find the amount of the given principal, for the time of the first payment, by simple interest. Then consider this amount as a new principal for the second payment, whose amount calculate as before. Proceed thus through all the payments to the last, always accounting the last amount as a new principal for the next payment. The reason is evident from the definition of compound interest. Otherwise, RULE 2. Find the amount of 1 pound for the time of the first payment, and raise or involve it to the power of whose index is denoted by the number of payments. Then that power multiplied by the given principal, will produce the whole amount: from which the said principal being subtracted, leaves the compound interest of the same. This is evident from the first rule. EXAMPLES. 1. To find the amount of 720l, for 4 years, at 5 per cent. per annum. Here 5 is the 20th part of 100, and the interest of 11 for a year is or 05, and its amount 1.05. Therefore, 2. To find the amount of 507 in 5 years, at 5 per cent. per annum, compound interest. Ans. 631 16s 34d. 3. To find the amount of 50l in 5 years, or 10 half-years, at 5 per cent. per annum, compound interest, the interest payable half-yearly. Ans. 641 Os 1d. 4. To find the amount of 50l in 5 years, or 20 quarters, at 5 per cent. per annum, compound interest, the interest payable quarterly. Ans. 641 2s Old. 5. To find the compound interest of 370l forborn for 6 years, at 4 per cent. Ans. 981 3s 44d. 6. To find the compound interest of 410l forborn for 2 years, at 45 per cent. per annum, the interest payable half-yearly. per annum. Ans. 481 48 114d. 7. To find the amount, at compound interest, of 2171 forborn for 24 years, at 5 per cent. per annum, the interest payable quarterly. Ans. 2421 13s 4d. ALLIGATION. ALLIGATION teaches how to compound or mix together several simples of different qualities, so that the composition may be of some intermediate quality or rate. It is commonly distinguished into two cases, Alligation Medial, and Alligation Alternate. ALLIGATION MEDIAL. ALLIGATION MEDIAL is the method of finding the rate or quality of the composition, from having the quantities and rates or qualities of the several simples given. It is thus performed : * Multiply the quantity of each ingredient by its rate or quality; then add all the products together, and add also all the quantities together into another sum; then divide the former sum by the latter, that is, the sum of the products by the sum of the quantities, and the quotient will be the rate or quality of the composition required. EXAMPLES. 1. If three sorts of gunpowder be mixed together, viz. 50lb at 12d a pound, 44lb at 9d, and 26lb at 8d a pound; how much a pound is the composition worth? * Demonstration. The rule is thus proved by Algebra. Let a, b, c be the quantities of the ingredients, and m, n, p their rates, or qualities, or prices ; then am, bn, cp are their several values, and am + bn + cp the sum of their values, also a + b + c is the sum of the quantities, and if r denote the rate of the whole composition, andr = (am+bn+cp)÷(a+b+c), which is the rule. Note. If an ounce or any other quantity of pure gold be reduced into 24 equal parts, these parts are called carats; but gold is often mixed with some base metal, which is called the alloy, and the mixture is said to be of so many carats fine, according to the proportion of pure gold contained in it: thus, if 22 carats of pure gold, and 2 of alloy be mixed together, it is said to be 22 carats fine. If any one of the simples be of little or no value with respect to the rest, its rate is supposed to be nothing; as water mixed with wine, and alloy with gold and silver. |