From 12 subtract. Here I say 7-3=4; so is the fractional part; then I say, 1 borrowed from 12, and 11 remain, so 114 is the difference or answer. NOTE. When an integer is given to be subtracted from a mixed number, you have only to subtract the given integer from the integral part of the mixed number; and to the remainder annex the fractional part. Thus, 92—5—43. RULE 5th. If one or both of the given fractions be compound, first reduce the compound fractions to simple ones, then reduce the simple fractions to a common denominator, and subtract the one numerator from the other. & of 3, and 4, 48, 38, and 1-888-35. Ans. 40 3 of 1, 3 of 1,, and 2,4%, and 4-13=2=1}• RULE 6th. 20 Ans. When the given fractions are of different denominations, first reduce them to the same denomination, then reduce the fractions now of one denomination, to a common denominator, and subtract the one numerator from the other; or, reduce each of the given fractions separately, to value, and subtract the one value from the other, and it is done. 3s. of ££ and 2, 18, then 188 l = £} } } = 14s. 4d. 13. From £ take d Ans. Ans. 21 3888-6s. 71d. Ans... N. B. The reason of the rules of subtraction is the same as in addition; for like kinds, or things, can only be subtracted from each other; and therefore, in subtraction the fractions must all have the same denominator, and be of the same denomination. MULTIPLICATION OF VULGAR FRACTIONS. In multiplication of Fractions it is unnecessary to reduce the given fractions to a common denominator, as in Addition and Subtraction, only if a mixed number be given, reduce it to an improper fraction; if an integer be given, reduce it to an improper fraction, by putting an unit for its denominator; if a compound fraction be given, you may either reduce it to a simple one, or instead of the preposition of, insert the sign of multiplication: then proceed by the following RULE. Multiply the numerators together for a new numerator, and the denominators together for a new denominator. NOTE.-Multiplication of Fractions is only the expression of a compound fraction. Thus, & multiplied by is the same as 2 of 3 as above directed in the Rule. EXAMPLES. 1. Multiply by == Ans. 2. Multiply by 5%. OPERATION. NOTE. Where fractions are to be multiplied, if the numerator of one fraction be equal to the denominator of another, or can be divided by any one equal measure, cancel them thus : swer required. Or, 3×3×== as before. 8 16 so is the an OBSERVATION 1st. If any number be multiplied by a proper fraction, the product will be less than the multiplicand; for multiplication is the taking of the multiplicand as often as the multiplier contains unity, and consequently, if the multiplier be greater than unity, the product will be greater than the multiplicand; if the multiplier be unity, the product will be equal to the multiplicand; and if the multiplier be less than unity, the product will, in the same proportion, be less than the multiplicand. Thus, supposing the multiplier to be or, the product, in this case, will be equal to one half or to one third of the multiplicand. OBSERVATION 2d. Mixed numbers may be multiplied without reducing them to improper fractions. Thus : In multiplying a fraction by an integer, you have only to multiply the numerator by the integer, the putting 1 for the denominator being only matter of form. And to multiply a fraction by its denominator is to take away the denominator, the product being an integer the same with, or equal to the numerator. Thus, X87, for X=-7. OBSERVATION 4th. If the numerators and denominators of two equal fractions be multiplied cross ways, the products will be equal. Thus, if, then will 3X12-9X4. OBSERVATION 5th. In multiplying fractions, equal factors above and below may be cancelled, or cut off. Thus, 1 of 2×2 of 3=1×××, and cancelling the factors 2, 3, 4, both above and below, the product is, in like manner to facilitate an operation, a factor above and another below may be divided by the same number. Thus, )(1 multiplied by).(= Ans. Or we may exchange one numerator for another. Thus, X ====, the same as before. 5 147 1 OBSERVATION 6th. To take any part of a given number is to multiply the said number by the fraction Thus of 320 is found to be thus: X320={ X320 — 14X4 0 200-200. In like manner all such fractions may be multiplied; as, for example, suppose it were required to multiply of 453. X45X303 303 303 X1111-301. Hence, to reduce a compound fraction to a simple one, is only to multiply the parts of it together. OBSERVATION 7th. If a multiplicand of two or more denominations be given, to be multiplied by a fraction, reduce the higher part or parts of the multiplicand to the lowest denomination, and then multiply in the following manner, viz. : admit it were required to multiply £8 103s. by 3, 1 say, £8=8×20s.=160s. and 160+103-1703s.633, and 6 3 130113s. £5 13s. 10d. or, without reducing, you may multiply the given multiplicand by the numerator of the fraction, and divide the product by the denominator. 12 = 1. Multiply by 4. Multiply 64 by 8. 5 Multiply 91 by 1 of 2. EXERCISES. 6. Multiply 12 by 3 of 3. 8. Multiply 13 by . 9. Multiply £3 12s. 6d. by 2. The reason of the rule may be shewn thus: X=1; for 13, and of is, and consequently of is. Or thus: assume two fractions equal to two integers, as & and g, equal to 2 and 3, and the product of the fractions will be equal to the product of the integers; for X-48-6, and 2X3-6. DIVISION OF VULGAR FRACTIONS. RULE. 1. If the fractions are single and have a common denominator, divide the one numerator by the other, and place the quotient over the denominator. 2. If a mixed number be given, reduce it to an improper fraction; if an integer be given, put an unit for its denominator; if a compound fraction be given, reduce it to a simple one, and work by the following RULE. Multiply cross-ways, viz. the numerator of the divisor into the denominator of the dividend, for the denominator of the quotient; and the denominator of the divisor into the numerator of the dividend, for the numerator of the quotient. OBSERVATIONS. 1. Instead of working as above taught, you may invert the divisor, and then multiply it into the dividend. Thus, in example 1, instead of 3)(13, you may say, X=1=1=17. 2. If any number be divided by a proper fraction, the quotient will be greater than the dividend; for in division the quotient shews how often the divisor is contained in the dividend; and consequently if the divisor be greater than unity, the quotient will be less than the divi; if the divisor be unity the quotient will be equal to the diviand if the divisor be less than unity the quotient will in the same proportion be greater than the dividend. Thus, supposing the divisor to be or, the quotient in this case will be double or triple the dividend. dend dend 3. To divide a fraction by an integer is only to multiply the integer into the denominator of the fraction, the numerator being continued. Thus: 7)( 4. A mixed number may sometimes be divided by an integer. Thus divide the integral part of the mixed number by the integer given, and if there be no remainder divide likewise the fraction of |