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93. A miller had a quantity of rye worth 6s. per bushel, and wheat worth 9s. per bushel; he wishes to make a mixture of them which shall be worth 8s. per bushel : what part of each must the mixture contain?

Analysis. The difference in their prices per bushel is 3s.; hence, the difference in the price of 1 third of a bushel of each is 1s. Now if 1 third of a bushel is taken from a bushel of rye, the remaining 2 thirds will be worth 4s.; and if 1 third of a bushel of wheat which is worth 3s., be added to the rye, the mixture will be worth 7s. Again, if of a bushel is taken from a bushel of rye, the remaining third will be worth 2s., and if of a bushel of wheat, which is worth 6s., be added to the rye, the mixture will be worth 8s.; therefore, of a bushel of rye added to of wheat will make a mixture of 1 bushel, which is worth 8 shillings; consequently the mixture must be rye and wheat; or 1 part rye to 2 parts wheat.

PROOF. Since 1 bushel of rye is worth 6s., bu. is worth of 6s., or 2s.; and as 1 bu. of wheat is worth 9s., bu. is worth of 9s., or 6s. ; and 6s.+2s.=8s.

Note. If we make the difference between the less price and the price of the mixture, the numerator, and the difference between the prices of the commodities to be mixed, the denominator, the fraction will express the part to be taken of the higher priced article; and if we place the difference between the higher price and the price of the mixture over the same denominator, the fraction will express the part to be taken of the lower priced article.

94. A goldsmith has a quantity of gold 16 carats fine, and another quantity 22 carats fine; he wishes to make a mixture 20 carats fine; what part of each will the mixture contain. Ans. of 16 carats fine, and of 22 carats fine.

302. Examples requiring a mixture of commodities of different values, like the last three, are commonly classed under a rule called Alligation.

OBS. Alligation is usually divided into medial and alternate. The

92d example is an instance of Medial Alligation; the 93d and 94th are instances of Alternate Alligation. Questions in the latter very seldom occur in practical life.

95. A grocer mixes 50 pounds of tea worth 4 shillings a pound, with 100 lbs. worth 7s. a pound: what is a pound of the mixture worth?

96. A milk-man mixed 30 quarts of water with 120 quarts of milk, worth 5 cents per quart: what is a quart of the mixture worth?

97. A farmer made a mixture of provender containing 30 bushels of oats, worth 25 cents per bushel; 10 bushels of peas, worth 75 cents per bushel, and 15 bushels of corn, worth 50 cents per bushel: what is the value of the whole mixture; and what is it worth per bushel?

98. An oil dealer mixed 60 gallons of whale oil, worth 314 cents per gallon, with 85 gallons of sperm oil, worth 90 cents per gallon: what is the mixture worth per gallon?

99. A grocer had three kinds of sugar, worth 6, 8, and 12 cents per pound; he mixed 112 lbs. of the first, 150 lbs. of the second, and 175 of the third together: what was the mixture worth per pound?

100. A goldsmith melted 10 oz. of gold 20 carats fine with 8 oz. 22 carats fine, and 4 oz. of alloy: how many carats fine was the mixture?

101. If 4 men reap 12 acres in 2 days, how long will it take 9 men to reap 36 acres?

Analysis.-If 4 men can reap 12 acres in 2 days, 1 man can reap of 12 acres in the same time ; and of 12 acres is 3 acres. Now if 1 man can reap 3 acres in 2 days, in 1 day he can reap of 3 acres, and of 3 is 1 acre. Again, if 1 acre requires a man 1 day, 36 acres will require him as many days as 1 is contained times in 36; and 36÷124 days. Now if 1 man can reap the given field in 24 days, 9 men will reap it in of the time; and 24-9=23. Ans. 9 men can reap 36 acres in 2 days.

Note. This and similar examples are usually placed under Compound Proportion, or "Double Rule of Three." If the analysis of

them is found too difficult for beginners, they can be deferred till review.

102. If 7 men can reap 42 acres in 6 days, how many men can reap 100 acres in 5 days?

103. If 14 men can build 84 rods of wall in 3 days, how long will it take 20 men to build 300 rods?

104. If 1000 barrels of provisions will support a garrison of 75 men for 3 months, how long will 3000 barrels support a garrison of 300?

105. If a man travels 320 miles in 10 days, traveling 8 hours per day, how far will he go in 15 days, traveling 12 hours per day?

106. If 24 horses eat 126 bushels of oats in 36 days, how many will 32 horses eat in 48 days?

107. A lad returning from market being asked how many peaches he had in his basket, replied that,, and of them made 52: how many peaches had he?

Analysis. The sum of,, and 1=13. (Art. 127.) The question then is this: 52 is 13 of what number? Now if 52 is 13, is 52-13-4; and 12 is 4 x 12-48. Ans. 48 peaches.

12 12

PROOF. Of 48 is 24; is 16; and is 12. Now 24 and 16 are 40, and 12 are 52.

303. This and similar examples are often placed under a rule called Position, or Trial and Error.

OBS. The shortest and easiest method of solving them is by Analysis.

108. A farmer lost of his sheep by sickness; were destroyed by wolves; and he had 72 sheep left: how many had he at first?

109. A person having spent and of his money, finds he has $48 left: what had he at first?

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110. After a battle a general found that of his army had been taken prisoners, were killed, had deserted, and he had 900 left: how many had he at the commencement of the action?

111. What number is that and of which is 80 ? 112. What number is that of which if and } be added to itself, the sum will be 110?

113. A certain post stands in the mud, in the water, and 10 feet above the water: how long is the post? 114. Suppose I pay $85 for of an acre of land: what is that per acre?

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115. A man paid $2700 for of a vessel: what is the whole vessel worth?

116. A gentleman spent of his life in Boston, of it in New York, and the rest of it, which was 30 years, in Philadelphia: how old was he?

117. What number is that of which exceeds of it by 10?

118. In a certain school of the scholars were studying arithmetic; algebra, and the remainder, which was 30, were studying grammar: how many scholars were there in the school?

119. A owns, and B of a ship; A's part is worth $650 more than B's: what is the value of the ship? 120. In a certain orchard are apple-trees:

peach trees; plumb-trees, and the remaining 15 were cherrytrees: how many trees did the orchard contain?

SECTION XII.

RATIO AND PROPORTION.

ART. 305. Ratio is that relation between two numbers or quantities, which is expressed by the quotient of the one divided by the other. Thus, the ratio of 6 to 2 is 6÷2, or 3; for 3 is the quotient of 6 divided by 2.

MENTAL EXERCISES.

Ex. 1. What is the ratio of 14 to 7? Ans. 2. 2. What is the ratio of 10 to 2? Of 16 to 4?

QUEST.-305. What is ratio?

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306. The two given numbers thus compared, when spoken of together, are called a couplet; when spoken of separately, they are called the terms of the ratio.

The first term is the antecedent; and the last, the consequent.

307. Ratio is expressed in two ways:

First, in the form of a fraction, making the antecedent the numerator, and the consequent the denominator. Thus, the ratio of 8 to 4 is written; the ratio of 12 to 3, 12, &c.

Second, by placing two points or a colon (:) between the numbers compared. Thus, the ratio of 8 to 4, is written 8 4; the ratio of 12 to 3, 12 : 3, &c.

OBS. 1. The expressions, and 8: 4 are equivalent to each other, and one may be exchanged for the other at pleasure.

2. The English mathematicians put the antecedent for the numerator and the consequent for the denominator, as above; but the French put the consequent for the numerator and the antecedent for the denominator. The English method appears to be equally simple, and is confessedly the most in accordance with reason.

3. In order that concrete numbers may have a ratio to each other, they must necessarily express objects so far of the same nature, that one can be properly said to be equal to, or greater, or less than the other. (Art. 280.) Thus a foot has a ratio to a yard; for one is three times as long as the other; but a foot has not properly a ratio to an hour, for one cannot be said to be longer or shorter than the other.

QUEST.-306. What are the two given numbers called when spoken of together? What, when spoken of separately? 307. How many ways is ratio expressed? What is the first? Second? Obs. How do the English express ratio? How do the French? In order that concrete numbers may have a ratio to each other, what kind of objects must they express?

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