Examples for Practice.

(1) If the expenses of 7 persons for 3 months amount to 70 guineas; what will be the expenditure of 10 persons for 12 months at the same rate?

Answer: £420.

(2) If 10 horses consume 7bush. 2pks. of oats in 7 days; in what time will 28 horses consume 3qrs. 6bush. at the same rate?

Answer: 10 days.

(3) If 10 men reap 20 acres of corn in 4 days; how many men can reap 70 acres in 10 days, at the same rate of labour?

Answer: 14 men.

(4) If 48 men can do a piece of work in 16 days of 9 hours each: in how many days of 12 hours each will 64 men be able to do a piece of work three times as great?

Answer: 27 days.

(5) If the carriage of 13cwt. 2qrs. 19lbs. for 35 miles come to £4. 17s. 6d.; what must be paid for the conveyance of 41cwt. 1lb. for 49 miles?

Answer: £20. 9s. 6d.

(6) If £20. in trade gain £16. in 15 months, what sum will gain £24. in 3 months, at the same rate?

Answer: £150.

(7) If 12 men can perform a piece of work in 20 days; required the number of men who could perform another piece of work four times as great in a fifth part

of the time.

Answer: 240 men.

(8) If with a capital of £1000., a tradesman gain £100. in 7 months, in what time will he gain £60. 10s., with a capital of £385.?

Answer: 11 months.

CHAPTER IV.

THE DOCTRINE OF FRACTIONS,

USUALLY TERMED VULGAR FRACTIONS.

68. DEF. ALL whole numbers or Integers, being supposed to be formed by the repetition of the unit, may therefore be regarded as the result of the multiplication of that element; but if the unit be considered capable of division into any number of equal portions, the quantities thence arising must be viewed in the light of broken magnitudes; and these are therefore termed Fractions, or more generally, Vulgar Fractions, in order to distinguish them from fractions of a different form, whose nature will be discussed in the next chapter.

NOTATION AND NUMERATION OF FRACTIONS.

69. DEF. 1. If we suppose the unit to be divided into 2, 3, 4, 5, &c., equal portions, one of the portions in each case is represented by,,,, &c., which may be regarded as the primitive Fractions of their respective denominations, and are called the Reciprocals of the natural numbers, 2, 3, 4, 5, &c. also, the fractions, 1, 1, †, &c., are read, one-half, one-third, one-fourth, one-fifth, &c.

70. DEF. 2. If two or more of these equal portions be taken together, the aggregates thence arising are expressed by repeating the unit as often as such portions are repeated, in the form of their sum, the number below the line remaining the same.

Thus, if the primitive fraction be taken twice, there will arise a new fraction expressed by: if be repeated thrice, there results a new fraction expressed by 2: again, if be taken four times, the new fraction will be ; and similarly of all the other primitive fractions: also, the fractions,,, &c., are read two-thirds, three-fourths, four-fifths. &c.: and all quantities of this form are called Simple Fractions.

71. DEF. 3. Hence, the number below the line denotes the number of equal portions into which the unit is supposed to be divided, and is therefore called the Denominator; and the number above the line expressing the number of such equal portions intended to be taken, is therefore termed the Numerator.

Thus, of the fraction, whose Terms are 5 and 7, the denominator 7 implies that the unit is supposed to be divided into seven equal portions; and the numerator 5 shews that five of such equal portions are here the object of our consideration: and hence it is also manifest, that the integer 5 is 7 times as great as the fraction; and 5 may therefore be expressed in a Fractional Form by .

72. From the last Article it follows, that if the numerator be less than the denominator, the value of the fraction is less than the unit; if the numerator be equal to the denominator, the value of the fraction is the unit; and if the numerator be greater than the denominator, the value of the fraction is greater than the unit.

73. DEF. 4. If the numerator be less than the denominator, the fraction is termed a Proper Fraction; but if the numerator be greater than the denominator, it is called an Improper Fraction: also, if the terms be equal to one another, we have merely the representation of the unit in the form of a fraction.

Thus, is a proper fraction, 11 is an improper fraction, and is a representation of the unit in a fractional form, being of the same value as §, %, ? &c.

74. We are hence enabled to find the results of the multiplication and division of a fraction by an integer, and these may be integers or fractions.

many parts of the unit are implied, as there are in.

be

If the fraction be divided by 5, the quotient will

2

7 x 5

2

35

; because the same numbers of parts are

indicated in and, and each part in the former is five times as great as each part in the latter, by Article (71).

Hence, to multiply and divide a fraction by a whole number, we have only to multiply the numerator and denominator by it, respectively.

75. What is called a Compound Fraction, may be replaced by a simple one, by similar reasoning.

A Compound Fraction is made up of two or more simple fractions connected by the word of, as 3 of 3 of † :

a simple fraction of the ordinary form: that is,

and from this, we infer that a compound fraction is equivalent to the simple fraction formed by multiplying together respectively the numerators and the denominators of its constituent simple fractions.

TRANSFORMATION OF FRACTIONS.

76. If the numerator and denominator of a fraction be multiplied or divided by the same number, the value of the fraction will not be altered.

For, if the fraction be multiplied by 5, the product is 15 and again if this be divided by 5, the quotient is, by the last Article but one: but since these two operations are the reverse of, and therefore neutralize, each other, it follows that

and also, that

15 3 15÷5

=

35 7 35÷5

Hence, a whole number may be expressed in the form of a fraction with any denominator we please: thus,

Also, a fraction may be transformed into another with a given denominator, provided it be a multiple of the denominator of the proposed fraction: thus, & may be transformed so as to have 96 for a denominator, because

for the Multiplication of a fraction by an integer, it appears to be immaterial whether the numerator be multiplied, or the denominator be divided, by it: and inasmuch as

for the Division of a fraction by a whole number, it amounts to the same thing whether we divide the numerator, or multiply the denominator, by it.

78. A quantity made up of two others, one of which is an integer and the other a fraction, may be represented in the form of a fraction alone.

Let us take 3, which is called a mixed quantity, and is intended to express the integer 3 and the fraction taken together, and is read three and four-fifths: then, since