99. What cost 1240 yds. of flannel, at 3s. 4d. per yard? 100. What cost 2128 lbs. of spice, at 2s. 6d. per pound? 101. What cost 5250 yds. of lace, at 6d. per yard?

102. What cost 56480 yds. of tape, at 1d. per yard?

471. Notwithstanding the law requires accounts to be kept in Federal Money, goods are frequently sold at prices stated in the denominations of the old state currencies.

When the price per yard, pound, &c., stated in those currencies, is an aliquot part of a dollar, the answer may be easily obtained in Federal Money.

TABLE OF ALIQUOT PARTS IN DIFFERENT STATE CURRENCIES.

Note.-1. In N. Y. currency 8s. make $1; in N. E. currency 6s. make $1. From example 103 to 119 inclusive, the prices are given in N. Y. currency ; from example 120 to 132 inclusive, they are given in N. E. currency. For the mode of reducing the different State currencies to each other and to Federal Money, see Section XVII.

103. At 1s. 4d. per yard, what cost 726 yds. of cambric?

Analysis.-If the price were $1 per yard, the cost would be $1726=$726. But 1s. 4d.=$; therefore the cost must be of $726, which is $121. Ans.

104. What cost 896 bu. of wheat at 6s. per bushel?,

Analysis.-6s. 4s.+2s. Now 4s. $; and 2s. of 4s. At $1 a bushel the cost would be $896.

Or, thus: 6s=$3; therefore the number of bu. minus of itself, will be the cost, and 896-224 († of 896)=672. Ans. $672.

105. What cost 752 yds. of balzorine, at 2s. 8d. per yard? 106. What cost 1232 yds. of calico, at 1s. 6d. per yard? 107. What cost 763 lbs. of pepper, at 1s. 3d. a pound? 108. What cost 1116 bu. of apples, at 1s. 4d. per bushel ? 109. What cost 1920 yds. of shirting, at 1s. 2d. per yard? 110. At 6s. a basket, what will 1560 baskets of peaches cost? 111. At 5s. 4d. a pound, what will 1200 lbs. of tea come to? Note.-2. Since 5s. 4d. is less than $1, it is plain 1200-400-$800. Ans. 112. At 7s. per yard, what will 432 yds. of crape cost? 113. At 6s. 8d. a pound, what cost 972 lbs. of nutmegs? 114. At 2s. 8d. a pair, what cost 864 pair of cotton hose? 115. At 11⁄2d. a yard, how much will 2800 yds. of tape come to? 116. What cost 1628 yds. of flannel, at 4s. per yard? 117. What cost 2560 bu. of oats, at 2s. per bushel ? 118. What cost 9600 lbs. of wool, at 2s. 6d. a pound? 119. What cost 3200 lbs. of sugar, at 6d. per pound? 120. What cost 600 yds. of damask, at 5s., N. E. cur., per yard? Nole.-3. 5s. N. E. cur. is less than $1; hence, 600—100-$500. Ans. 121. What cost 2500 bu. of potatoes, at 1s. 6d. per bushel? 122. What cost 1440 yds. of gingham, at 2s. per yard? 123. How much will 4848 chickens cost, at 1s. apiece? 124. How much will 1680 slates cost, at 1s. 6d. apiece? 125. How much will 920 turkeys cost, at 4s. 6d. apiece? 126. What cost 4860 lbs. of butter, at 1s. 1d. per pound? 127. What cost 1260 melons, at 8d. apiece?

128. What cost 2340 lbs. of tea, at 4s. a pound?

129. What cost 240 bu. of peas, at 4s. 6d. per bushel? 130. What cost 720 pair of gloves, at 5s. 3d. a pair? 131. What cost 360 bushels of corn, at 3s. per bushel? 132. What cost 7686 lbs. of butter, at 1s. per pound? 133 What cost 960 yds. of silk, at 5s. per yard? 134. What will 75 lbs. of butter cost, at $16.80 per cwt.? 135. What will 125 lbs. of wool cost, at $36 per hundred? 136. What will 15 cwt. of hemp cost at $60 per ton? 137. What will 2500 lbs. of iron cost, at $72 per ton? 138. What cost 14 acre of land, at $120 per acre?

RATIO AND PROPORTION.

472. In comparing numbers or quantities with each we may inquire, either how much greater one of the num r quantities is than the other; or how many times one of contains the other. In finding the answer to either of these es, we discover what is called the relation between the two =rs or quantities.

3. The relation between the two quantities thus compared. wo kinds:

st, that which is expressed by their difference.

ond, that which is expressed by the quotient of the one diby the other.

4. RATIO is that relation between two numbers or quantihich is expressed by the quotient of the one divided by the Thus, the ratio of 6 to 2 is 6÷2, or 3; for 3 is the quoof 6 divided by 2.

The relation between two numbers or quantities denoted by their dif . is sometimes called arithmetical ratio; while that denoted by the quothe one divided by the other, is called geometrical ratio. Thus 4 is the etical ratio of 8 to 4; and 2 is the geometrical ratio of 8 to 4.

as the term arithmetical ra'io is merely a substitute for the word difere term difference, in the succeeding pages. is used in its stead; and when rd ratio simply is used, it signifies that which is denoted by the quotient. one divided by the other, as in the article above.

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

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

T.-472. In how many ways are numbers or quantities compared? 474. What is 475. What are the two given numbers called when spoken of together? When of separately?

476. 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, is 12: 3, &c. The expressions, and 8: 4, are of the same import, and one may be exchanged for the other, at pleasure.

OBS. 1. The sign (:) used to denote ratio, is derived from the sign of division, (÷) the horizontal line being omitted. 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, while it is confessedly the most in accordance with reason.

2. 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. 294.) 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.

477. A direct ratio is that which arises from dividing the antecedent by the consequent; as 6÷2. (Art. 474.)

478. An inverse, or reciprocal ratio, is the ratio of the reciprocals of two numbers. (Art. 160. Def. 10.) Thus, the direct ratio of 9 to 3, is 9: 3, or ; the reciprocal ratio is, or÷ 1=3; (Art. 229;) that is, the consequent 3, is divided by the antecedent 9.

Note.-The term inverse, signifies inverted. Hence,

An inverse, or reciprocal ratio is expressed by inverting the fraction which expresses the direct ratio; or when the notation is by points, by inverting the order of the terms. Thus, 8 is to 4, inversely, as 4 to 8.

QUEST.-476. In how many ways is ratio expressed? The first? The second? Obs. Which of the terms do the English mathematicians put for the numerator? Which do the French? In order that concrete numbers may have a ratio to each other, what kind of objects must they express? 477. What is a direct ratio? 478. What is an inverse of reciprocal ratio? How is a reciprocal ratio expressed by a fraction? Ilow by points?

79. A simple ratio is a ratio which has but one antecedent ne consequent, and may be either direct or inverse; as 9:3, }.

30. A compound ratio is the ratio of the products of the sponding terms of two or more simple ratios.

Thus,

The ratio compounded of these is 72: 12=6.

. 1. A compound ratio is of the same nature as any other ratio. The s used to denote the origin of the ratio in particular cases.

The compound ratio is equal to the product of the simple ratios.

31. From the definition of ratio and the mode of expressing the form of a fraction, it is obvious that the ratio of two numis the same as the value of a fraction whose numerator and minator are respectively equal to the antecedent and conset of the given couplet; for, each is the quotient of the numerdivided by the denominator. (Arts. 474, 185.)

8. From the principles of fractions already established, we may, thereleduce the following truths respecting ratios.

ST.-479. What is a simple ratio? 480. What is a compound ratio? Obs. Does it n its nature from other ratios? What is the term used to denote ?

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