146. Equal factors in a numerator and a denominator may also be cancelled. Ex.-Reduce & of of 35 to a simple fraction. Throw the whole number 35 into a fractional form. Cancel 3 (that is, divide by 3) in the first denominator and second numerator. Cancel 7 (that is, divide by 7) in the second denominator and third numerator. Then multiply the remaining factors. 1. Multiply 3, 4, and g together. together. Ans. 27 70° Ans. 4. Ans. 27 56 or denominator disappears by cancelling, 1 (not 0) is left in its place. 148. Cancelling is dividing. Hence, by cancelling before we multiply, we save the trouble of dividing after we multiply, to reduce the result to its lowest terms. 149. GENERAL RULE.-1. To multiply one fraction by another, or to reduce a compound fraction, first cancel factors common to any numerator and denominator; then multiply the numerators together for a new numerator, and the denominators for a new denominator. 146. In what other way may we cancel? Give an example.-147. When we cancel, what operation do we perform? When a numerator or denominator disappears by cancelling, what is left?-148. What trouble do we avoid, by cancelling before we multiply?-149. Give the general rule for the multiplication of fractions. MULTIPLICATION OF FRACTIONS. 89 2. Whole numbers occurring in a compound fraction must first be reduced to a fractional form, and mixed numbers to improper fractions. EXAMPLES FOR THE SLATE. 1 19 1. Multiply together,, 14, and 1. 36 10 7. Reduce of of 8. Reduce 2 5 9. Reduce of § of 1 of 41. 16 10. Reduce & of 7 of 10 of 12. MIXED NUMBER X MIXED NUMBER. 150. RULE. To multiply two or more mixed numbers together, reduce them to improper fractions, and proceed as in multiplication of fractions. EXAMPLE.—Multiply 43, 54, and 11 together. Reduce the mixed numbers to improper fractions. Then, cancelling, we get 11, or 351. 150. Recite the rule for multiplying two or more mixed numbers together. 2. What will 15 yards of velvet cost, at 41 dollars a yard? Ans. $675. 32 son, 3. A person having 423 acres of land, left 3 of it to his What was the son's share? Ans. 254 acres. 4. A farmer has three wheat fields, of 4 Their average yield is 333 bushels to the acre. yield of the whole? What is the Ans. 4251 bu. 5. General Putnam lived to be 72 years old. Patrick Henry attained of that age; how old was he at the time of his death? 7 6. How much flour must be laid in for a garrison of 355 men, to allow each man 563 pounds? 327 7. The British House of Commons contains 654 members. 34 of this number are from England and Wales; how many does that make? Ans. 496 members. 8. How many yards are there in a bale of linen, containing 56 pieces, if there are 253 yards in each piece? 9. If a clock ticks sixty times in a minute, how many times will it tick in 153 hours, there being sixty minutes in an hour? Ans. 56160 times. 10. If 680 persons subscribe for a work in three volumes, costing half a guinea a volume, what is the whole amount of the subscription? 11. A owns of a factory. He sells half his share to B, who in turn sells of his share to C. What part of the factory belongs to C? Ans.. 12. What will be the cost of three boxes of oranges, allow ing 96 oranges to the box, at 12 cents apiece? 1 13. Multiply 53, 51, and together. 19 Ans. 11. DIVISION OF FRACTIONS. 91 Division of Fractions. FRACTION WHOLE NUMBER. 151. To divide any number of equal parts by 2, we may either take half as many such parts, or make each part half as great. EXAMPLE.-Divide by 2. Take half the number of parts. Half of is . Or, make each part half as great. A tenth is half as great as a fifth. Hence half of is 4. 4 can be reduced to . The two answers agree. 10 Now, in the first case, we divided the numerator by 2. In the second case, we multiplied the denominator by 2. The former mode is better, because it brings the fraction at once in its lowest terms. Hence the following rule. 152. RULE.-1. To divide a fraction by a whole number, divide its numerator by the whole number, if this can be done without a remainder; if not, multiply its denominator. 2. To divide a mixed number by a whole number, reduce the mixed number to an improper fraction; then proceed as above. EXAMPLES.-1. Divide by 6. 5 16 If the numerator 5 contained 6 exactly, we should divide it by 6. As it does not, we multiply the denominator. 2. Divide 2 by 6. Reduce 24 to an improper frac tion, 19. As the numerator 18 contains 6 exactly, divide it by 6. 5 5 Ans. 151. To divide any number of equal parts by 2, what may we do? Illustrate these two methods, in dividing by 2. Which method is better?-152. Give the rule for dividing a fraction by a whole number. 3 is contained in 1, 7 times. Hence, in it is contained 3 of 7 times, or 21 times. But is twice as great as, and half as many times.of 21 of is. hence is contained only Answer, 21, or 2. Now, what have we done to the dividend, to produce 21? We have multiplied its numerator by the denominator of the divisor, and multiplied its denominator by the numerator of the divisor. Or, in other words, 2 0 we have inverted the divisor, and then mul- 3/3 × 1 = 71 tiplied the fractions. Hence the rule. 155. RULE.-1. To divide one fraction by another, multiply the dividend by the divisor inverted. 2. Whole and mixed numbers must first be reduced to improper fractions. 154. Divide by 2. What have we done to the dividend g, to produce this result?-155. Give the rule for dividing one fraction by another. |