2. If 6 men do a piece of work in 4 days, in how many days will 8 men do it?

8 m. : 6 m.:: 4 days.

4

8)24

Ans. 3 days.

Here, because the time is required, we make 4 days the third term; and because 8 men will'do a piece of work in less time than 6 men, we write the greater number, 8, as the first term, and the other, 6, as the second.

Remarks. The reason of the Rule may be easily explained. Thus, in the first example, when 4 yards cost 12/, one yard will evidently cost 12-4-3/; and when 1 yard costs 3/, 6 yards will cost 6x3=18/. To discover the answer, then, it is plain, that both multiplication and division are necessary, and that the result is the same, whether we perform the one process first or the other. In the 2d example, it is plain, that the product of the second and third terms, (that is, 6×4=24 days,). is the time in which one man would perform the work, consequently, that 8 men will do it in part of that time, or 3 days. The same manner of reasoning will apply to any other instance whatever.

2. The first and second terms must always be of the same kind; and the third term of the same kind with the fourth, or answer. 3. In some cases, it is better to multiply and divide as in Compound Multiplication and Division, than to reduce the 3d term to its lowest name.

4. If any number occur to the memory which will divide the first and second, or first and third terms without a remainder, it will shorten the work to divide these terms by that number, and use the quotients thence arising instead of the original terms.

The following example will illustrate the last two remarks. Find the value of 80 yds. at the rate of £15: 17: 6 for 16 yds. 16 yds.: 80 yds. :: £15 17 6

Here, we divide the first and second terms by 16, and then work with the quotients 1 and 5, which evidently have the same ratio to each other as 16 and 80.

EXERCISES.

1. If 4 lb. of prunes cost 4/, what is the value of 20 lbs ? 2. If 4 yards of linen cost 14/, what is the value of 26 yards? 3. If a ship's crew have water to serve them 15 days, at the rate of 30 gallons a-day, what must their daily allowance be that it may serve them 25 days?

4. If 12 men consume a certain quantity of provisions in 15 days, how long will the same quantity serve 20 men at that rate? 5. How many barrels of 32 gallons each, will hold as much as 48 barrels of 42 gallons each?

6. A piece of calico, 25 inches wide, is valued at 2/1 per yard, what should be the price per yard of another piece of the same quality, 30 inches wide?

7. When butter sells at 1/7 per 22 oz., how much is it per 16 oz.? 8. If it require 33 yards of calico, 25 inches wide, to make curtains

for a bed, how many yards of calico 30 inches wide will it require?

9. If the penny loaf weighs 9 oz. 2 dr. when wheat sells at 25/ per boll, what ought it to weigh, when wheat sells at 35/ per boll? 10. What do a man's wages amount to in 143 days, at £28 ayear?

11. How many yards of broad cloth, at 15s. per yard, are equal in value to 24 reams of paper, at 17/6 per ream?

12. Received 140 gallons of whisky for 84 gallons of rum, at 15s. per gallon, what does the whisky cost me per gallon? 13. How many lbs. at 2/9 are equal in value to 110 lbs. at 4/6? 14. A butcher buys a piece of linen, measuring 26 yards, at 2/7 per yard, how much beef, at 10/8 per stone, must he give in

return?

15. If 8 bushels of wheat yield 6 bushels of flour, what should a sack of flour, or 5 bushels, cost, when the market price of flour is 63/ per qr., and the expense of grinding 2/ per qr.? 16. Suppose it is found by trial that a common dipped candle, weighing 2 oz. 13 dr., lasts 5 hours 19 minutes, and that a moulded candle, of the same weight, lasts 7 hours 20 minutes, what should a stone of the moulded cost, when a stone of the dipped sells at 13/4 ?

17. A Russian silver ruble contains 278 grains of pure silver,

what is its value if 11 oz. 2 dwt. of pure silver be worth 66/? 18. An American dollar contains 3714 grains of pure silver, what is its value if 11 oz. 2 dwt. of pure silver be worth 66/? 19. Find the value of 475 gallons of port wine, at £65:11 per pipe of 138 gallons.

20. Find the excise duty on 1386 gallons of port wine, at £44: 11 per tun of 252 gallons.

21. Find the value of an English ell of Holland, at 4/6 per

mish ell.

Fle

22. Find the value of 1 lb. of pure gold, at the rate of 444 gui

neas per 11 oz.

23. Find the value of 281bs. of sugar, at the rate of 9/7 for 11lbs. 24. Find the value of 3 quarters of broad cloth, at 17/9 per yard. 25. Find the value of 4 cwt. 3 qrs. 14 lbs. of cheese, at 65/4 per cwt.

26. Find the value of 15 casks, containing net 53 cwt. 1 qr. 21 lbs. of Montreal potashes, at 45/ per cwt.

27. Find the value of 5 bundles hemp, weighing 3 tons 6 cwt. 2 qrs. 7 lbs., at £63: 10 per ton.

28. Find the value of 1 qr. 2 nails of velvet, at 18/6 per yard. 29. Find the value of a yard, at 21/6 per English ell.

30. Find the value of a hhd. of sugar, weighing net 12 cwt. 3 qrst at 9/9 per 14 lbs.

21 lbs,

31. Find the value of a piece of sheet-lead, measuring 68 square feet, 8lbs. to the square foot, at 42/6 per cwt.

32. Find the value of 54 barrels of flour, at 58/6 per sack; 7 sacks being equal to 10 barrels.

33. Find the value of a pack of wool, (that is 240 lbs.) at 21/9 per stone of 14 lbs.

34. Find the value of flour per sack, when it sells at 10 dollars, each 4/6 per barrel.

35. Bought a hhd of sugar, weighing net 14 cwt. 2 qrs. 14 lbs. for £64: 7, what did it cost me per cwt. ?

36. How much money at 3 per cent. will yield as much interest as £490, at 4 per cent. ?

37. If a sum of money yields £17: 10 of interest at 3 per cent., what will it yield at 5 per cent. in the same time?

38. Find the interest of £316: 10: 6 for a year, at 5 per cent. 39. Find the interest of £275: 10 for a year, at 4 per cent. 40. Find the interest of £712:15: 6 for a year, at 34 per cent. 41. Find the commission on £512: 16: 6, at 24 per cent. 42. Find the insurance on £340: 15: 6, at 3/ per cent.

43. Find the duty on 10 bales of cotton wool, weighing net 2710 lbs. at 16/8 per 100 lbs.

44. Find the value of 1725 stones of hay, at 64/ per 100 stones. 45. Find the net weight of 38 cwt. 3 qrs. 10 lbs., allowing 4 lbs. of tare per 100 lbs.

46. If 34 yards of broad cloth, 1 yard wide, make a suit of clothes, how much cloth yard wide will it require?

47. If muslin yard wide costs 3/6 per yard, what should be charged for cloth of the same quality yard wide?

48. If 54 bolls of barley weigh 97 stones 15 lbs. Scotch troye, what will 20 bolls of the same weigh?

49. If 95 yards of broad cloth cost £7:2:9, what will 33 yards of the same cost?

50. If 10 cwt. 14 lb. of sugar cost £39: 14:04, what will 27 cwt. 3 qrs. of the same cost?

51. Received £7: 16 for part of a piece of broad cloth, containing at first 37 yards, value £29: 12; how much remains unsold? 52. A steeple projected a shadow to the distance of 33 yards, when a 4 feet staff, perpendicularly erected, cast a shadow 5 feet 6 inches; what was the height of the steeple ?

33. Suppose the arms of a deceitful balance to be to each other as 12 to 114, what weight will be required at the end of the shorter arm, to counterbalance a weight of 46 lbs. suspended from the end of the longer arm?

54. Suppose the arms of a deceitful balance to be to each other as 10 to 10, and suppose a weight of 35 lbs. hangs from the end of the shorter arm, what weight hung from the end of the longer arm will produce an equilibrium?

55. If 1 lb. of bread a-day be the allowance to a seaman, and if 14 lb. of flour yield 17 lb. 6 oz. of bread, how much flour will a crew of 56 men consume in a year?

56. A traveller walks 24 miles a-day, and after he has advanced 42 miles, another follows him, who walks 32 miles a-day, in what time will he overtake him?

57. How many yards of carpet, yard wide, will cover a floor 25 feet long, and 18 feet wide ?

38. How many yards of paper 23 inches wide, will cover a wall 30 feet long, and 94 feet high?

59. If 3 cwt. 13 lbs. of soap cost £14: 2:9, what will 7 cwt. 2 qrs. 25 lbs. cost?

60. If 5 cwt. 2 qrs. 15 lbs. of ashes cost £7:3: 54, how much may be bought for £9: 10:54 ?

61. Bought 2 cwt. 2 qr. 14 lbs. of sugar for £13:2:7, at what must I retail it per lb. to clear 46/ on the whole?

62. Bought 52 ells of broad cloth, at 20/6 per ell, how may I retail it per yard, to clear 7 guineas on the whole?

63. A bankrupt's debt amounts to £1565: 10, and his effects to £853: 15: 6, how much will his creditors receive per £? 64. How much water must be mixed with 84 gallons of gin, at 24/ per gallon, to reduce the price to 21/ per gallon?

65. How much water must be mixed with 120 gallons of rum, at 21/ per gallon, to reduce the price to 18/ per gallon?

66. Find the value of 25 yards of broad cloth at 23/9 per Eng

lish ell.

67. At what price per yard must I retail linen which cost me 2/9/ per yard, to clear 3d. on each shilling which it cost me? 68. If a retailer, who charges 3d. of profit on every shilling prime cost, draws £1000 in the course of a year, what is his whole gain?

69. If a retailer, who has 3d. of profit in every shilling he draws, draws £1000 in the course of a year, what is his whole gain? 70. If a retailer has 2d. of profit in every shilling he draws, to what extent must he deal to clear £100?

71. When wheat sells at 47/6 per quarter, what is it per boll; a quarter being 17,203 cubic inches, and a boll 8789 cubic inches?

72. When oats sell at 29/6 per quarter, what is that per boll of 12,822 cubic inches ?

73. When barley sells at 26/4 per boll, what is it per quarter? 74. When wheat sells at 35/6 per boll, what is it per quarter? 75. Sold 3 yds. 3 qrs. 2 nails for 21/114, from a piece of cloth, containing at first 25 English ells 4 qrs.; what is the value of the remainder at that rate?

76. A. has 200 yards of linen, worth 2/6 per yard, which he barters with B. at 2/7, taking in return yarn at 3/9 per spindle, which D. commonly sells at 3/4, ready money; how much

yarn does A. receive for his linen, and which of the two has the better bargain ?

COMPOUND PROPORTION.

WHEN the answer to a question in Proportion depends on seve ral circumstances considered collectively, it may be found by dividing the question into as many statings as its several particulars require, and working these separately as before. But such questions may be solved with greater ease and expedition by the following

RULE. Make that number the third term, which is of the same kind with the number required. Write any term of the supposition, and another of the same kind in the demand, as the first and second terms, and write the remaining terms, in similar pairs, directly un der these; considering, as you put them down, from the nature of the question, as in Simple Proportion, which term of each pair ought to stand first. The numbers being thus stated, multiply together the first term, and all that stand under it; in like manner, multiply together the second term, and all that stand under it; then multiply the latter product by the third term, and divide the result by the former product, the quotient will be the answer, in the same name with the third term.

Note. When it is necessary to reduce any term, the corresponding term must also be reduced to the same denomination, as in Simple Proportion.

EXAMPLE.

If 12 men cut down 32 acres of corn in 18 days, how many men will it require at that rate, to cut down 128 acres in 36 days?