191. When the fractions are such as will not reduce to known quantities without remainder. RULE. Reduce the given fractions to fractions of the highest denomination mentioned, (Art. 183.) then reduce these to a common denominator, (Art. 180.) add the numerators together, as in Art. 187, and reduce the sum to its proper quantity, (by Art. 186.) which will be the answer2. 5 8 I first reduce shill. and d. to fractions of a pound, which is the 7 9 This rule will be readily understood; for it is plain, from what has been shewn in the preceding notes, that in order to add fractions together, they must be reduced first to parts of the same whole, and then to parts of the same denomination; after which the sum is evidently found (as before) by adding all the new numerators together, placing the common denominator under the sum, and reducing this fraction to its equivalent value in known denominations. tions of a pound, these I reduce, to a common denominator, and add as in Art. is next reduced to its lowest terms, and lastly to its pro SUBTRACTION OF VULGAR FRACTIONS. 192. When the given fractions are both simple. RULE I. Reduce the fractions to a common denominator, beginning with the numerator of the greater fraction. II. Subtract the lower new numerator from the upper, and under the remainder place the common denominator; this fraction being reduced to its lowest terms will be the answer3. As fractions of different denominations cannot be added, so neither can they be subtracted; we must first reduce them to a common denominator, and then it is plain that their difference is found by taking the difference of the new numerators, and placing it over the common denominator. Thus in Ex. 1. 193. When there are mixed numbers, compound or complex fractions. RULE I. Reduce mixed numbers to improper fractions, compound and complex fractions to simple ones, reduce these to a common denominator, and proceed as before. 63 63 or as in the example. Nothing can be plainer than this rule; for if one fraction be subtracted from another of the same denomination, the remainder is evidently a fraction of the same denomination with both; thus, if two fifths be taken from three fifths, the remainder will be one fifth; and the same of other fractions. II. If the answer be an improper fraction, reduce it to its equivalent whole or mixed number ". 13 First, I reduce the mixed number to an improper fraction Secondly, the compound fraction to a simple one 15 24 Thirdly, I reduce these two to a 252 common denominator, and subtract, which gives the fraction 96 13 16 The two complex fractions are first reduced to the simple ones and these are next reduced to a common denominator; and all the rest as before. 20 33 b This rule is sufficiently plain from what is shewn in the note on the similar rule in Addition of Fractions, (Art. 189.) 194. To subtract mixed numbers, without reducing them to improper fractions. RULE I. Reduce the fractional parts to a common denominar, and having subtracted the less whole number from the reater, subjoin the difference of the fractions to the difference f the whole numbers for the answer. II. But if the lower new numerator is greater than the upper, btract it from the common denominator, add the remainder the upper numerator, and set the sum over the common deominator for the fractional part; then carry 1 to the less whole umber before you subtract it from the greater. * The first part of the rule evidently supposes that the whole number and action to be subtracted is each less than the whole number and fraction from hich they are respectively to be taken; now it is plain, that if 23 be taken rom 32, the remainder is 1; and it will be equally so when applied to other imilar examples. But with respect to the second part of the rule, where the lower new numeator is the greater, subtracting it from the common denominator is equivalent borrowing 1, (as in simple subtraction); thus in Ex. 14, where 6 36 is to 45 I say, 36 from 10 I cannot, I therefore borrow 45, (or which is equal to 1,) then 36 from 45, and 9 remain; this added to the 0 gives 19, viz. 19 ; then, because I borrowed 1 (or 45) in the fraction, I ust carry 1 to the subtrahend of the whole numbers; wherefore carrying 1 to |