miles a day; B follows at the rate of 25 miles a day; in what time and what distance will he overtake A? answer 20 days, and 500 miles. 21 If 58 gallons of water, in one hour, fall into a cistern containing 230 gallons, and by a pipe in the cistern, 35 gal. lons run out in an hour; in what time will it be filled? answer 15h. 20min. 22 A certain cistern has four pipes; by the first it will be filled in 10 minutes, by the second in 20, by the third in 40, and by the fourth in 80; in what time will all four, running together, fill it ? answer 5min. 20sec. 23 Astronomers compute the earth's orbit, or track which it describes round the sun in 365 days 6 hours, to be about 596900000 miles; how far then, per minute, must we be carried through the firmament by this wonderful motion? ansaver 1134+miles. 24 Isaac Newton, and others, have found, by nice experi. ments, that sound flies at the rate of 1 142 feet per second, and a person in health has about 75 beats of the artery or pulsations in a minute; now the breadth of a river is required, at one side of which A, firing a gun, B, directly opposite at the other, counts six pulsations at his wrist between seeing the flash and hearing the report? answer 5481ft. or I mile 201ft. 25 If the report of a piece of ordinance be heard one minute and three seconds after the flash was observed; the distance is required? answer 13 miles 5 furlongs. T THE DOUBLE RULE OF THREE. HE Double rule of three is that, wherein five numbers or terms are given, to find a sixth, three of which are a supposition, and two a demand; and is either direct or in verse. RULE for Stating. 2 Set the two terms of the supposition, which are like those of the demand, one under the other, in the first place; that of the same kind with the term sought in the second, and the two demanding terms in the third place, with the two correspondent correspondent terms of the supposition and demand in the same line, and of one denomination; as in the subsequent examples. viz. 2 Suppose 4 men in 12 days mow 48 acres; how many answer 128 acres. I If three men in 4 days eat 5lb. of bread, how much will suffice 6 men for 12 days? If 3m.7 51b. { 6m. 12d. 2 If 3 men cat 5lb. in 4 days; in how many days will 6 men consume 30lb. 6m. To know whether the stating be direct or inverse : Consi der the upper pair of extremes, and the lower, each separately with the middle term, as a stating of the single rule, and try them as taught in that rule: if both lines be direct, the stating is in direct proportion; but of inverse, if either pair of the extremes be so. Thus, the first example above is di rect, and the second inverse. DIRECT PROPORTION. RULE. Divide the continual product of the two last extremes and middle term by that of the two first, and the quotient will be the sixth term, or answer. PROOF. By two statings of the single rule of three. Note. If either of the two first terms, or both, will divide, or can be divided by any of the three last, or by any other number without remainder, the operation may be abbreviated by cancelling them, and using their quotients or aliquot parts in their stead. EXAMPLES. 1 If three men in four days eat 5lb. of bread; how much will suffice 6 men for 12 days? If 3m. 6m. 51b. 4d. 12. 72 If m { acres can 8 men mow in 16 days? 3 If 12 oxen in 16 days eat 20 acres of grass; how many acres will serve 24 oxen 48 days? answer 120 acres. 4 If 10 bushels of oats be sufficient for 18 horses 20 days; how many bushels will serve 60 horses 36 days, at that rate? answer 60 bushels. 5 If 56lb. of bread be sufficient for 7 men 14 days; how many pound will suffice 21 men 3 days? answer 36lb. 6 If 8 men have 3/ 45. for 4 days work; how much ought 48 men to receive for 16 days? answer 761 165. ans. 80dols. 7 If 700 dols. in half a year raise 14 dols. interest; what will be the interest of 400 dols, for 5 years? 8 If 112 acres of grass be mowed by 16 men in 7 days; how many acres may 24 men mow in 19 days? answer 456 acres. 9 If 161 18s. be the wages of 16 men for 8 days; what sum will 32 men earn in 24 days? answer 1011 85. 10 If 751. in 9 months amount to 781 7s 6d. at what rate per cent. is the interest computed? answer 61. per cent. II Suppose the wages of 6 persons for 21 weeks be 120/. what will be the hire of 14 persons for 46 weeks? answer 6131 6s 8d. 12 What is the interest of 2591 135 5d. for 20 weeks, at 5 per cent? answer 41 195 10d. 13 If 2 men can do 12 rods of ditching in 6 days; how many rods may be done by 8 men in 24 days? answer 192 rods. 14 If the carriage of 8C.wt. 128 miles cost 6,40; what must be paid for the carriage of 4C.wt. 32 miles? answer 80cts. 15 If 200lb. be carried 40 miles for 40cts. how much must be paid at that rate for the carriage of 20200lb. 60 miles? answer 60,60 16 If the freight of 9 hogsheads of sugar, each weighing 12 hundred weight, for 20 leagues, cost 161. what must be paid for the freight of 50 casks of ditto, each weighing 24 hundred weight, 100 leagues? answer 921 115 10d. INVERSE INVERSE PROPORTION. RULE. Transpose the inverse extremes, that is, set that of the first place under the third, and that in the third under the first; then work as in direct proportion Note. See the note in direct proportion. EXAMPLES. 1 If 7 men can reap 84 acres of wheat in 12 days; how many men can reap 100 acres in 5 days ? If 844.7 7m. { 100A. gd. Inverse term 12 2 If 4 dollars be the hire of 8 men for many days must 20 men work for 40 dols. 3 If 4 men have 24 shillings for three many men will earn 41 16s. in 16 days? three days; how answer 12 days. days work; how answer 3 men. what principal in 12 months will gain 61. 4 Suppose the interest of 3331 6s 8d. for 9 months be 151. answer 100%. 5 If 2colb. be carried 40 miles for 40cts; how far may 20200lb. be carried for 60,60? answer 60 miles. 6 If 145 men can make a wall 32 feet high, and 40 feet long, in 8 days; in how many days can 68 men build a wall 28 feet high, of the same length answer 14days, 11h. + 7 If a footman, when the days are 14 hours long, can travel 276 miles in 16 days; in how many days can he travel 852 miles, when the days are but 12 hours long? 7 If answer 57 days 7 hours. + 8 If rs men eat 3 shillings worth of bread in 6 days, when wheat is sold at nine shillings per bushel; how many days will 30 men require to eat 13s 4d. worth, when wheat is at 6s. per bushel? answer 20 days. 9 If 100%. principal in 12 months gain 87. interest; what principal will gain 8/ 12s. in 5 months? answer 2581. 10 Suppose 1001. will defray the expences of 5 men for 22 weeks and 6 days; how long will 12 men be spending 1501. answer 14 weeks 2 days. Application. 1 If 7 bushels of malt be sufficient for 7 persons 4 months, how many bushels will serve 46 persons 10 months? answer 115 bushels. 2 How many men must be employed to reap 240 acres in 12 days, if 36 men can reap 60 acres in 5 days? + answer 60 men. 3 If 5 men make 300 pair of shoes in 40 days; how many men may make 900 pair in 60 days? answer IO men. 4 A porter having received 42 shillings for the carriage of 3C.wt. 150 miles; how much ought he to have for the conveyance of 7C. 2qr. 14lb. 50 miles? answer 35s 7d. 5 A person having engaged to remove 8000C.wt. a certain distance in 9 days, with 18 horses, in 6 days he removed 4500C.wt. how many horses will be required to remove the remainder in the remaining 3 days? answer 28 horses. 6 If 20 hundred weight be carried 50 miles for 51. how much will forty hundred weight cost, to be conveyed 100 miles? answer 201. 7 A farmer having sown 48 bushels, found that it produced 576 bushels the first year, now supposing he sows 240 bushels of grain each year for 6 years successively; what will be his whole increase at the expiration of the last year? answer 17280 bushels.. 8 If 12 men in 6 days reap 80 acres; in how many days will 25 men reap 200 acres? answer 7 days. 9 Ar |