Page:Encyclopædia Britannica, first edition - Volume I, A-B.pdf/470

 §38 A R I T H M E T I C K. ed decimal fraflions : and fraftions having any other de- The reafon of the rule appears by reverfing the openominator are called vulgar fradions. ration; for if the numerator be divided by the denomi1. A proper fradtion is that whofe numerator is lefs nator, it will quot the integer, or whole number. than its denominator, and confequently is in value lefs Pros. IV. To reduce a compound fradtion to a fimple one. than unity; as y. 2. An improper fradtion is that whofe numerator is e- Rule. Multiply the numerators continually for the qual to or greater than its denominator; and confequent- numerator of the fimple fradtion ; and multiply the denoly is in value equal to or greater than an unit; as -f. minators continually for its denominator. 3. A fimple fradtion is that which has but one numeExa m pl e s. rator, and one denominator; and may be either proper or improper ; as ^ or Ex, 2. ’ 4 of y of d = 4. A compound fradtion is made up of two or more Fx.i. From this problem may be deduced a method fimple fradtions, coupled together with the particle of, of Cor. a fradtion of a lefier denomination to a fracand is a fradtion of a fradtion; as y bf or -i- of y of tionreducing of a greater denomination ; namely. 5. A mixt number confifts of an integer, and a frac- Form a compound fradtion, by comparing the given tion joined with it; as fradtion with the fuperior denominations; and then reBecaufe in moft cafes fradlions can neither be added nor fubtradted, till they be reduced, we begin with re- duce the compound fradtion to a fimple one. dudtion. Examples. 1. What fradtion of ar pound Sterling is ^of a penny? Ttedudion of Vulgar Fraflions. •^d. is ^ of Problem I. To reduce an improper fradtion to an 2. What fradtion ofTaT C.of isy^-L.of a pound ? integer, or mixt number. T 18. is -J- Of vy - = yyyC. Rule. Divide the numerator by the denominator, the quot gives integers ; and the remainder, if there be Prob. V. To reduce a fradtion of a greater denomiany, placed over the divifor or denominator, gives the nation to a fradtion of a lefler denomination. fradtion to be annexed. Rule. Multiply the numerator of the given fradtion, Ex A m pl E s. as in redudtion of integers defcending ; and the produdt X. *35 = 85 integers, there being no remainder. is the numerator, to be placed over the denominator of the given fradtion. 2. 4*.7= 544-, the remainder being 5. Exam p l e s. 3.  the remainder being 48. Prob. II. To reduce a mixt number to an improper Here, as in redudtion defcending, multiply the numerator 3 by 20, becaufe 20 {hillings make a pound; as fradtion. Rule. Multiply the integer by the denominator; to under. z. the produdt add the numerator: The fum is the nume,rator of'the improper fradtion; and the denominator is 3 * 19 — {hilling. the fame as before. Examples. 4 2. What fradtion of a penny is f L. ? 4 7 for Z. 1. 54b-= 1 ; 54X8 = 432 4 X 20 X 12 _ 057 0 J +5 5 Numerator 437 The reafon of this rule will appear by obferving, that 2. 98^1= HI1; for 98 X 14= 1372 every fradtion may be confidered in two views. Thus, + 10 may either be confidered as exprelfing three fourths of one unit, or as denoting the fourth part of three units. Numerator 1382 Now, if the unit be a pound Sterling, the fradtion, in Pros. III. To reduce a whole number to a fradtion the latter view, will denote the fourth part of three of a given denominator. and by reducing the numerator L. 3 to {hilRule. Multiply the whole number by the given de- pounds; lings, we have 7 s0 ; and again reducing 60 {hillings to nominator; and place the produdt by way of numerator pence, we have -| d. equal to s. or to £L. over the given denominator. Prob. VI. To find the value of a fradtion. Examples. Rule. Reduce the numerator to the next inferior 1. Reduce 9 to a fradtion whofe denomination is 5. ifdenomination nothing remain, is the value complete. 9X5=45; fotbe fradtion is ^. If there be any remainder, it is the numerator of a 2. Reduce 36 to a fradtion whofe denominator is 4. fradtion whofe denominator is the divifor. T his frac36 X 4 = 144 ; fo the fradtion is 3. Reduce 8 to a .fradtion whofe denominator is 1. tion may either be annexed to the quotient, or reduced to value, if there be any lower denomination. 8X1=8; fo the fradtion is Examp.
 * of-£ = -r'r.
 * divide by the denominator; and the quot,