COMPOUND DIVISION, ETC.] 



MATHEMATICS. ARITHMETIC. 



443 



a book on Arithmetic, exclusively with a view to mer- 

 cantile practice. Our endeavour is to prepare you for a 

 course of mathematical study; and we therefore wish 

 you to cultivate habits of thought and reflection to 

 know what you are actually about, and not to feel con- 

 tented by merely following a rule. We shall not insist 

 upon any marked departure from the customary forms 

 of expression, in the practical directions for working an 

 example; but must insist upon accuracy of thought, 

 whatever want of precision in language custom may 

 authorise. 



To divide a compound quantity by a number, the rule 

 is this : 



RULE. Commence with the highest denomination, and 

 take the proposed part of it ; reduce what is over to the 

 next denomination, and carry the result to the next 

 term of the dividend ; take the proposed part of the 

 sum, reducing what is over, and carrying as before ; and 

 so on, to the end. 



Thus, if the 7th part of 22 15s. 9<J. ,. a. 

 be required, we find it as in the mar- 7)22 15 9 



gin : the 7th part of 22 is 3 aud 



20s. over: this, carried to the 15s., 3 5 



gives 35s. ; the 7th part of which is 



5s., and there is nothing to carry: the 7th part of 9<f. is 



IV. ; and 2d., or 8 farthings, over ; the 7th part of which 



is 1 farthing, and 1 of a farthing ; hence, the 7th part of 



the proposed sum is 3 5s. Ijd. + 1 /. 



If the divisor exceed 12, we must proceed, upon the 

 same principle, by long division, unless it can be decom- 

 posed into convenient factors ; when the operation need 

 to consist only of successive steps like that explained 

 above. 



When the divisor, instead of being an abstract number, 

 is a concrete quantity, of the same kind as the dividend, 

 the rule is as follows : 



RULE. Reduce both dividend and divisor to the 

 lowest denomination found in either, and then perform 

 the division exactly as in the case of mere numbers : the 

 quotient will denote the number of times the smaller 

 quantity is contained in the greater. 



For example, let it be required to divide 63 7s. by 

 13 2s. 3d. Then, as pence is the lowest denomination 

 that occurs, we reduce />oth quantities to pence, and then 

 divide as in the margin: the quotient 

 shows that the smaller sum is con- 

 tained in the larger between 4 ami 

 5 times : it is contained in it 4 times 

 and a fractional part of a time, 

 represented by f J, which is nearly 

 another time, but not quite. You 

 may shorten the work a little by 

 3147) 15204(4 reducing the two quantities not 



12588 to pence, but to three-pences as 



shown below ; observing, that as 4 



2616 three-pences make 1 shilling, we 

 multiply the number of shillings 



by 4, and take in the odd threepence. From this mode 

 of working, we should conclude that the dividend con- 

 tains the divisor 4 times and a part of a time, denoted 

 by the fraction -f3*, s , which differs from the former 

 fraction only in appearance not in value; for if we 

 wish to express that one number is to be divided by 

 another, we may, as you are aware, 

 do so by writing the Litter below the 

 former, or by writing twice, three 

 times, &c., the latter below twice, 

 three times, &c. the former. This is 

 pretty obvious, since the quotient of 

 dividend and divisor is the same, 

 whatever number both be multiplied 

 by. It will be seen that the upper 1049) 6068(4 



and lower numbers of the first frac- 4196 



tion are only those of the second, 

 each multiplied by 3. 872 



There is no room in this treatise 



for many examples. We shall, however, give two. 



first is to show that 33 19s. 6d. divided by 13 gives 



L- lls. 6</. for quotient ; the second is to show that the 



same sum divided by 13 gives 2? f r quotient. In 

 working the second example, you had better reduce to 

 sixpences, not to pence. 



Fractions. What has preceded suffices to convey a 

 general, and, we hope, a pretty accurate notion of the 

 arithmetic of integral quantities. We are now to show 

 how the fundamental operations are to be applied to 

 fractions. It has Been impossible to avoid all allusion 

 to fractions in the foregoing part of the subject, because 

 they force themselves upon our notice even when ope- 

 rating upon integers; but the arithmetic of fractions 

 remains to be explained, and, indeed, the formal defini- 

 tion of a fraction to be given. In strictness, a fraction 

 is a part of a whole that is, it is less than the quantity 

 of which it is said to be a fraction. Thus, -J, |, * <, are 

 strictly fractions proper fractions. The first denotes 

 a third part of unit, or 1 ; the second a fifth part of 2 ; 

 the third a forty-third part of 26 ; each part being less 

 than one whole. But 4, J-, $$, are also called fractions, 

 though four-thirds, seven-fifths, sixty-four forty-thirds, 

 are all greater than one whole, as is plain. Fractions 

 such as these, where the upper number, called the 

 numerator, is not less than the lower, called the deno- 

 minator, are said to be improper fractions. You will 

 readily see why these terms, numerator and denominator, 

 are so applied : the upper number enumerates, or states 

 the number of parts of that particular d< ,, 

 indicated by the lower number. Thus, * moans three of 

 the parts called fourths: if it were J of 1, then, since 

 one- fourth is 5s., three-fourths, or j, would be 15s., and 

 so on. Instead of reading this fraction three-! 

 may, if we please, say tliree >\ four. Three 



pounds divided by the numlxsr 4, is evidently the same 

 as three-fourths of one pound ; and any fraction niny be 

 viewed in either of these two ways : thus it is matter of 

 indifference whether we call |, Jil-e-scvcnths, or 5 <//' 

 by 7. A moment's reflection will convince that live- 

 sevenths of anything, is the same as a seventh part of 

 five such things; for a seventh part of one of them added 

 to a seventh part of another, then again this sum 

 increased by a seventh part of another, aud so on, till a 

 seventh part of each of the five has been t.ikrn, ami all 

 these sevenths added, will be equal to all five divided by 7 



The fractional not.ition is perfectly general any num. 

 her may be expressed in it ; a whole number, or an in > 

 as well as a fraction properly so called. Thus 6, 8, Ac., 

 may be written f , f , <tc. ; and it is sometimes convenient 

 to write integers this way. Here the denominator is 

 . or 1 ; but we may express an integer in the form of 

 a fraction with any denominator. Thus, if 7 for denomi- 

 nator be chosen, the two numbers, 6, 8, may be written 

 y,*, 6 , as is evident: you have only to multiply the num- 

 ber by the chosen denominator, and to place the factor, 

 thus used as a multiplier, underneath that is, as a 

 divisor. The numerator and denominator are called the 

 terms of the fraction ; and when an integer is united to 

 a fraction, the whole is called a mixed number. Thus, 

 2J; 3|, <tc., are mixed numbers. 



To reduce a Mixed Number to an Improper Fraction. 



The rule is this : multiply the integer by the denomi- 

 nator of the fraction ; add the product to the numerator, 

 and put the denominator underneath. Thus, 2J^ = J, 

 3 > = y> ; for 2 is evidently $, and -f- $ = %. In like 

 manner 3 is V , and y + f = V anc ^ so on - Hero 

 are other ' examples : 6J- = V 4 = V ' ^'u^v 6 ' 

 12, r = yJ 1 . To accomplish the contrary purpose 

 that is, 



To reduce an Improper Fraction to a Mixed Number, 



You have only to perform the division indicated by the 

 denominator, and to annex to the quotient the fractional 

 correction as in common division. Thus, ! 4 S = 5J, 

 , = 4J, 7 6 = 8", and so on. 



To reduce Fractions with Different Denominnfors, to others 

 of the same Value, with Equal Denominatnrs. 



This is one of the most important operations in the 

 arithmetic of fractions ; for till fractions appear with a 



