FRACTIONS. (Continued from page 226.) 186. To Subtract Common Fractions. Rule.—Reduce the fractions if necessary to equivalent fractions having a common denominator, fond the difference of their numerators, and write it over the common denominator. Example. (<)\ 11—77 7 — 3 5 (3) Tvv _ Tvt = Tvt Compare the following: 77 175ths-35 175ths = 42 175ths. Find the difference of— Fractions. 187. To Subtract One Mixed Number From Another When The Fraction In The Subtrahend Is Greater Than The Fraction In The Minuend. Example. 584 = 58T'T It is greater than ff, therefore we take 1 32f = 32i4 unit from the 8 units, change it to 24ths, and TVff oei 7 add it to the 9 24ths. i erence „ fj + & = |}. H — if = n 2 units from 7 (8 — 1) units = 5 units. 3 tens from 5 tens = 2 tens. Fractions. 188. To Multiply A Fraction By An Integer. Multiply Jv by 6. 6 times 77T are £f = If. 6 times T'T = J = If. 1. Observe that by the first operation we obtain f f; that in || there are 6 times as many parts as there are in fa, and that the parts are of the same size as those in fa. 2. Observe that by the second operation we obtain \; that in J there are the same number of parts as there are in fa, and that the parts are 6 times as great as those in fa. Note The 7 of ,», may be regarded as a dividend; the 24, as a divisor, and fa itself as a quotient. In ||, we have a dividend 6 times as great as that in fa, the divisor remaining unchanged. In | we have a divisor 1 sixth as great as that in fa, the dividend remaining unchanged. Multiplying the dividend or dividing the divisor by any number multiplies the quotient by the same number. Rule.—To multiply a fraction by an integer, multiply its numerator or divide its denominator by the integer. Fractions. 189. To Divide A Fraction By An Integer. Divide f by 3. One third of « = f. One third of \ = ^T. One third of f = 5«T = f. 1. Observe that by the first operation we obtain ?; that in f there are 1 third as many parts as there are in f, and that the parts are of the same size as those in f. 2. Observe that by the second operation we obtain s6r; that in /t there are the same number of parts as there are in ?, and that the parts are 1 third as great as those in ?. Note 1.—The 6 of 5 may be regarded as a dividend; the 7 as a divisor, and the f itself as a quotient. In f we have a dividend 1 third as great as that in ?, the divisor remaining unchanged. In j6! we have a divisor 3 times as great as that in f, the dividend remaining unchanged. Dividing the dividend or multiplying the divisor by any number divides the quotient by the same number. Rule.—To divide a fraction by an integer, divide its numerator or midtiply its denominator by the integer. I. Find the quotient. (See p. 245, problems 15 and 16 ) Fractions. 190. To Multiply By A Fraction. $6 multiplied by 3, means, take 3 times $6. $6 x 3 = $18. S6 multiplied by 2, means, take 2 times $6. $6 x 2 = $12. $6 multiplied by 2*, means, take 2|- times $6; or 2 times $6 + |of$6. $6x2^ $15. $6 multiplied by \, means, take £ of $6. $6 x \ = $3. $6 multiplied by -|, means, take f of $6. $6 x | = $4. To The Teacher.—Require the pupil to examine the preceding statements until he clearly understands that to multiply by a fraction is to take such part of the multiplicand as is indicated by the fraction. Thus: to multiply 48 by J is to take three fourths of 48; that is, three limes 1 fourth of 1^8. It will thus be clear that multiplication by a fraction involves both multiplication and division. Example I. Example II. Multiply 24 by £. Multiply \ by f. 1 fourth of 24 is 6. 1 fourth of f is ^. 3 fourths of 24 are 18. 3 f. .rths of | are 29T. Example III. Example IV. Multiply 275£ by f. Multiply 346f by 2 J, 1 fourth of 275| is 6S-&. Two times 346| = 692f 3 fourths of 275| are 206TV 1 half of 346| = 173£. 692| + 173£ = 866 Ans. Rule.— To multiply by a fraction, divide the multiplicand by the denominator of the fraction and midtiply the quotient thus obtained by the numerator of the fraction. Observe that in practice we may, if more convenient, multiply the multiplicand by the numerator of the fraction, and divide the product thus obtained by the denominator. To multiply 12 by | we may take 3 times 1 fourth of 12 or 1 fourth of 3 t'mes 12, as we choose. |