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By Proportion, Rule II.
dif. err. dif. As 5 : 20 :: 2
As 5: 25 :: 2 2
10 "Add the Position, 8
Add the Position, 6
Ans. 16 Dem.—It is easy to perceive, that the fourth number here must be added to the position, because by increasing the quantity of whiskey, the quantity of brandy is lessened, and consequently lessens the result, which is too great. The reason of the assumed proportion will appear evident by considering that the result varies with the position; and that the errour of the result is ever proportional to the errour of its position; and the difference of any result is to the errour of its position; so that the errour of any two results has the same proportion to the difference of the position from which they flow, as the difference of any other two results has to the difference of their position. But the difference of the errours is equal to the difference of the results, therefore, as the difference of the errours is to either errour, so must the difference of the positions be to a fourth number, that is, the errour of the position.
2. A labourer was hired for 20 days, at 6 shillings per day, for every day he worked; but with this condition, that for every day he played, he should forfeit 2 shillings; at the expiration of the time he received for his services 4 pounds 8 shillings; what number of days did he work ?
Ans. 16 days. 3. Two gentlemen, A and B, have both the same income. A saves } of his; but B, by expending $50 a year more than A, at the end of 4 years finds himself $100 in debt; what does each receive and spend per annum ?
Ans. Each receives $125 per annum; À spends $100 á year, and B spends $150.
4. There was a fish caught in Black River Bay, near Sackets Harbour, in 1831, whose head was 151. inches long, its tail was as long as the head and half the body, and the body was exactly the length of both head and tail; what was the length of the whole fish ?
Ans. 10 feet 4 inches. 5. What number is that, which being increased by its 7, its 1. and 5 more, will be just doubled ?
Ans. 20. 6. A gentleman has 2 fine horses, and a carriage worth 8400; now if the first horse be harnessed in it, he and the
carriage will be triple the value of the second; but if the se. cond be put in, he and the carriage will be worth 7 times the value of the first horse; what is the value of each horse ?
Ans. $80 the value of the first, $160, the second 7. Friend John, who had in credit liv'd,
Though now reduc'd, a sum receiv'd-
8. A gentleman dying left his two sons $20,000, (the one 10 years old and the other 15,) to be so divided that the share of each being put out on interest at 5 per cent, would amount to equal sums when they respectively became 21 years of age; what was the share of each ? $912249, the younger son's share.
$1087737, the share of the elder son.
QUESTIONS ON DOUBLE POSITION. What does Double Position teach? A. It teaches to resolve certain questions by means of two suppositions and false numbers. How do you obtain the errours ? A. By assuming any two convenient numbers, and proceeding with each of them separately, according to the conditions of the question, as in Single Position. After obtaining the errours, how do you proceed to obtain the answer by Proportion? A. By stating the question thus: as the difference of the errours, if alike, or their sum if unlike, is to either errour, so is the difference of the positions, to a fourth proportional number, which being added to, or subtracted from, the position producing the errour, gives the number required.
PRACTICAL QUESTIONS. 1. Three brothers on counting their money, found that Joseph had $20, George five dollars more than Joseph, and John as many as the other two; what number of dollars had the three?
Ans. $90. 2. The Arabian or Indian method of notation was first known in England about the year 1150; how long is it since to the present year, 1834 ?
Ans. 684 years. 3. There are two numbers, the greater of them is 73 times 109; and their difference 17 times 28; what is their sum and product ?
15438 their sum. Ans.
59526317 their product. 4. Suppose nine thousand men march in a column of seven hundred and fifty deep; how many march abreast? Ans. 12.
5. A merchant would ship 360 bushels of corn in barrels which hold 3 bushels and 3 pecks each; how many barrels does he need?
Ans. 96 barrels. 6. A man can perform a certain journey in 351 days, travelling 13 hours a day; how long will it require to perform the same journey when he travels only 117 hours a day.
Ans. 4065 days. 7. A man bought 180 oranges at the rate of 2 for a penny, and 180 more at the rate of 3 for a penny; after which, he sold them out at the rate of 5 for 2 pence; did he gain or lose by the bargain?
Ans. He lost 6 pence. 8. Two men, A-and B, being on opposite sides of a wood, which is 536 yards about, they begin to go round it, both the same way, at the same instant of time; A goes at the rate of 11 yards per minute, and B at the rate of 34 yards in 3 minutes; what number of times must the wood be gone round before B overtakes A ?
Ans. 17 times. 9. It is required to divide 600 acres of land among three men, A, B and C; so that B may have 100 acres more than A, and C 64 more than B. Ans. A 112a. B 212a. C 276a.
10. A gentleman divided his fortune among his three sons, giving A £9 as often as B £5, and to C but £3 as often as B £7, and yet C's dividend was £2584; what was the whole estate ?
Ans. £19466 2s. 8d. 11. The clocks in Italy go on to 24 hours; then how many times does a clock strike, in that country, in performing one complete revolution of the index?
Ans. * 300 times,
12. What length must be cut off from a board 94 inches broad, to contain a square foot, or as much as 12 inches in length and 12 in breadth ?
Ans. 15 inches, 13. If 14 men can finish a job in 15 days; how many men must be added to complete it in 2 days? Ans. 91 men.
14. The top of a liberty-pole, being broken off by a gust of wind, struck the ground on a horizontal plane, at the distance of 10 feet from the foot of the pole; what was the height' of the whole pole, allowing the length of the broken piece to be 26 feet?
Ans, 50 feet. 15. *Suppose a pole 90 feet high, to stand on a horizontal plane; at what height must it be cut off so that the top may fall touching the ground 30 feet from the bottom, and the other end of the part cut off may rest on the stump or upright part?
Ans. 40 feet. * From the square of the length of the pole, (that is, the sum of the parts forming the hypotenuse and perpendicular,) take the square of the base, (that is, the square of the distance between the bottom of the pole and the top resting on the ground,) then divide the remainder by twice the length of the pole, and the quotient will express the height at which the pole must be cut off,
16. There are three horses belonging to different men, employed to draw a load from Watertown to Albany for $35; A and B are supposed to do it of the work, A and City and B and C te of the work. They are to be paid proportional; will you by your work find out what each man is entitled to receive?
A's share $13379.
14357 17. A man has a room 197 feet long, and 121 feet broad; he wishes to have the floor laid with hard bricks, each 6 inches long, and 3 inches wide; how many bricks will it take?
Ans. 1950 bricks. 18. Three towns, A, B, and C, are so situate that B lies 30 miles directly east of A, and C lies 40 miles directly south of A; what is the distance from B to C? Ans. 50 miles.
19. In working a sum' in Division, I found the remainder to be 649, the quotient was 113, the divisor was a sum eq to both and 24 more; what was the dividend ?
4 7 2
20. A circular fish-pond is to be made in a garden, that shall take up just half an acre; what length of cord will it require to reach from the centre to the circumference ?
Ans. 277 yards. 21. How many 3 inch cubes may be cut out of a 12 inch cube?
Ans. 64 cubes. 22. A person goes to a tavern with a certain sum of money in his pocket, where he spends 2 shillings; he then borrows as much money as he had left, and going to another tavern, he there spends 2 shillings also; then borrowing again as much money as he had left, he went to a third tavern, where he likewise spent 2 shillings; and thus again repeating the same at a fourth tavern, he then had nothing remaining; what sum had he at first?
Ans. 3s. 9d. 23. What is the area of a right angled triangle, the three sides of which are 30, 40 and 50 rods? Ans. 600 rods.
24 A stationer sold quills at 11s. per thousand, by which he cleared of the money, but quills growing scarce, he raised them to 13s. 6d. per thousand; what did he clear per cent by the latter price?
Ans. £96 75. 3 d. 25. When Hymen's golden knot was tied
That made friend Delia thine,
Ans. Delia's age was 16, yours 32 years. 26. A father left his son a fortune, 1 of which he ran through in 8 months; of the remainder lasted him 12 months longer; after which he had only £820 left ; what sum did the father bequeath his son ?
Ans. £1913 6s. 8d. 27: A, leaves Exeter at 10 o'clock in the morning for London, and goes at the rate of 2 miles an hour without in termission; B sets out from London for Exeter, at 6 o'clock the same evening, and rides 3 miles an hour constantly: the