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each day. Now if the number of hours had been the same in both cases, the proportion would have been,
10 : 12 :: 240 to the answer ; or, if a man in 10 days, travel 240 miles, how far will he travel in 12 days ? The answer is 288 miles.
But as the number of hours is different in the two cases, we must substitute another proportion, and say, if a man, travelling 12 hours a day for a certain number of days, will travel 288 miles, how far will he go if he trayvel 16 hours per day? The proportion in this case will be
12 : 16 :: 288 to the answer, 384. The question may likewise be resolved in this way. By travelling 10 days, when the days are 12 hours long, he travelled 120 hours, and by travelling 12 days, when the days are 16 hours long, he would travel 192 hours, which may be reduced to this proportion,
120 : 192 :: 240 : 384 You will see, by the first mode of resolving the question, that 240 miles has to the fourth term or answer, the same proportion that 10 days has to 12 days, and that 12 hours has to 16 hours ; stating this in form of proportion, it will be,
10 : 12 12 : 16
:: 240, which will give the Ans. 384. Thus it appears that 240 is to be multiplied by the product of 12 by 16, and divided by the product of 10 by, 12, or the product of 12 by 16 is to be multiplied by 240, and divided by the product of 10 by 12.
Again, if 12 men, working 8 hours a day, spend 24 days in digging a ditch, 48 feet long, 12 feet wide, and 6 feet
deep, how many days will it take 18 men, of the same ability, working 9 hours a day, to dig a ditch 112 feet long, 8 feet wide, and 4 feet deep?
This question appears very complicated, but can be resolved by proportion, as follows,
18 : 12
In this example the answer is to be in days, consequently 24 must be the third term; as 18 men can perform the work in less time than 12 can, 18 must be the first term and 12 the second ; and as it will not require so many days to perform the same work when the days are 9 hours long, as it will when they are but 8 hours long, 8 must be the second term, and 9 the first; and as it will take longer to dig a ditch 112 feet long, 8 feet wide, and 4 feet deep, than it will to dig one, 48 feet long, 12 feet wide, and 6 feet deep, 112 must be the second term, and 48 the first. This question may be stated by first finding the contents of each ditch in feet, when the statement will be,
18 : 12
9 : 8 :: 24. 3456 : 3584
From what you have been taught, you will readily upderstand the following rule for the
DOUBLE RULE OF THREE.
66 Make the number, which is of the same kind with the required answer, the third term ; and of the remaining numbers, take any two that are of the same kind, and place one for a first term, and the other for a second term, according to the directions in simple proportion; then, any other two of the same kind, and so on, till all are used ; lastly, multiply the third term by the product of the second terms, and divide the result by the product of the first terms, and the quotient will be the fourth term, or answer required."
Pup. I think that I understand this sufficiently to perform any question belonging to the double rule of three.
Tut. You will frequently, find questions in this rule which will require close thought to state them correctly. The only difficulty will be in stating the question, and this should excite you to obtain a thorough knowledge of the rule.
If a man travel 130 miles in 3 days, when the days are 12 hours long, in how many days, of 10 hours each, may he travel 360 miles ?
What is the single rule of three ?
Why do you multiply the second and third terms together, and divide by the first ?
When the answer is to be less than the third term, which of the other two terms must be the second terms?
When the first and second terms consist of different denominations, what do you do?
When the third term consists of different denominations, what denomination will the answer be in ?
What is compound proportion, or double rule of three ?
Examples for Practice. 1. If 100 workmen can finish a piece of work in 12 days, how many are sufficient to do the same work in 3 days ?
Ans. 400. 2. If 6 horses eat 21 bushels of oats in 3 weeks, how many bushels will 20 horses eat in the same time?
Ans. 70 bushels.
3. If 20 horses eat 70 bushels of oats in 3 weeks, how many bushels will 6 horses eat in the same time?
Ans. 21 busbels 4. If the moon move 13 deg. 10 min. 35 sec. in one day, in what time does it perform one revolution ?
Ans. 27d. 7h. 43m. 5. If 365 men consume 75 barrels of provisions in 9 months, how much will 500 men consume in the same time ?
Ans. 1024 barrels. 6. If I give $6 for the use of $100 for 12 months, what must I give for $357,82 the same time? Ans. $21,469.
7. How many tiles of 8 inches square will lay a floor 20 feet long, and 16 feet broad?
Ans. 720. 8. If I buy 76 yards of cloth for $113,17, what did it cost per El English ?
Ans. $1,861. 9. If 3 horses and 4 oxen be worth 9 cows, how many cows will 6 horses and 8 oxen be worth? Ans. 18.
10. If a staff, 5 ft. 8 in. in length, cast a shadow of 6ft. how high is that steeple whose shadow measures 153 ft. ?
Ans. 1441 ft. 11. Bought 4 pieces of Holland, each containing 24 Ells English, for $96 ; how much was that per yard ?
Ans. $0,80. 12. A bankrupt owes in all $920, and his money and effects are but $607,50; what will a creditor receive on $11,333 ?
Ans. $ 7,083. 13. Bought 126 gallons of rum for $110, how much water must be added to it, to reduce the first cost to 75 cents per gallon?
Ans. 20 gal. 14. If a person, whose rent is $145, pays $12,63 of taxes, how much should a person pay whose rent is $378 ?
Ans. $32,925. 15. If a board be 9 inches broad, how much in length will make a square foot ?
Ans. 16 inches. 16. How many yards of sarsenet, 3 qrs. wide, will line9 yards of cloth of 8grs. wide ?
Ans. 24 yards. K2
17. Lent a friend 292 dollars for 6 months ; some time afterwards he lent me 806 dollars ; how long may I keep it to balance the favour?
Ans. 2 months, 5 days. 18. How many yards of matting, 2 ft. 6 in. broad, will cover a floor, that is 27 ft. long, and 20 ft. broad ?
Ans. 72. 19. How many yards of cloth, 3 qrs. wide, are equal in measure to 30 yards, 5 qrs. wide ? Ans. 50 yards.
20. How many yards of canvass that is an ell wide, will line 20 yards of say that is 3 qrs. wide ? Ans. 12.
21. A wall, that is to be built to the height of 27 ft. was raised 9 feet by 12 men in 6 days ; how many men must be employed to finish the wall in 4 days, at the same rate of working ?
Ans. 36. 22. If, when the days are 13 hours long, a traveller perform his journey in 351 days; in how many days will he perform the same journey, when the days are 110 hours long?
Ans. 4093 days. 23. How many yards of cloth may be bought for $195,75 of which 91 yards cost $11,02 ? Ans. 168yds. 3qrs.
24. If I buy 57 yards of cloth for 49 guineas, at 28 shillings each ; what did it cost per Ell English ?
Ans. $5,012. 25. If the distance from Worcester to Templeton be 31 iniles ; I demand how many times wheel, whose circumference is 15 feet, will turn round in performing the journey?
Ans. 10560 times. 26. What will 373 gross of buttons come to at 13 cents per dozen ?
Ans. $58,50. 27. A farm, containing 125 acres, 3r. 27pls. is rented at $11,50 per acre ; what is the yearly rent of that fapm ?
Ans. $1447,065. 28. If jo of a ship cost $1163, what is the whole worth ?
Ans. $2658,285. 29. A merchant bought 9 packages of cloth at 3 guineas for 7 yards; each package contained 8 parcels, each parcel 12 pieces, and each piece 20 yards; how many dollars came the whole to, and how much per yard ?
Aps. $34560 the whole ; $2 per yd.