2 4 2 8. From 3-of-take of of Diff. 5 3 17 3 45 194. To subtract mixed numbers, without reducing them to improper fractions. RULE I. Reduce the fractional parts to a common denominator, and having subtracted the less whole number from the greater, subjoin the difference of the fractions to the difference of the whole numbers for the answer. II. But if the lower new numerator is greater than the upper, subtract it from the common denominator, add the remainder to the upper numerator, and set the sum over the common denominator for the fractional part; then carry 1 to the less whole number before you subtract it from the greater. The first part of the rule evidently supposes that the whole number and fraction to be subtracted is each less than the whole number and fraction from which they are respectively to be taken; now it is plain, that if 23. be taken from 3, the remainder is 1-; and it will be equally so when applied to other similar examples. But with respect to the second part of the rule, where the lower new numerator is the greater, subtracting it from the common denominator is equivalent 36 to borrowing 1, (as in simple subtraction); thus in Ex. 14, where 6 is to 10 45 be taken from 12, I say, 36 from 10 I cannot, I therefore borrow 45, (or 45 45 which is equal to 1,) then 36 from 45, and 9 remain; this added to the 45 19 ; then, because I borrowed 1 (or 45) in the fraction, I must carry 1 to the subtrahend of the whole numbers; wherefore carrying 1 to 13 subtracted as before, I next subtract the whole numbers 1 from 4, and 3 re mains; to this I subjoin the fraction for the answer. Having first reduced the fractions and 45 to a common denominator, 9 secondly, I subtract 36 from 45, and add 10 to the remainder, which gives 19, this placed over the common denominator 45, gives for the fractional part; 19 45 I then carry 1 to the whole number 6 is 7, this I subtract from the whole num19 ber 12, and 5 remains for the whole number; wherefore 5quired. is the answer re 45 6 makes 7, which taken from 12 leaves 5; all this is evident, being exactly the process of simple subtraction. 9 Then 14 12+7=9, therefore the fraction; also carry --- 14 1 to the 4 is 5, then 18-513 the whole number; 195. To subtract a proper fraction from a whole number. RULE I. Subtract the numerator of the fraction from the denominator, and place the remainder over the denominator for the fractional part. 2. Subtract 1 from the whole number, prefix the remainder to the fractional part, and it will give the answer d. d This rule depends on the same considerations with the preceding, as may be seen by an attentive examination of the 20th example, in which it is required to take --) from which taking 9 from 8; now here we borrow 1, or 9 9 pensate by (subtracting or) lessening the 8 by the 1 we borrowed. RULE. Reduce the fractions to their proper quantities, then subtract by the rules of Compound Subtraction. • The grounds of this rule are explained in the note on the similar rule in Addition of Fractions, Art. 190. 3 5 32. From 33 weeks take 4- hours. Diff. 3w. 2d. 23h. 8 ? 22m. 30". 197. When the fractions will not reduce to known quantities without remainder. RULE. Reduce the given fractions to fractions of the greatest denomination mentioned, reduce the latter to a common denominator, subtract the less numerator from the greater, as in Art. 192, then having placed the remainder over the common denominator, reduce this fraction to its proper quantity, by Art 186. f This rule depends on the same principles with the similar rule in Addition of Fractions, Art. 191. It will, in some cases, be more convenient to reduce the fraction of the greater denomination to an equivalent one of the less, and proceed according to the latter part of the rule. |