8. A cannon ball, at its first discharge, flies about a mile in 8 seconds, and light is 8 min. in travelling to us from the sun, a distance of about 95,000,000 mls. Required the ratio of these two velocities. Ans. 1,490,1964 to 178000000

51

9. A, B, and C own lots of ground of equal size, but of different values. If A's and B's be taken together, the average price per acre is twice the price per acre of A's alone; if B's and C's be taken together, the average price per acre is three times that per acre of C's; and if all three be put together the average price per acre is $23. What is the value per acre of each?

Ans. A's $15. B's $45. C's $9.

10. A man was engaged to work for 40 days, on condition, that for every day he labored, he should receive $1, and for every day he was idle he should forfeit 50 cts. At the end of the time he was $5 in debt. How many days was he idle? Ans. 30.

11. A man bought a cask of wine, but in conveying it home, of it leaked out. He sold the remainder at $2.50 per gal. and for the whole, received exactly what he paid for the cask at first. What price per gal. did he pay ? Ans. $2.

12. A garrison of 2,000 men was supplied with provisions for a year, but at the end of 2 months 300 were marched away; 3 months after 500 were added to them, and afterwards, 100 were added at every month's end until the provisions were exhausted. How long did they last? Ans. 9 mo. 0 d. 11 h.

13. Suppose a garrison of 2,000, diminished and increased as often as in the last example, and by the same numbers, and suppose at first, each man was allowed 40 oz. a day. Then suppose that after every change of numbers, the daily allowance was also changed, so that the stock of provisions might still last a year from the commencement of the time. What would be the successive allowances ? 14. At what time between 4 and 5 o'clock, are the hour and minute hands of a time piece together? Ans. 4 o'clock, 21 m. 49. 9 sec. 15. What time is it, when the time past from noon is of the time onward to midnight? Ans. 3 o'clock.

16. Soldiers, marching at quick time, make about 120 paces, of 28 inches, per minute, or 2 paces per second. At this rate, how long will a detachment be in arriving at a fort, 20 miles distant, allowing a halt of 1 hour? Ans. 7 h. 17 m. 84 sec.

17. A and B are exactly opposite to each other on a circular road, 536 yds. around. A goes 11 yds. per minute, and B, 34 yds. in 3 min. the same way. How long before they are together.

Ans. 1 d. 2 h. 48 m.

18. A person having one year spent more than his income, found that by saving of it annnally afterwards, he could in 4 years make good the deficiency, and have $20 besides. What was his income? Ans. $1,200.

19. A person, after spending and of his money, has $60 re. maining. What had he at first? Ans, $144.

NOTE. In the course of these examples may be found several of the nature of those, which are commonly classed under the rule of POSITION. The few,

immediately preceding and following this note are among this number. The principle of Position cannot be well explained without the use of Algebra; and when it is considered that the problems, which it is employed to solve, may be investigated in a manner much more simple and certainly much more easily comprehended, viz by fractions it will be seen with what propriety we have discarded it. It may be well however to mention the distinction between those classes of questions, which have been arranged under the separate heads, single and double Position.

The former of these classes comprehends those cases, in which, after a number has been increased or diminished, by itself, or by parts of itself, the result is giv. en, and the number required. In this case it will be seen, that the number required will be greater or less, according as the number given is greater or less.-. That is, the number required, has a fixed ratio to the number given. By Position, then, the solution would be as follows. Suppose the answer to be any number you choose to take, and increase or diminish that number, according to the terms of the question. The result, which you thus obtain, will, of course have the same ratio to its supposition, which the number in the question has to the answer which is then found by Simple Proportion.

The other class comprehends those cases, in which, after a number has been increased or diminished by some other number or numbers, which are not known parts of the former number, the result is given, and the number required. In this case, two suppositions are necessary; but as the principle of the operation cannot be clearly explained to pupils, acquainted only with Arithmetic, we omit further particulars.

20. A and B commenced playing together with equal sums of money. A won $20, but directly after lost of what he then had ; when B's money was twice A's. What had each at first. A. $100.

21. A and B have equal incomes; but B spends $50 yearly more than A, who saves of his. In 4 years, B is $100 in debt. the income of each? Ans. $125.

What is

22. A and B lay out equal sums in trade. A gains $126, and B loses $87; A then gains of what he has, and B loses of what he has. A's money is then 3 times B's. What had each at first?

Ans. $300.

23. A fish has a head 9 in. long, a tail as long as his head and half his body, and a body as long as his head and tail. Required his whole length. Ans. 6 ft.

24. A merchant has three pieces of broadcloth of equal lengths. The average price of the first and second together is $7 per yd., and the average price of the whole together is of the third. What is the price of each? A. $6, $8, $16.

25. A person, after spending $20 more than of his income, had remaining $30 more than of it. What was his income? A. $200. 26. A man bought equal numbers of oranges and lemons, obtaining of the former, 3 for a shilling and of the latter 2 for a shilling. He sold the whole at the rate of 5 for 2 shillings, and, by so doing, lost 6 shillings. How many of each did he buy? A. 180.

27. HARTFORD and ALBANY are about 100 miles distant from each other. A leaves the former place at 26 minutes past 9 in the morning, and B, the latter, at 4 minutes before 3 in the afternoon; they meet at 4 minutes before 7 in the evening, when it appears that the distance A travelled before B started, is 12 mls, more than he has tra yelled since. How far did each travel per hour?

A. A, 8 mls. B, 6 mls.

28. In a lottery formed by combinations of 3, from 60 numbers, how many prize tickets having 3 drawn numbers on them, how many having 2, and how many having 1? A.84; 1,836; 11,475.

29. A merchant, having annually increased his estate by $100 more than of it, at the end of 4 years found it amounted to $10,342.18. What had be at first? A. $4000.

30. Divide 1,200 acres of land among A, B, and C, so that B may have 256 more than A's share, and C may have 270 more than B's share. A. A, 312, B, 412, C, 476.

31. A sets out from HARTFORD for BOSTON at 7 minutes past 8 'clock in the morning, and some time afterwards, B sets out from BOSTON for HARTFORD. They meet after B has been travelling 5 hours, when it is found that the distance travelled by A, before B started, is to the distance he has travelled since, as 23 to 22. what time did B start, and at what time did they meet.

At

A. B started 8 m. before 2, afternoon-They met 22 m. past 7. 32. If the numerator of a certain fraction be increased by 1, the value will be, but if the denominator be increased by 1, it will be }. Required the fraction. A...

33. A's age is double of B's, and B's is triple of C's: also, A's and B's together are 126. Required the several ages.

A. A's 84, B's 42, C's 14. 34. Divide the number 90 into 4 such parts, that the first, increased by 2, the second, diminished by 2, the third, multiplied by 2, and the fourth, divided by 2, shall all be equal.

A. 18; 22; 10; and 40.

35. A person at play lost of his money, and then won 3 shillings. He then lost of what he had, and then won shillings: finally he lost of what he then had, and found he had but 12 shillings left. What had he at first ? A. £1.

36. A, B, C and D were to share $100,000, as follows. A was to have, B, C, and D the rest. But C's and D's shares having become forfeit, it is required to divide the whole fairly between A and B. Ans. A $57,142.8574. B $42,857.1426.

37. A holds an estate on which B and C have claims as follows. If B applies alone, within a certain time, he is to have, and A, }. If C applies alone, he is to have, and A, . They both apply together, by which A lost $2,400 more than if B had applied alone. What would have been A's share if C had applied alone? A. $2,100.

38. On the centre of a dial-plate turn an hour hand, a minute hand, second hand, and a hand revolving once in 30 days, intended to mark the day of the month. They all start together from the figure 12. When will they next be together?

A. In 30 days exactly, at the same figure. 39. A, B and C draw prizes: A draws $200; B, as much as A and a third of what C draws; and C, as much as A and B both. What is the amount of the prizes? A. $1,200.

40. A gentleman bought several gallons of wine for $94; and after using 7 gals. himself, found that of the remainder was worth $20. How many gals. were there at first? A. 47.

41. A farmer has two flocks of sheep, each containing the same number. From the first he sells 39, from the second 93, and then finds twice as many remaining in the former as in the latter. How many sheep were there in each flock at first? A. 147.

The first year A gains

42. Two pieces of cloth of different lengths but of the same price per yd. were bought, the one for £5, and the other for £6; 10. If 10 yds. be added to the length of each, the sums will be as 5 to 6. Required the length of each piece. A. 20 and 26. 43. A and B begin trade with equal sums. $40, and B loses $40 The second year A loses at the end of the first, and B gains $40 less than that A had lost. B had then, twice as much as A. each begin with? A. $320.

of what he had twice the sum What sum did

44. Four places are situated in the order of the letters A, B, C, D. The distance from A to D is 34 miles. The distance A to B, is to the distance from C to D, as 2 to 3; and of the distance from A to B, added to half the distance from C to D, is 3 times the distance from B to C. What are the several distances?

Ans. A to B=12; B to C=4; C to D=18. 45. A person bought two casks of wine, one of which held 3 times as much as the other. From each he drew 4 gals. when there were 4 times as many gals. left in the larger as in the smaller? How many gals. in each at first? A. 12 and 36.

46. In a quantity of gunpowder the nitre was 10 lb. more than } of the whole, the sulphur, 4 lbs. less than of the whole, and the charcoal 2 lb. less than of the nitre. What was was the quantity of powder? A. 69 lb.

47. What time is it, when the time past, since noon, is of the time onward to midnight? A. 4 o'clock, 9 m. 1311 sec.

46. A shepherd being asked the number of sheep he had in his flock said, if I had as many more, half as many more, and 7 sheep and a half, I should have 500. How many had he? A. 197.9 49. A person in a tavern borrowed as much money as he had already, and spent 1 shillling. He then went to a second, and borrowed as much as he then had, and spent 1 shilling. Doing the same, likewise at a third and fourth, he had nothing left. What had he at first? A. 11 d.

50. A company at a tavern, on settling their bill, found that if there had been three persons more, they would have had a shilling less apiece to pay; and if there had been two less, they would have had a shilling more apiece to pay. Required the number of per、 sons, and each man's bill. A. 12 persons.-5s. each.