149. When the price is pounds, shillings, pence, and farthings. RULE. Multiply the given number by the pounds, take parts for the rest of the price, and proceed as in the former rules".

105. What will 571 cwt. of sugar cost, at 21. 12s. 9d. per cwt. ? OPERATION.

S.

150. When the given number consists of a whole number and

parts.

RULE. Work for the whole number according to the direc tions given in the former rules, and add in such a part of the price as the question requires ".

b If the given number be multiplied by the pounds each costs, the product will evidently be the value for the pounds; the reasons on which the method of taking the aliquot parts for the remainder of the price is founded, have been already explained.

The price of will evidently be (half the price of 1, or) half the given

113. What is the value of 1094 dozen of wine, at 21. 6s. 6d. per dozen?

114. 237 yards of cambric, at 12s. 4d. per yard?

115. 142 at 2l. 10s. 6d. Answer 359l. 16s. 3d. 116. 2734 at 7s. 6d. Answer 102l. 9s. 4d.4.

117. 2344 at 17s. Answer 1991. 10s. 9d.

118. 120 at 5l. 15s, 5d. Answer 6961. 16s. 6d.4.

151. When the given number is of several denominations. RULE. Multiply the given price by the highest denomination of the given number, and take the remaining denominations of the given number in aliquot parts of the highest, and of one another, and work as in the preceding rules".

price; in like manner the price of any part will evidently be the same part of the given price: the rule is therefore manifest.

The reason of the rule will appear from an examination of the 119th Example; where, since I cwt. costs 87. 12s. 4d. it is plain that 15 cwt. will cost 15 times 87. 12s. 4d. (or 1297. 5s.); 1 quarter will cost of 87. 12s. 4d. (or 21. 3s. 1d.), and 14 lb. will cost half what a quarter costs, (or 11. Is. 6d.†); and that these several quotients being added together, the sum will be the

answer.

The Examples under this rule should be proved by The Rule of Three,

127. What must be paid for 59cwt. 2qr. 24lb. of salt, at 21. 17s. 4d. per cwt.? Ans. 171l. 3s. 7d.4.

128. At 2l. 9s. 6d. per cwt. what must be given for 22cwt. 3qr. 21lb. of currants? Ans. 56l. 15s. 4d.‡.

129. What is the yearly rent of 145 acres, 1 rood, 32 poles of land, at 201. 10s. 6d. per acre? Ans. 29851. 7s. 2d.4.

130. What will 1234hdd. 28gal. 3qt. of port wine cost, at 601. per hhd.? Ans. 740671. 7s. 7d.1.

FRACTIONS.

152. A fraction is a number which denotes one or more parts of unity; the number 1 being supposed divisible into any number of equal parts at pleasure: whatever expresses any assigned number of those parts is called a fraction.

Fractions are usually divided into Vulgar, Decimal, Duodecimal, and Sexagesimal f.

VULGAR FRACTIONS.

153. Vulgar Fractions are those which express the parts of unity, into whatever number of equal parts it may be supposed to be divided.

154. A Vulgar Fraction is denoted by two numbers placed one over the other, with a small line between them, thus, . 155. The number below the line is called the denominator; it shews how many parts the unit is supposed to be divided into.

156. The number above the line is called the numerator; it shews how many of the aforesaid parts are to be understood by the fraction.

Thus in the above fraction, the number 7 is the denominator, and shews that is supposed to be divided into 7 equal parts; 4 is the numerator, and shews that 4 of those parts are to be understood by the fraction; that is, the value of the fraction is four sevenths, or four of such equal parts of which seven just make the number 1.

The word fractio is derived from the Latin frango, to break, and is a name descriptive of the numbers included under it. Decimals and Duodecimals will be distinctly treated of; Sexagesimals (or Sixtieths) will likewise be explained when we treat of Practical Geometry, Trigonometry, &c. The term vulgar comes from vulgus, the common people. Vulgar Fractions mean fractions that admit of any denomination whatever.

157. When the numerator is less than the denominator, it is evident that the fraction expresses fewer parts than 1 is supposed to be divided into; consequently the value of the fraction is less than 1: such a fraction is called a proper fraction; thus, half),

(one

(two thirds), (three elevenths), &c. are proper fractions. 158. When the numerator is equal to the denominator, the fraction expresses just so many parts as 1 is supposed to be divided into; consequently its value will be equal to 1. In like manner, when the numerator is greater than the denominator, the fraction expresses more parts than 1 is divided into, and its value is greater than 1. In either case the fraction is called an improper fraction; thus, (four fourths), (five fifths), 7 (seven thirds), (twenty-one ninths), &c. are improper fractions.

159. But this division is not confined to unity; we may conceive the parts themselves susceptible of a similar division; every fraction may be subdivided into other parts or fractions, and these into others, and so on without end: the expression denoting a fraction arising from such a division and subdivision of unity is called a compound fraction; thus, of (one half of one half), of (two thirds of three fifths), of of (one fourth of two sevenths of three eighths); such expressions as these consisting of two or more fractions, with the word of interposed between them, are, as is evident, meant to express a part of a part, or parts of parts, and are called compound fractions.

160. A simple fraction is that which is expressed by one numerator and one denominator, and both whole numbers.

161. When a whole number and a fraction are connected, so that both together form but one number, such an expression is called a mixed number; thus, 1 (one and two sevenths), 5 (five and eight ninths), 14 (fourteen and one sixth), &c. are mixed numbers.

162. When either the numerator or the denominator is a mixed number, or when both are mixed, the fraction is called a complex fraction; thus, -(three and one half, fourths), (two, three and

2

3층

(two and two thirds, three and four fifths ), &c. are complex

fractions.

When both terms are complex, this reading will scarcely be intelligible; perhaps it will be better to read these fractions in the following manner; viz.

2

three and one half by four; two by three and five sixths; 35

two thirds by three and four fifths; and the like of others.

163. A common measure of two or more numbers is such a number as will divide each without leaving any remainder; and the greatest common measure is the greatest number possible that will divide each without remainder.

164. A common multiple of two or more numbers is a number which each of them will divide without remainder; and the least common multiple is the least number that each of them will so divide.

REDUCTION OF VULGAR FRACTIONS.

165. Reduction of Fractions is the changing them from one form to another without altering their values. By the operations of reduction, fractions are expressed in the most convenient form for the readily understanding of their values, and likewise prepared for adding, subtracting, multiplying, and dividing.

166. To find the greatest common measure of two numbers. RULE I. Divide the greater number by the less, and divide the divisor by the remainder.

II. Proceed in this manner, díviding continually the last divisor by the last remainder, until nothing remains: the last di

There are some other denominations of fractions, as continued fractions, used for approximating to indeterminate ratios in small numbers; vanishing fractions, the properties of which are best explained by Fluxions, &c.

h Previous to entering on the Reduction of Fractions it will be proper to remark, that one fraction is always equal to another when the numerator of the first is to its denominator as the numerator of the second is to its denominator; thus, is equal to or to or to or to 10 or to 50% or to 100, &c. for

2

Hence it follows, that since numbers have the same ratio to one another that their like multiples or like parts have respectively, both terms of any fraction may be either multiplied or divided by any (the same) number without altering the original value of the fraction; thus both terms of 20 may be multiplied by 7,9, &c. or divided by 10, 5, &c. and the results +10, +70, 4, 7, &c. will be equal to each other and to the given fraction 8.

40 180 2