of the second piece is the same, or 608 sq. rods, and its length is 25 rods; its width must be the quotient of 608 divided by 25, which is 24 rods.

25

37. If a piece of cloth is 9 feet long and 3 feet wide: how long must be a piece of cloth that is 23 feet wide, to contain the same number of yards?

38. If it take 44 yards of carpeting, that is 11 yards wide, to cover a floor: how many yards, of of a yard wide, will it take to cover the same floor?

39. If a piece of wall-paper, 14 yards long and 11⁄2 feet wide, will cover a certain piece of wall, how long must another piece be, that is 2 feet wide, to cover the same wall?

40. If it takes 5.1 yards of cloth, 1.25 yards wide, to make a gentleman's cloak: how much serge, § of a yard wide, will be required to line it?

41. If 6 men can build a wall 80 feet long, 6 feet wide, and 4 feet high, in 15 days, in what time can 18 men build one 240 feet long, 8 feet wide, and 6 feet high?

ANALYSIS. The solid measure of the wall is found by multiplying the three dimensions together; 80 × 6 × 4 = 1920 cu. ft. In 15 days, 1 man can do

of 1920 320 cu. ft.; and in 1 day,

he will do of 320 cu. ft. 320. In 1 day, 18 men can do 18 times. 330 = 384 cu. ft.; and it will take 18 men as many days to build the wall, as 384 cu. ft. is contained times in the solid measure of the second wall: 240 × 8 × 6=11520 cu. ft.. and 11520÷÷384 30 days.

80 × 6 × 4

of 1920 of 320

320 15

= 1920 cu. ft.:

=

× 18 =

240 × 8 × 6

11520

384

OPERATION.

320 cu. ft. =

solid measure of first wall. work done in 15 days by 1 man. work done in 1 day by 1 man. 384 cu. ft. = work done in 1 day by 18 men. 11520 cu. ft. solid measure of second wall.

320 cu. ft. =

= 30 days

time for 18 men to do the work.

Or,

240X8X6 of 1 of 80 X6 X4 X 18 =

1

42. If 96 lbs. of bread be sufficient to serve 5 men 12 days, how many days will 57 lbs. serve 19 men?

43. If a man travel 220 miles in 10 days, traveling 12 hours a day in how many days will he travel 880 miles, traveling 16 hours a day?

44. If 9 men pay $135 for 5 weeks' board, how much must 8 men pay for 4 weeks' board?

45. If 12 men reap 80 acres in 6 days, in how many days will 25 men reap 200 acres?

46. If 4 men are paid 24 dollars for 3 days' labor, how many men may be employed 16 days for $96 ?

47. A wall, to be built to the height of 27 feet, was raised to the height of 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?

48. Two men bought a horse for $150: one paid $90, and the other, $60; they sold the horse, and gained $75: what did each gain?

90

150

ANALYSIS.-Each must have the same part of the gain that the money which he paid, is of the whole money paid. One paid $90, which is = 3 of $150, and he ought to receive of the gain; and the other paid $60, which is = 3 of $150, and he ought to receive of the gain. 3 of $75 and of $75 = $30 = gain of the other.

60

150

=

$45

= gain of

one;

49. Three persons bought 2 barrels of flour for 15 dollars. The first one ate from them 2 months, the second 3 months, and the third 7 months: how much should each pay?

50. If two persons engage in a business, where one ad

vances $875 and the other $625, and they gain $300, what is each one's share?

51. A, B, and C sent a drove of hogs to market, of which A owned 105, B 75, and C 120; on the way, 60 died: how many must each lose?

52. A man who has only $50, owes $75 to A, $150 to B, and $100 to C: how much ought he to pay to each?

53. A can do a piece of work in 4 days, and B can do the same in 6 days: in what time can they both do the work, if they labor together?

5

ANALYSIS. Since A can do the work in 4 days, in 1 day he can do of the work, and B can do of the work in 1 day: both can, in 1 day, do the sum of and: ¦ + } since in 1 day they can do of the work, it will require, for the whole work, as many days as is contained times in 1.

12

=

1÷矗

÷ + } = 1 2 = 1/2 = what both can do in 1 day; 1 ÷ 11⁄2 = 1 × 12 = 12 = 23 days (required for A and B to

5

12

12 =

do the whole work.

54. A can build a shed in 6 days, and B can build it in 5 days: in what time can they, by working together, build the shed?

55. A father earns, in 9 days, $18, and his son earns the same amount in 15 days: in what time could they, together, earn the amount ?

56. A laborer can dig a trench in 25 days, but with the assistance of a second laborer, he digs it in 16 days: in what time would the second laborer, alone, have dug it?

57. A can build a wall in 16 days, and B can do it in 21 days; they both worked on the wall; after working 5 days, B left it: in what time could A finish the work?

58. A can build a wall in 18 days, and B can do it in 24 days; A worked alone for 6 days, and was then assisted by B: in what time was the work finished?

59. If a barrel of flour would last a family 6 weeks, and if it would last a second family 8 weeks: how long would of the barrel last both families?

60. Divide $500 between 3 persons, giving to one $1, as often as to the second $4, and to the third $.

61. Divide $176.40 among 3 persons, so that the first shall have twice as much as the second, and the third three times as much as the first: what is each one's share?

62. A person bought 3 lots of ground for $6000; he paid $150 more for the second than for the first, and $350 more for the third than for the second: what was the cost of each?

63. Three men hire a pasture, for which they pay 66 dollars. The first puts in 2 horses for 3 weeks; the second, 6 horses for 2 weeks; the third, 9 horses for 13 weeks: how much ought each to pay?

ANALYSIS. The pasturage of 2 horses for 3 weeks, would be the same as the pasturage of 1 horse 2 times 3 weeks, or 6 weeks; that of six horses, 2 weeks, the same as for 1 horse six times 2 weeks, or 15 weeks; and that of 9 horses 13 weeks, the same as 1 horse for 9 times 11 weeks, or 12 weeks. The three persons had an equivalent for the pasturage of 1 horse for 6+15 +12=33 weeks; therefore, the first must pay; the second, 33; and the third, 13 of 66 dollars.

15

64. Two persons, A and B, enter into partnership, and gain $175. A puts in 75 dollars for 4 months, and B puts in 100 dollars for 6 months: what is each one's share of the gain?

65. Three men engage to build a house for 580 dollars. The first one employed 4 hands; the second, 5 hands; and . the third, 7 hands. The first man's hands worked 3 times as many days as the third, and the second man's hands twice as many days as the third man's hands: how much must each receive?

ALLIGATION.

350. ALLIGATION is the process of mixing substances in such a manner that the value of the compound may be equal to the sum of the values of the several ingredients.

ALLIGATION MEDIAL.

351. ALLIGATION MEDIAL is the process of finding the mean price of a mixture, when the quantity of each simple and its price, are known.

1. A merchant mixes 8 lb. of tea, worth 75 cents a pound, with 16 lb., worth $1.02 a pound: what is the price of the mixture per pound?

OPERATION.

8 lb. at 16 lb. at

75 cts.

$ 6.00

$1.02 = $16.32

24

ANALYSIS-The quantity, 8 lb. of tea, at 75 cents a pound, costs $6; and 16 lb., at $1.02, costs $16.32: hence, the mixture, = 24 lb., costs $22.32; and the price of 1 lb. of the mixture is found by dividing this cost by 24: Hence, to find the price of the mixture,

24 ) 22.32

Rule.-I. Find the cost of the entire mixture:

$0.93

II. Divide the entire cost of the mixture by the sum of the simples, and the quotient will be the price of the mixture.

Examples.

1. A farmer mixes 30 bushels of wheat, at 5s. per bushel, and 72 bushels of rye, at 3s., with 60 bushels of barley, at 2s.: what is the price of 1 bushel of the mixture?

2. A wine-merchant mixes 15 gallons of wine, at $1 per gallon, with 25 gallons of brandy, worth 75 cents per gallon: what should be the price of a gallon of the compound?

3. A grocer mixes 40 gallons of whisky, worth 31 cents per gallon, with 3 gallons of water, which costs nothing: what should be the price of a gallon of the mixture?

4. A goldsmith melts together 2 lb. of gold, of 22 carats