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weight, including what was lost, and how much had they left to subsist on ?
Ans. The whole weight, 147000lb. ; left to subsiston, 126000lb.
65. If 2000 soldiers, after losing one seventh part of their bread, had each 12 ounces a day for 12 weeks, what was the whole weight of their bread, including that lost, and how much might they have had per day, each man, if none had been lost ?
Ans. The whole weight was 147000lb. ; the loss, 21000lb.; had none been lost, they might have had 14 ounces per day.
66. If .85 of a gallon of wine cost $ 2.72, how much will .25 of a gallon cost ?
Ans. $ 0.80. 67. If 61.3 pounds of tea cost $ 44.9942, what is the price per lb. ?
Ans. $0.73,4. 68. What is the value of .15 of a hogshead of lime, at $2.39 per hhd. ?
Ans. $ 0.35,85. 69. If .75 of a ton of hay cost $15, what is it per ton ?
Ans. $ 20. 70. How many yards of carpeting that is half a yard wide will cover a room that is 30 feet long and 18 feet wide ?
Ans. 120 yards. 71. If a man perform a journey in 15 days when the day is 12 hours long, in how many days will he do it when the day is but 10 hours long?
Ans. 18 days. 72. If 450 men are in a garrison, and their provisions will last them but 5 months, how many must leave the garrison that the same provisions may be sufficient for those who remain 9 months ?
Ans. 200 men. 73. The hour and minute hands of a watch are together at 12 o'clock; when will they next be together?
Ans. lh. 5m. 27 3 sec. 74. A and B can perform a piece of work in 55 days, B and C in 6 days, and A and C in 6 days; in what time would each of them perform the work alone, and how long would it take them to do the work together ?
Ans. A would do the work in 10 days; B, in 12 days; C, in 15 days; A, B, and Ç, together, in 4 days.
75. A, B, and C can perform a piece of work in 4 days, B can do it in 12 days, C can do it in 15 days; in what time would A and B perform the labor ?
Ans, 56 days. 76. How many bricks 8 inches long, 4 inches wide, and 2 inches thick, will it require to build the walls of a house which is 46 feet long, 28 feet wide, and 25 feet high, and the walls to be 18 inches thick ?
Ans. 143,775 bricks.
77. Lent a friend $ 200 for 12 months, on condition of his returning the favor ; how long ought he, to lend me $150 to requite my kindness ?
Ans. 16 months. 78. If 5 oxen or 7 cows eat 341 tons of hay in 87 days, in what time will 2 oxen and 3 cows eat the same quantity of hay?
Ans. 105 days. 79. If 360 men be placed in a garrison, and have provisions for 6 months, how many men must be sent away at the end of 4 months that the remaining provision may last them 8 months longer?
Ans. 270 men. 80. My tailor informs me it will take 104 square yards of cloth to make me a full suit of clothes. The cloth I am about to purchase is 18 yards wide, and on sponging it will shrink 5 per cent. in width and length. How many yards of the above cloth must I purchase for my "new suit”? , Ans. 616, yd.
DOUBLE RULE OF THREE.
COMPOUND PROPORTION is the method of performing such operations in Proportion as require two or more statements.
EXAMPLES. 1. If a man travel 117 miles in 30 days, employing only 9 hours a day, how far would he go in 20 days, travelling 12 hours a day?
The distance to be travelled depends on two circumstances, - the number of days the man travels, and the number of hours he travels in each day.
We will first suppose the hours to be the same in each case ; the question will then be, — If a man travel 117 miles in 30 days, how far will he travel in 20 days ? This will lead to the following proportion. 30 days : 20 :: 117 miles : ****2 = 78 miles.
That is, if we multiply 117 by 20, and divide the product by 30, we obtain the number of miles he will travel in 20 days, which is 78.
Now, if we take into consideration the number of hours, we must say, - If a man, travelling 9 hours a day for a certain number of days, has travelled 78 miles, how far will he go in the same time, if he travel 12 hours a day? This will furnish the following proportion. .
9 hours : 12 hours : : 78 miles : 2 = 104 miles, the answer to the question.
By this mode of resolving the question, we see that 117 miles have, to the answer 104 miles, the proportion that 30 days have to 20 days, and that 9 hours have to 12 hours. Stating this in Compound Proportion, we have
30 : 20 ..11: 104 miles, the answer.
9:12 Thus it appears that if 117 be multiplied by both 20 and 12, and the product be divided by 30 times 9, the quotient will be 104 miles; or if we multiply 117 by 20, and divide the product by 30, and then multiply this quotient by 12 and divide by 9, it will produce the same answer as before.
This question may be performed by analysis thus : — If he travel 117 miles in 30 days, in one day he will travel to of 117 miles, which is 4 miles ; and, travelling 9 hours a day, he will in one hour travel f of miles, which is 18 miles; and in a day of 12 hours he will travel 12 times 3 miles, which is 156 miles; and in 20 days he will travel 20 times 156 miles, which is 104 miles, the answer, as before.
The answer to the above question might have been obtained by dividing the third term by the product of the two ratios which the first two terms have to the second terms; that is, by the ratio of 30 to 20, which is : =; and of 9 to 12, which is i. Thus,
117= 3x = 117= =12x5=936 = 104 Ans. 2. If 6 men in 16 days of 9 hours each build a wall 20 feet long, 6 feet high, and 4 feet thick, in how many days of 8 hours each will 24 men build a wall 200 feet long, 8 feet high, and 6 feet thick ?
In stating this question, there are several circumstances to be taken into consideration ; the number of men employed,
the length of the days, length of the wall, and its height and breadth.
As the answer to the question is to be in days, we make the days the third term.
Were all the circumstances of the question alike, except the number of men and the number of days, the question would consist in finding in how many days 24 men would perform the same labor that 6 men had done in 16 days; that is, if 6 men had built a certain wall in 16 days, how many days would it take 24 men to perform the same labor ? This would furnish the following proportion.
24 men : 6 men :: 16 days : 6229 = 4 days. Or, if this were the question, - If a certain number of men, by laboring 9 hours a day, perform a piece of work in 16 days, how many days would it take the same men to do the labor by working 8 hours a day? - the following would be the proportion.
8 hours : 9 hours :: 16 days : 9x18 = 18 days. Or, if this were the question, - If a certain number of men build a wall 20 feet long in 16 days, how long would it take the same men to build a wall 200 feet long? - the following would be the statement.
20 feet : 200 feet :: 16 days : 16X200 = 160 days. Or, if only the days and height of the wall were considered, this would be the statement.
6 feet : 8 feet :: 16 days : 8x18 = 214 days. Lastly, were we to consider only the days and the thickness of the wall, it would furnish the following statement.
4 feet : 6 feet :: 16 days : 6X16 = 24 days. We see, by this mode of resolving the question, that 16 days must have to the true answer the ratio compounded of the ratios
That 24 men have to 6 men ;
That 4 feet have to 6 feet.
24 men : 6 men ? • 8 hours : 9 hours 20 feet : 200 feet >:: 16 days : 90 days = Answer. 6 feet : 8 feet
4 feet : 6 feet The continued product of all the second terms by the third term, and this divided by the continued product of the first terms, will produce the answer. Thus 6X9X200X8X6X16 .
= 8294 = 90 days, Answer. 2AX8X20X6X4
OPERATION BY CANCELLING.
24 X 8 X 20 X 6X4 = 90 days, Ans.
3. If 5 compositors in 16 days, 11 hours long, can compose 25 sheets of 24 pages in each sheet, and 44 lines in a page, and 40 letters in a line, in how many days 10 hours long may 9 compositors compose a volume, to be printed on the same letter, consisting of 36 sheets, 16 pages to a sheet, 50 lines to a page, and 45 letters in a line ?
Ans. 12 days.
: 16 days : 12 days, Ans.
OPERATION BY CANCELLING,
3 € $ $ 4 $ X IX X 36 X 16 X $ø X 45 X 16
9 X 1Ø x 25 x 24 x 44 x 40 = 12 days, Ans.
RULE. — Make that number which is of the same kind as the answer required the third term; and of the remaining numbers, take any two that are of the same kind, and consider whether an answer depending upon these alone would be greater or less than the third term, and place them as directed in Simple Proportion. Then take any other two, and consider whether an answer depending only upon them would be greater or less than the third term, and arrange them accordingly; and so on,