XV. COMPOUND PROPORTION, OR The RULE of FIVE, IS so called, from its having five numbers or terms given to find a sixth. Here, if the proportion be direct, the sixth term must bear such a proportion to the fourth and fifth as the third bears to the first and second. But if the proportion be inverse, then the sixth term must bear such proportion to the fourth and fifth as the first bears to the second and third, or as the second bears to the first and third. The three first terms are a supposition, the two last a demand. RULES. 1. Let the principal cause of gain, loss, or action, &c. be put in the first place. 2. Let that which denotes time, distance of place, &c. be in the second place, and the remaining one in the third place." 3. Place the other two terms which move the question underneath those of the same name. 4. If the blank or term sought fall under the third term, multiply the first two terms together for a divisor, and the last three for a dividend: the quotient arising from them will be the answer, or sixth term. 5. If the blank fall under the first or second term, multiply the third and fourth terms together for a divisor, and the other three for a dividend: the quotient arising from them will be the answer. PROOF. By two Statings. EXAMPLES. (1) If 6 men can mow 72 acres of grass in 12 days, how many men can mow 120 acres in 4 days? (2) Suppose 2 bushels of wheat will be sufficient for a family of six persons 12 days, how many bushels will serve 36 persons 4 days? (3) Suppose the salary of 6 persons for 21 weeks is 120l. what will be the salary of 14 persons for 46 weeks? (4) If, for the carriage of 40 cwt. 100 miles, I give 91. 5s. what weight can I have carried 125 miles for 70l. 10s. 34d. at the same rate? (5) An usurer put out 120l. to receive interest for the same: but when it had continued 9 months, he took it up, received for the principal and interest 125l. 8s. I demand at what rate per cent. per annum he received? (6) What is the interest of 2597. 138. 5d. for 20 weeks, at 51. per cent. per annum? (7) If a quantity of provisions serve 1400 men 20 weeks, at the rate of 14 ounces per day each man, how many men will the same provisions maintain for 8 months, at the rate of 8 ounces per day each? (8) Suppose 8 men earn 57. in 5 days, how many men will earn 10 guineas in 12 days? (9) Suppose 1401. would defray the expenses of five men for twenty-four weeks and four days, how long would twelve men be in spending 2001. at the same rate? (10) What money, at 341 per cent. per annum, will clear 387. 10s in a year and a quarter's time? (11) If a sack of coals be the allowance of 7 poor people for a week, how many poor belonged to that parish, which, when coals were 36s. per chaldron, had 417. to pay in 6 weeks on that account? QUESTIONS for Exercise at leisure Hours. (12) A. and B. are on opposite sides of a wood, 135 toises or fathoms about. They begin to go round it both the same way at the same instant of time; A. goes 11 toises in 2 minutes, and B. 17 in 3. The question is, how many times will they surround this wood before the nimbler overtakes the slower? (13) If a lever, 40 effective inches long, will, by a certain power thrown successively thereon, in 13 hours raise a weight 104 feet, in what time will two other levers, each 18 effective inches long, raise an equal weight 73 feet; the force of straight levers being in direct proportion of their lengths? (14) A weight of 1lb. laid on the shoulder of a man, is no greater burden to him than its absolute weight, or 24 ounces. What difference will he feel between the said weight applied near his elbow, at 12 inches from the shoulder, and in the palm of his hand 28 inches therefrom? and how much more must his muscles then draw to support it at right angles; that is, having his arm extended right out ? (15) In giving directions for making an Italian chair, the shafts whereof were settled at 11 feet between the axle-tree, whereon is the principal bearing, and the backband, by means of which the weight is partly thrown upon the horse; a dispute arose whereabout on the shafts the centre of the body of this machine should be fixed. The coachmaker advised this to be done at 30 inches from the axle; others were of opinion, that at 24 it would be a sufficient encumbrance to the horse. Now, admitting the two passengers, with their baggage, ordinarily to weigh 2 cwt. a piece, and the body of the vehicle to be about 70lb. more; what will the beast, in both those cases, be made to bear more than his harness? (16) Suppose a person to travel 152 miles in 7 days, when the days are 12 hours long, how many days will he be in travelling 576 miles, when the days are 16 hours long? (17) A water-tub holds 147 gallons; the pipe usually brings in 14 gallons in 9 minutes; the tap discharges, at a medium, 40 gallons in 31 minutes. Supposing these both to be carelessly left open, and the water to be turned at two in the morning; the servant at five, finding the water running, shuts the tap, and is solicitous in what time the tub will be filled after this accident, in case the water continue flowing from the main? (18) If the scavenger's rate, at 1d. in the pound, amcunt to 6s. 7d. where they ordinarily assess of the rent, what will the king's tax for that house be, at 4s. in the pound, rated at the full rent? (19) If, when port wine is 17 guineas the hogshead, a company of 45 people will spend 207. therein, in a certain time; what is wine a pipe, when 13 persons more will spend 637. in twice the time, drinking with equal moderation? (20) There is an island 73 miles round, and three men start together, to travel the same way about it. A. travels 5 miles a day, B. 8, and C. 10. When will they all come together again? (21) A certain man hired a labourer on this condition,—that for every day he worked he should receive 1s. but for every day he was idle he should be fined 8d. When 390 days were past, neither of them was indebted to the other. How many days did he work, and how many days was he idle? (22) A. lent his friend B. fourscore and eleven guineas from the 11th of December to the 10th of May following; B. on another occasion, let A. have 100 marks, from September the 3d to Christmas following. Quere, How long ought the person obliged to let his friend use 401. fully to retaliate the favour? (23) A man hired a labourer for 40 days, on condition that he should have 20d. for every day he worked, and forfeit 10d. for every day he idled; at last he received 21. 1s. 8d. for his labour. How many days did he work, and how many was he idle? XVI. PRACTICE, SO called from the general use it is of to all persons concerned in trade and business. All questions in the Rule of Three, where the first term is unity or one, may be performed by this rule; Which is by taking aliquot or even parts, by which means many tedious reductions may be avoided. But as there are a great variety of such parts, so many, therefore, are the ways of applying them, that it would be an endless task to give all the easy methods of operation adapted to particular cases; so I shall only give the general rules, with a sufficient number of examples to illustrate them. In order to perform expeditiously, it will be necessary that the learner get by heart the following TABLE. Of a Pound Of a Sh. Of a Ton. Of a Hun. Of a 2r. of a C. s. d. id. cwt. gr. lb. lb. =2 1.8= 2 2 3 4: 4 10 5 68 10 Case 1. When the price is less than a penny. RULE. Divide by the aliquot parts that are in a penny, then by 12 and 20, which will give the answer. EXAMPLES. (1) 2107 at d. (2) 1470 atd. (3) 1276 yds. at d. per yd. Case 2. When the price is less than a shilling. RULE. Take the aliquot part or parts that are in a shilling, add them together, and the sum will be the answer in shillings, &c. which, divided by 20, as before, will give pounds, &c. |