196, 197. SINGLE RULE OF THREE. 129 Now, since the product of the extremes is equal to that of the means, 4 ripes at equals 6 times 12, or, according to che second arrangement, 12 vimos 6. Hence, if 12 times 6, or 72, be divided by 4, the first extreme, the quotient, 18, is evidently the other extreme, or the value of x. 196. 3. If 4 men can do a piece of work in 6 days, in low many days can 8 men do it? By analyzing the example, we find that 4 men 6 days=1 man 24 days, and 1 man 24 days=8 men 3 days. 8 then is the answer. Moreover it i's obvious, that if 4 men ean do a piece of work in 6 days, twice the number or men will do it in half the time, or 3 days; and generally the greater the number of men, the less the time, and the reverse; and also, the the longer the time, the less the number of men, and the reverse. In the above example, the ratio of the men, 4 to 8=2, but the ratio of the times, 6 to 3 Now if we invert the first ratio, it becomes, 8 to-4=; and wo equal ratios, and consequently a proportion : i. e. 8:4::6 :3, or 8:6:: 4:3. By the question the proportion would stand, 8:6 :: 4:x; then 8x=4x6, and s=243, Ans. Where more requires less or less requires more, that is, when one of the ratios is inverted, as ex. plained in this article, it is denominated inderse proportion; otherwise it is called direct proportion. 197. When three terms of a proportion are given, the operation by which the fourth is found, is called the Single Rule of Three. All quescions, which can be solved by the single rule of three, must contain three given numbers, two of which are of the sanie kind, and the other of the kind of the required answer; and from an examination of the preceding analysis, it will be seen that the given number, which is of the same kind as the answer, may always be one of the means in the proportion; and, since the proportion is not altered by changing the places of the means, (195) it may always be regarded as the first mean, or the middle one of the three given terms. Now if the conditions of the ques. tion require the answer to he greater than the given number of the same kind, or sirst mean, the other mean must obviously be greater than the first extreme; but if the answer be required to be less, the second mean inust be less than the first extreme. Hence we have the following general RULE. 198. Write down the given number, which is of the same kind as the answer, or number sought, for the second term. Consider whether the answer ought to be greater, or less, than this number; and if greater, write the greater of the other two given numbers for the third term. and the less for the first term; but if less, write the least of the other two given numbers for the third term, and the greater for the first. Multiply the second and third terms together, and divide the product by the first, the quotient will be the answer. NOTE. Before stating the question, the first and third terms must be reduced to the same denomination, if they are not alread middle term to the lowest denomination mentioned in it. The answer will be in the same denomination as the second term, and may be brought to a higher by reduction, if necessary. QUESTIONS FOR PRACTICE. 4. If 15 bushels of corn cost 8. If 8lb. 40%. of tobacco cost $7.50, what will 25 bushels 58. 60. what will 2416. 12oz. cost? cost? bu. $ cts. bu. Jb. oz. s. d. 1b. Oz. 15 : 7.50 :: 25 84 5 6 24 12 25 16 12 16 3750 132oz. 661. 156 -24 396oz. By analysis. -'If 15 bushels ofl oz. d. oz. born cost $7.50, one bushel will | 132 : 66 : : 396. cost ($7.50 *15=) $0.50, and if ! 66 one bushel cosi 90.50, 25 bushels will cost ($0.50x25=) $12.50. 2376 The pupil should be required to solve the following questions by a. 2376 nalysis as well as by the rule of three. 132 ) 26136 ( 198d. 5. If $7.50 buy 15 bushels 16s. 6d. Ans. Here the several terins must he of corn, what will $12.50 buy ? reduced to the lowest denomina$ cts. bu. $ cts. tions mentioned, before stating the 7.50 : 15 : : 12 50 question. 15 9. If 8 acres produce 176 bushels of wheat, what will 31 6250 acres produce ? 1250 Ans. 748 bushels. bu. 17.50 ) 187.50 ( 25 Ans. 10. A borrowed of B 250 This is the reverse of the prece | dollars for à months; afterring example, and therefore proves | wards B borrowed of A 300 Jollars; how long must he 6. If a family of 12 persons keep it to balance the former spend 5 bushels of wheat in 4 favor ? Ans. 5mo. 25d. weeks, how much will last them Jl. A goldsmith sold a tanka year, allowing 52 weeks to ard weighing 39oz. 15pwt. for a year ? £10 12s. what was it per oz.? w. bu. W. oz. pwt. £ 4: 5 :: 52 Ans. 65 bush. 39 15:10 12:: 1 Ans. 58. 4d. 7. If 9lb. of sugar 6s. what: 12. If the interest of $100 will 251b. cost? Ans. 16s. 8d. for 1 year be 6 dolls.-what will When there is a remainder after | be the interest of 336 dollars dividing the product of the second and third terms by the first, reduce ! for the same time ? jt to the next lower denomination, and divide as before, :: 336 Ans. $20.16. 198. SINGLE RULE OF THREE. 131 13. If 100 men can do al 22. If a staff 4ft. 6in. in lengh piece of work in 12 days, how cast a shadow 6 feet, what is inany men can do the same in the height of a tree whose sha3 days ? Ans. 400 men. | dow measures 108 feet? 14. If 100 dollars gain 6 dol . Ans. 81 feet. lars in one year, in what time 23. If the earth revolve on will a sum of money double at its axis 366 times in 365 days, that rate, simple interest ? in what time does it perform 1 $ yr. $ revolution ? 6:1:: 100 Ans. 164 yrs. i rey.. ds. rev. 366 : 365 ::1 15. If $100 gain $6 in 12 Aus. 23h. 56m. 4s. nearly. * months, in how many months will a sum of money double at 24. Bought 4 bales of cloth, that rate, simple interest ? each containing 6 pieces, and $ mo. | each piece containing 27 yds. 6:12:: 100 Ans. 200mo. i at £16 4s. per piece; what is 16. If $100 gain 6 dollars in the value of the whole, and 365 days, in how many days, the price per yard ? will a sum of money double at Ans. £388 16s. and 12s. per that rate, simple interest? yard. Ans. 6093} days. 25. If a hogshead of rum cost 17. A owes B £296 17s. but $75.60, how much water must becoming a bankrupt, can pay be added to it to reduce the only 7s.6d. on the pound; price to 1 dollar per gallon ? how much will B receive ? Ans. 12 gal. Ans. £111 6s. 4d. 2qrs. 26. If a board be. 9 inches 18. If 1 dozen of eros cost wide, how much in length will 101 cents, what will 250 eggs make a square foot ? Cost? Ans. $2.187. Ans. 16in. 19. If a penny loaf weigh 27. How many yards of pa9oz, when wheat is 6s. 3d. per pe per 3 quarters of a yard wide, bushel, what ought it to weigh Ich will paper a room that is 24 when, wheat is 8s. 2 d. per yards round, and 4 yds. high? bnshel ? Ans. 6oz. 13drs. Ans. 128 yards. 20. How many yards of flan 1 28. If a man spend 75 cents nel 5grs. wide, will line 20 per day, what does he spend yards of cloth 3qrs. wide ? per annum ? Ans. $273.75. Ans. 12 yards. 29. A garrison of 500 men 21. If a person at the equator | has provisions for 6 months; be carried by the diurnal mo- i how many, must depart that tion of the earth, 25000 miles there may be provisions for in 24 hours, how far is he car- i | those who remain, 8 months ? ried in a minute ? Aps. 125. Ans. 1738 miles. * This is called a sideral day. 30. The salary of the Presi: 1 39. The earth is 3606 in dent of the United States is circumference, and revolves $25000 a year; what is that on its axis in 24 hours; how per day? Ans, $68.493. far does a place move in one 31. If a field will feed 6 cows minute in lat. 44o, a degree in 91 days, bow long will it feed that latitude being about 50 21 cows? Ans. 26 days. miles ? Abs. 123 miles.' h. m. deg. m. m. 32. A lends B 66 dollars for 24 X 60 : 360 X 50 :: 1 1 year; how much ought B to 40. If the earth perform its lend A for 7 months, to bal. diurnal revolution in 24 hours, ance the favor? in what time does a place on Ans. $113.142. its surface move through one 33. At $1.25 per week, bow degree? Ans. 4 minutes. many weeks' board can I have 360° : 24 :: 1° for 100 dolls.? Ans. 80 weeks. i 41. There is a cistern which 34. If my watch and seal be has a pipe that will empty it worth 48 dollars, and my watch in 6 hours; how many such be worth 5 times as much as pipes will be required to empty my seal, what is the value of it in 20 minutes ? the watch? Ans. $40. Ans. 18 pipes. 6 : 48 : : 5 42. What is the value of 35. A cistern containing | 642 dollars against an estate 230 gallons, has 2 pipes; by which can pay only 69 cents one it receives 0 gallons per on the dollar? Ans. $442.98. hour, and by the other discharges 35 gallons per hour; in 1 43. If 6352 stones of 3 feet wbat time will it be filled ? | long complete a certain quan Ans. 15h. 20m. tity of walling; how many 36. What will 39 weeks' ! stones of 2 feet long will raise a like quantity ? Ans. 9528. board come to at $1.17 per week? Ans. $45.63. 44. Suppose 450 men have 37. If 40 rods in length and provisions for 5 months, how 4 in breadth make 1 acre, how many must depart, that the many rods in breadth, that is provisions may serve those 16 rods long, will make 1 acre? who remain 9 months ? Ans. 10 rods. Ans. 200 men. 38.. How many men must be 45. A person's annual in. employed to finish in 9 days, i come being £146, how much what 15 would do in 30 days? | is that per day? Ans. 8s. Ans. 50 men. - 199, 200, 201. COMPOUND PROPORTION, 133 2. Compound proportion. ANALYSIS. 199. 1. If a person can travel 36 miles in 4 days, when the days are 3 hours long, how far can he travel in 2 days, when the days are 12 hours long? I. If a person can travel 96 miles in 4 days, he can travel (96--4=) 24 miles in 1 day, anci, if he can travel 24 in a clay, which is 3 hours long. he can travel (24-6=) 3 miles in 1 hour, and if he can travel 3 miles in an hour, he can travel, w:en the days are 12 lows long, (12 X3=-) 36 miles in 1 day. or (36x2=) 72 miles in 2 days, which is the answer. II. It must be evident that the distances traveller! by a person going all the time at the same rate will be in proportion to the iimes in which ihey are travelled. Iu this case, 4 days, which are o leurs long, are equal to (84=) 32 hours, and 2d. 12 hours long equal (12*2=) 24h. and hence we have this proportion, 32h. :: 96m. :: 24h. : X, or the distance travesled in the 2 days, wbich we find to be 72 miles as before. Ill. It will be obvious, in the above question, that the distance travell. rus on two circumstances, viz. the number of days and the length of the days. Now, supposing the days bad all been of the same length, we should have bad this proportion, viz. 4d. : 96m. :: 2d. : 2, or the distance travelled in 2 days; or, supposing the number of days had heen the same in both cases, the proportion would stand, 8b. : 96m. : : 12h. : x, or the distance travelled, when the days are 12 hours long, Uniting these proportions together, we have 40.2 3h. { : 96m. : : ia : x, by which it appears i hat 36 is to be multiplied by 2 and 12, or (2x12=) 24, and divided hy 4 and 3, or (4*3=) 32, which is the same as the second method of solving the question. 200. 2. If 12 men can make 9 Tods or fence in 6 days, when the days are 10 hou's long how many men will be required to make 18 rods of fence in 4 days, when the days are 8 hour's long? In this question, the number of days and their lengil being supposed to be the same in both cases, we should have this proportion, gros. i 12 men : : 18 : x, or the number of mon required to build the 18 rods--supposing the number of rods to be the same in both cases, and the days to be in equal length, weshould have this proportion, 4d. : 12 men:: 6d. : x, or ile number required to build the fence in 4 days, and supposing the number of rods and also the number of days 10 lie the same in both cases, we should have this proportion, 8 hours :12 men :: 101. : X, or the number required, when the days are 8 hours long. These three proportions canbined, we have Ords. ) 18rds.) :: Cris. : %, an. 12 10h. hy which it appears that 9:4 8 :12 :: 15 % 6 10:, and multiplying the product, the ihirdrems by the second, and dividing by the product of the first terms, we find the value of x io le 45 men, which is the answer, DOUBLE RULE OF THREE. 201. A proportion, which is formed by the combination of two, or more, simple proportions, as ju the preceding esam |