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, § 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. $8.82. 4. Add of a week, of a day, and an hour together? 5. Add of a dollar, of a dollar, and together? 6. Add of a yard, of a foot, and gether? Ans. 2d. 14th. of a dime Ans. $1.45. of an inch to Ans. 1 ft. 4 in. 1 barley corn. CASE XI, To add compound fractions together, connected by the preposition or (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. 1. Example,-Add of of, and of of together? Ans.. Operation, = xx% reduced is. Now, it is plain, that of of of the first compound is equal to , and × × of the second compound is equal to 'reduced is equal to, which added to the sum is as required. 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? 4. Bring $167 to eighths? Ans. 79 66 81 66 100 3 ( 185 8 66 175 2 3. Bring 33 cts. to thirds? 5. Bring $87 to halves? 6. Bring 14 to an improper fraction? TO MULTIPLY FRACTIONS. CASE I. When the fractions are proper. 66 101 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 -B be one mile, yard, &c. 1 1 2 The square of 1 is 1, because 1 × 1 is 1, and is, hence, × = 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. ILLUSTRATION. 27 yards. In the rectangular or parallelo- A, gram A B C D, the length of the side A B is 10 yards, and the length of the line A C is 73 yards, the line A B is divided into 21 parts, C and the line A C into 15 equal parts, which are drawn at right angles to each other, consequently, there are 315 rectangles in the whole figure A B C D, and every four of these make 1 square yard, this is manifest from the following example: therefore, 10 × 7 = 2 × 15 = 315 X 78 as required. = CASE III. To multiply a whole number by a fraction. RULE.-Multiply the whole number by the numerator of the fraction, and divide the product by the denominator, the quotient will be the result. From what has been already stated, it is evident, that the multiplication of a whole number by a fraction implies the taking some part of it; for instance, if we multiply 4 by, agreeably to the rule 4 x == 2, and 9 = 3, &c. Multiply 35 by . Multiply 84 by 4. APPLICATION OF FRACTIONS TO SHORT ACCOUNTS.' 1. Multiply 114 by 11 cts. Example, 2 x2 = 529 Ans. $1.32. 2. What will 74 lbs. come to at 8 c. pr. lb? 3. What will 44 lbs. come to at $3 per lb? 4. What will 19 yards come to at $3 of a dollar? Ans. $7.403. 5. What will 2 yards come to at $ of a dollar? Ans. $2.40§. 6. What will 63 lbs. of tea cost at 65 cts. per pound? Ans. $4.523. SUBTRACTION OF FRACTIONS. CASE I. If the fractions have a common denominator. RULE.-Subtract the lesser numerator from the greater, and under the remainder write the common denominator, and reduce the fraction if necessary. 1. Example From $ take? Ans. of a doll. or 50c. If the denominators of the fractions are unlike. RULE.-Find a common denominator according to Case VI. Addition, ("Second Method.") 1. Example-From take 4? Ans. . Here the denominators of the fractions are in the ratio By Case I, or by Case VI. Addition, find a common denominator; thus, by cross multiplication. 11 x7 = 77 common denominator, the result is 4. 1. From $1% take?? 2. From $3 take? Ans. $ or .02 c. 66 $1 or .50c. 3. From $1 take? (Here the ratio is as 4 to 1.) When the fractions have a unit for a numerator. RULE. Write the difference of their denominators over their product. |