19. If a field feed 6 cows 91 days, how long will it feed 21 cows? :' Ans. 26 days. 20. There is a cistern having 1 pipe which will einpty it in 10 hours; how many pipes of the same capacity will empty it in 24 minutes? Ans. 25 pipes. 21. Lent a friend $292 for 6 months; sometime afterwards he lent me 8806: how long may I keep it to balance the favour? Ans. 2 mo. 5 days. 22. If of a gallon of brandy cost 95 cents, what will cost? 23. If of a hogshead of molasses cost $16,00, what will the whole hogshead cost? Ans. $20,00. 24. If of a yard of broadcloth cost $6,00, what will cost? Ans. $2,00. 25. A smith bought of a ton of Russia iron for $25,35; what would be the price of a ton at that rate ? Ans. $91,26. 18 7 Ans. 0,19ets. COMPOUND RULE OF THREE. 196. In the Rules of Three, which have already been explained, the quantity sought, and quantity of the same kind, expressed in the enunciation, have to each other a simple ratio, and determined by that of two other quantities, which, in like manner, are expressed in the enunciation of the question. But in the Compound Rule of Three, the ratio of the quantity sought to the quantity of the same kind, expressed in the enunciation, is not determined by the simple ratio of two other quantities only, but by many simple ratios, which are compounded (187) after the examination of the question. When the ratios have been compounded, the rule is reduced to a simple Rule of Three. The following examples will explain what has been said on this subject. EXAMPLE I. Thirty men have built 132 rods of wall in 18 days; how much will 54 men build in 28 days? In this question, it is plain, that the answer does not depend upon the number of men only, but also upon the number of days. To have regard to both, we must consider, that 30 men labouring 18 days will not perform more work than 18 times 50, that is, 540 men, who labour during one day. In like manner, 54 men labouring 28 days, will not accomplish more than 28 times 54, or 1512 men, who labour one day. The question is then changed to this: 540 men have built 132 rods of wall, how much will 1512 men build in the same time? That is, we must seek the fourth term of a proportion, which commences with these three; hours. hours. róds. 540 1512 :: 132 : Multiplying 1512 by 132, and dividing the product by 540, we find for answer to the question, 1693 rods. EXAMPLE II. A man walking 7 hours in a day, was 30 days traveling 230 leagues; if he walk 10 hours in a day, how many days will it take him to go 600 leagues; traveling always with the same speed. If he walk the same number of hours a day in each case, it is plain the increase of time will be in proportion to the increase of distance; but as in the second case, he travels each day a greater number of hours, less time will be required in performing the journey. Hence the operation to be performed requires two operations, one in the Rule of Three Direct, the other in the Rule of Three Inverse. The question, however, may be reduced to a question in the simple Rule of Three, by considering, that to travel 30 days, 7 hours in a day, is to travel 30 times 7, or 210 hours; hence the question is changed to this; if 230 leagues have been travelled in 210 hours, how long will it take to travel 600 leagues? When the number of hours, which is the answer to this question, has been found, the number of days demanded will be found by dividing that sum by 10, because the man travelled ten hours in a day. Hence the fourth term must be sought of a proportion, whose three first terms are the following; leagues. leagues. hours. :30 : 600 :: 210 : We find, that the fourth term required is 54719 hours; which, divided by 10, the number of hours the man travelled in a day, give 541 days for the answer sought. give P EXAMPLES FOR PRACTICE. 1. If 6 men in 8 days eat 10lb. of bread, how much will 12 men eat in 24 days? Ans. 60lbs. 2. If £100 in 12 months gain £5, how much will £400 gain in 3 months? 3. If 4 men in 12 days mow 48 acres, can 8 men mow in 16 days? Ans. £5. how many acres Ans. 128A. 4. If a family of 9 persons spend $450 in 5 months, how much would be sufficient to maintain them 8 months, if 5 more were added to the family? Ans. $1120. 5. If 10 bushels of oats will be sufficient for 18 horses 20 days, how many bushels will serve 60 horses 36 days, at the same rate? Ans. 60 bushels. 6. If 30 men perform a piece of work in 20 days, how many men will execute another piece, 4 times as large, in a fifth part of the time? Ans. 600 men. 7. What principal will gain $315 in 7 years, at 6 per cent. per annum ? Ans. $750. 8. A wall, that is to be built to the height of 27 feet, has been raised 9 feet in 6 days, by 12 men; how many men will it take to complete the wall in 4 days? Ans. 36 men. 9. If the carriage of 8cwt. 128 miles cost $12,80, what must be paid for the carriage of 4cwt. 32 miles ? Ans. $1,60cts. 10. If 6 men build a wall 20 feet long, 6 feet high, and 4 feet thick, in 16 days; in what time will 24 men build one 200 feet long, 8 feet high and 6 feet thick ? Ans. 80 days. 24 men : 16 :: 6 feet high 6 men 200 feet long 6 feet thick 11. An usurer put out $75 at interest, and at the end of 8 months received for principal and interest $79; at what rate per cent. did he receive interest? Ans. 8 per cent. 12. If the freight of 12cwt. 2qrs. 6lb. 275 miles, cost $27,78; how far may 60cwt. 3qrs. be carried for $234,78P Ans. 480 miles. 13. If 4 men spend $60, going 300 miles; what will be sufficient for the expense of 20 men, and one boy 700 miles, allowing the boy one half a man's expenses ? Ans. $717,50. 14. If 50 men build a bridge in 144 days, what number of men will it take to build a like bridge in 270 days? Ans. 10 men. 15. A man lent a friend $600, for 6 months, for which he received 89 interest; what sum will gain the same interest in 2 months? Ans. $1800. RULE OF FELLOWSHIP. 197. The Rule of Fellowship is so called, because it is used for dividing among many associates the proât or loss of their partnership. Its object is to divide a given number into parts, which have known relations to each other. The rule used for this purpose is founded upon the principle established in article (186), that if many equal ratios are given, the sum of all the antecedents is to the sum of all the consequents, as one antecedent is to its consequent. From this principle we deduce the following example. EXAMPLE I. It is required to divide 120 into three parts, which have to each other the same ratios as the numbers 4, 3, 2; the enunciation furnishes these two proportions. 43 as the first part, is to the second. But it has been shown (186), that the sum of the antecedents of many equal ratios, is to the sum of the consequents, as one antecedent is to its consequent. Hence it is inferred, that 9, the sum of the three parts, which are proportional to those sought, is to 120, their sum, as any one of these three proportional parts, is to the part of 120, which answers to it. The rule for performing this operation is as follows; 1st. Add the given proportional parts into one sum. 2d. Make as many proportions in the Rule of Three as there are parts to be found. Sd. Let each proportion have for its first term, the sum of the given proportional parts, for its second term, the number proposed to be divided, and for the third term, one of the given proportional parts. Thus in the question under consideration, the three following propertions must be made in the Rule of Three. 9: 120 :: 4: 9: 120 :: 3: 9: 120 :: 2: Of which we find (179), that the fourth terms are 531, 40, and 26 which have to each other the ratios required, and together compose the number 120. We may remark that it is not absolutely necessary to make as many applications of the Rules of Three as there are parts to be found; for the last may be dispensed with, by subtracting from the given number, the sum of the other parts when found. EXAMPLE II. Three persons are to share a prize vessel. The first has been at £20,000 cost in her capture; the second at $60,000; and the third at $120,000. The prize is valued at $800,000; how much ought each man to receive? It is plain, the question requires $800,000 to be divided into parts which have to each other the same ratios as 20,000, 60,000, 120,000; or (170) as 2, 6, 12; because each person ought to receive in proportion to his expenditure. It is necessary, then, to add the three proportional parts 2, 6, 12, and make the three following proportions, or two only. 20 800,000 8 2: to the first part. 20: 800,000 :: 20: 800,000 These three parts will be $80,000, $240,000, and $480,000. More complicated questions may be solved on the same principles, as in the following example. Three persons entered into partnership; the first put in $3000 for 6 months; the second, $4000 for 5 months; |