465. Examples like the preceding are commonly arranged under the rule of Bankruptcy.

Note.-A bankrupt is a person who is insolvent, or unable to pay his just debts.

43. A bankrupt owes $5000, and his property is worth $3500: how much can he pay on a dollar?

44. A man died owing $16400, and his effects were sold for $4100 what per cent. did his estate pay ?

45. If a man owes A $6240, B $8760, and C $9000, and has but $11500, how much will each creditor receive?

46. If I owe $48000, and have property to the amount of $32000, what per cent. can I pay?

47. What per cent. can a man pay, whose liabilities are $120000, and whose assets are $45000?

48. What per cent. can a man pay, whose liabilities are $1500000, and whose assets are $150000?

466. It often happens in storms and other casualties at sea, that masters of vessels are obliged to throw portions of their cargo overboard, or sacrifice the ship and their crew. In such cases, the law requires that the loss shall be divided among the owners of the vessel and cargo, in proportion to the amount of each one's property at stake.

The process of finding each man's loss, in such instances, is called General Average.

OBS. The operation is the same as that in solving questions in bankruptcy and partnership.

49. A, B and C, freighted a ship from New York to Liverpool; A had on board 100 tons of iron, B 200 tons, and C 300 tons, in a storm 240 tons were thrown overboard: what was the loss of each ?

50. A packet worth $36000 was loaded with a cargo valued at $65000. In a tempest the master threw overboard $25250 worth of goods: what per cent. was the general average?

51. A steam ship being in distress, the master threw of the cargo overboard; finding she still labored, he afterwards threw overboard of what remained. The steamer was worth

$120000, and the cargo $240000: what per cent. was the general average, and what would be a man's loss who owned of the ship and cargo?

52. A man mixed 25 bushels of peas worth 6s. a bushel, with 15 bushels of corn worth 4s. a bushel, and 20 bushels of oats worth 3s. a bushel: what was the mixture worth per bushel? Analysis. 25 bu. peas at 6s.-150s., value of the peas;

The mixture=60 bu.

and 270s., value of whole mixture. Now if 60 bu. mixture are worth 270s., 1 bu. mixture is worth of 270s.; and 270s.÷60=44s. Ans.

PROOF.-60 bu. at 44s.=270s., the value of the whole mixture.

467. The process of finding the value of a compound or mixture of articles of different values, or of forming a compound which shall have a given value, is called Alligation. Alligation is usually divided into two kinds, Medial and Alternate.

OBS. 1. When the prices of the several articles and the number or quantity of each are given, the process of finding the value of the mixture, as in the last example, is called Alligation Medial.

2. When the price of the mixture is given, together with the price of each article, the process of finding how much of the several articles must be taken to form the required mixture, is called Alligation Alternate, Alligation Alternate embraces three varieties of examples, which are pointed out in the following notes.

53. If you mix 40 gallons of sperm oil worth 8s. per gallon, with 60 gallons of whale oil worth 3s. per gallon, what will the mixture be worth per gallon?

54. At what price per pound can a grocer afford to sell a mixture of 30 lbs. of tea worth 4s. a pound, and 40 lbs. worth 7s. a pound?

55. If 120 lbs. of butter at 10 cts. a pound are mixed with 24 Ibs. at 8 cts. and 24 lbs. at 5 cts. a pound, what is the mixture worth?

56. A tobacconist had three kinds of tobacco, worth 15, 18, and 25 cents a pound: what is a mixture of 100 lbs. of each worth per pound?

57. A liquor dealer mixed 200 gallons of alcohol worth 50 cts. a gallon, with 100 gallons of brandy worth $1.75 a gallon: what was the value of the mixture per gallon?

58. A grocer sells imperial tea at 10s. a pound, and hyson at 4s. what part of each must he take to form a mixture which he can afford to sell at 6s. a pound?

Note.-1. It will be observed in this example that the price of the mixture and also the price of the several articles or ingredients are given, to find what part of each the mixture must contain.

Analysis. Since the imperial is worth 10s. and the required mixture 6s., it is plain he would lose 4s. on every pound of imperial which he puts in. And since the hyson is worth 4s. a pound and the mixture 6s., he would gain 2s. on every pound of hyson he puts in. The question then is this: How much hyson must he put in to make up for the loss on 1 lb. of imperial? If 2s. profit require 1 lb. of hyson, 4s. profit will require twice as much, or 2 lbs. He must therefore put in 2 lbs. of hyson to 1 lb. of imperial.

PROOF-2 lbs. of hyson, at 4s. a pound, are worth 8s., and 1 lb. of imperial is worth 10s. Now 8s.+10s.-18s. And if 3 lbs. mixture are worth 18s., 1 lb. is worth of 18s., which is 6s., the price of the mixture required.

59. A farmer has oats which are worth 20 cts. a bushel, rye 55 cts., and barley 60 cts., of which he wishes to make a mixture worth 50 cts. per bushel: what part of each must the mixture contain?

Analysis.-The prices of the rye and barley must each be compared with the price of the oats. If 1 bu. oats gains 30 cts. in the mixture, it will take as many bu. of rye to balance it, as 5 cts.

(the loss per bu.) are contained times in 30 cts., viz: 6 bu. Again, since 1 bu. oats gains 30 cts., it will take as many bushels of barley to balance it, as 10 cts. (the loss per bu.) are contained times in 30 cts., viz: 3 bu. Hence, the mixture must contain 2 parts of oats, 6 parts rye, and 3 parts barley.

60. If a man have four kinds of sugar worth, 8, 9, 11, and 12

cents a pound respectively, how much of each kind must he take to form a mixture worth 10 cents a pound?

Note.-2. In examples like the preceding, we compare two kinds together, one of a higher and the other of a lower price than the required mixture; then compare the other two kinds in the same manner. In selecting the pairs to be compared together, it is necessary that the price of one article shall be above, and the other below the price of the mixture. Hence, when there are several articles to be mixed, some cheaper and others dearer than the mixture, a variety of answers may be obtained. Thus, if we compare the highest and lowest, then the other two, the mixture will contain 1 part at 8 cts.; 1 part at 9 cts.; 1 part at 11 cts.; and 1 part at 12 cts. Again, by comparing those at 8 and 11 cts., and those at 9 and 12 cts. together, we obtain for the mixture 1 part at 8 cts. ; 2 parts at 11 cts. ; 2 parts at 9 cts.; and 1 part at 12 cts. Other answers may be found by comparing the first with the third and fourth; and the second with the fourth, &c.

61. A goldsmith having gold 16, 18, 23, and 24 carats fine, wished to make a mixture 21 carats fine: what part of each must the mixture contain?

62. A farmer had 30 bu. of corn worth 6s. a bu., which he wished to mix with oats worth 3s. a bu., so that the mixture may be worth 4s. per bu.: how many bushels of oats must he use?

Note.-3. In this example, it will be perceived, that the price of the mix ture, with the prices of the several articles and the quantity of one of them are given, to find how much of the other article the mixture must contain.

Analysis.-Reasoning as above, we find that the mixture (without regard to the specified quantity of corn) in order to be worth 4s. per bu., must contain 2 bu. of oats to 1 bu. of corn. Hence, if 1 bu. of corn requires 2 bu. of oats to make a mixture of the required value, 30 bu. of corn will require 30 times as much; and 2 bu. X30 60 bu., the quantity of oats required.

63. A merchant wished to mix 100 gallons of oil worth 80 cts. per gallon, with two other kinds worth 30 cts. and 40 cts. per gallon, so that the mixture may be worth 60 cts. per gallon: how many gallons of each must it contain?

64. A merchant has Havana coffee at 12 cts. and Java at 18 cts. per pound, of which he wishes to make a mixture of 150 lbs., which he can sell at 16 cts. a pound: how much of each must he use?

Note.-4. In this example, the whole quantity to be mixed, the price of the mixture, and the prices of the several articles are given, to find how much of each must be taken.

Analysis.-On 1 lb. of the Havana it is obvious he will gain 4 cts., and on 1 lb. of the Java he will lose 2 cts.; therefore to balance the 4 cts. gain he must put in 2 lbs. of Java; that is, the mixture must contain 1 part of Havana to 2 parts of Java. Now if 3 lbs. mixture require 1 lb. Havana, 150 lbs. mixture, (the quantity required,) will require as many pounds of Havana as 3 is contained times in 150, viz: 50 lbs. But the mixture contains twice as much Java as Havana, and 50 lbs X2=100 lbs.

Ans. 50 lbs. Havana, and 100 lbs. Java.

65. It is required to mix 240 lbs. of different kinds of raisins, worth 8d., 12d., 18d., and 22d. a pound, so that the mixture may be worth 10d. a pound: how much of each must be taken?

66. If 10 horses consume 720 quarts of oats in 6 days, how long will it take 30 horses to consume 1728 quarts?

Analysis. Since 10 horses will consume 720 qts. in 6 days, 1 horse will consume of 720 qts. in the same time; and of 720 qts. is 72 qts. And if 1 horse will consume 72 qts. in 6 days, in 1 day he will consume of 72 qts., which is 12 qts. Again, if 12 qts. last 1 horse 1 day, 1728 qts. will last him as many days as 12 qts. are contained times in 1728 qts., viz: 144 days. Now if 1 horse will consume 1728 qts. in 144 days, 30 horses will consume them in of the time; and 144 d.÷30=4%.

Ans. 30 horses will consume 1728 qts. in 44 days.

468. This and similar examples are usually placed under the rule of Compound Proportion, or Double Rule of Three.

67. If 15 horses consume 40 tons of hay in 30 weeks, how many horses will it require to consume 56 tons in 70 weeks?

68. If 8 men can make 9 rods of wall in 12 days, how long will it take 10 men to make 36 rods?

69. If 35 bbls. of water will last 950 men 7 months, how many men will 1464 bbls. of water last 1 month?