Add and }, here 4 and 8 make 12 RULE-Add the denominators together for a new numerator, and multiply them together for a new denomi nator. CASE VII. When the numerators are alike and more than a unit. RULE. Add the denominators together, and multiply their sum by the common numerator, and the product will be a new numerator; also, the product of the denominators will be a common denominator. Add and, here 4 and 7 make 11, which multiply by the numerator 3, which is common to both. & × 8 = 45, as above. , here and THIRD METHOD. by multiplying the numer ators alternately by the denominators. 3 x8 = 24 3 × 7 : 21 Add and. Here the ratio between the denomi nators is as 1 to 7; therefore, × 7 = and make RULE.-Find a common denominator by reducing the fractions to the lowest terms. Multiply all the divisors together 4 x 5 x 2 = 40 common denominator. CASE IX. To add mixed fractions whose numerators and denominators are unlike. Add $15 $35 (3) = The operation can be performed thus, by cross multiplication, 24 +20= 1 reduced, from whence the following Rule is deduced: multiply each numerator by all the denominators, except its own, for a new numerator, and all the denominators together for a new denominator. Add 1, 1, §, and together, (say dollars.) To add mixed or compound fractions. 1. Add of a day together? 2. Add of a year, day, of an hour, and of an hour, and of a minute Ans. 16h. 48m. 18s. of a month, ✈ of a week, 1⁄2 of a of a minute together? Ans. 4m. 1w. 1d. 8h. 5m. 48s. 3. Add of an eagle, of a dollar, of a dime, and of a cent? Ans. $8.824. 4. Add of a week, of a day, and an hour together? Ans. 2d. 144h. 5. Add of a dollar, of a dollar, and together? of a dime Ans. $1.451. of an inch to 6. Add of a yard, of a foot, and gether? CASE XI. To add compound fractions together, connected by the preposition of (see Def. 9.) GENERAL RULE. Multiply the numerators together for a new numerator, and the denominators together for a new denominator. Reduce the fractions, and then add them together agreeably to Case VIII. or IX. Operation, X X = 1% reduced is. Now, it is plain, that of % of of the first compound is equal to , and × 3 × of the second compound is equal to 4, which added to the sum is To reduced is equal to as required. Operation, × 1 = √6 = of a dollar or 10c. × 1 of of a dollar? Ans. 50c. To reduce mixed fractions to parts, or to an improper fraction. (See 11th Definition.) RULE.-Multiply the whole number by the denominator of the fraction, and add the numerator to the product for the numerator of the fraction sought, under which will be the given denominator. Example.-Reduce 17 dollars to half dollars. ILLUSTRATION. = It is well known that two half dollars are equal to one dollar; consequently, as 1 dollar is to 2 halves, 17 units or 17 dollars will contain 17 times as much, to which if we add one-half we get 35 halves for the required answer. 1. Bring $19 to quarters? 2. Bring $20 to quarters? 3. Bring 33 cts. to thirds? 4. Bring $167 to eighths? 5. Bring $87 to halves? 6. Bring 14 to an improper fraction? TO MULTIPLY FRACTIONS, CASE I. When the fractions are proper. RULE.-Multiply the numerators together for a new numerator, and the denominators together for a new denominator, ILLUSTRATION. It is manifest, that when a number is multiplied by 1, the product is equal to the multiplicand; therefore, when a number is multiplied by a fraction, which is less than 1, the product must be less than the multiplicand. Example 1.-Multiply by ? Ans.. From the analysis of Geometry, we find, that if a line be divided into 2 equal parts, the square of the whole line is 4 times the square of half the line: thus, let the line A 1 -B be one mile, yard, &c. The 2 1 2 square of 1 is 1, because 1 x 1 is 1, and is, hence, × 1 = 1 of 1. CASE II. squared When the multiplier and multiplicand are both mixed numbers. RULE.-Bring them to improper fractions, agreeably to Case XII. (Addition,) this done, multiply the numerators together as before, for a new numerator, and the denominators together for a new denominator; divide the new numerator (so called) by the new denominator, and the result will be the product of the mixed numbers. |