97. TO MULTIPLY A FRACTION BY A FRACTION OR MIXED NUMBER. PROBLEM. -— Multiply by ; or, what is of ? - ILLUSTRATIONS. T 1 of 3 = 1. TH ANALYSIS.-1. If is divided into 4 equal parts, each part is of as there are 3 thirds in a unit, each of which may be divided into 4 such parts, the unit contains 12 such parts, and each part is 2. Since ofis,of. 3. Since of is, of of ==}. RULE.-Multiply the numerators together for a new numerator, and the denominators for a new denominator. NOTE. - Mixed numbers may be changed to improper fractions when more convenient, or when the operation can be shortened by cancellation. Fractions with the word "of" between them are sometimes called "compound fractions." The word "of" is equivalent to the word "times," or the sign of multiplication; thus, of = { times ; or, ק. ORAL EXERCISES. 1. Multiply by; or, what is of ? 2. What is of 11? of? 8 of 18? of 1? 3. Multiply 51 by ; or, what is of 5? 4. If of a pie be cut into 2 equal pieces, what part of the whole pie is 1 piece? If cut into 3 equal pieces? 5. A girl, having of an orange, gave of it to her brother: what part of the orange did she give him. ANALYSIS. of an orange is equal to : of of an orange is 1. Hence she gave him of an orange. W of X? By . By . By f. By By 8. By 44. VII. . What is V of 1? W of 2? X of 3? ORAL AND WRITTEN EXERCISES. 15. If a pound of rice is worth $1, what is of a pound worth? W. 61 pounds? 9 pounds? 16. W. How much grain can be put into 71⁄2 bags, if each bag holds 24 bushels? 22 bushels? 17. W. What will 4 pounds of sugar cost, at 83 cents per pound? At 93 cents? At 111⁄2 cents? 10 18. What will 114 pounds of tea cost, at $1 per pound? At $1 per pound? W. At $? At $? At $13? 19. What will 228 bushels of corn cost, at 1 dollar a bushel? At $1 per bushel? At $11? * Change mixed numbers to improper fractions, and cancel when possible. WRITTEN EXERCISES. 20. What will of a pound of sugar cost at 19} cents per pound? FORMULA. 19 cents = 14 BY CANCELLATION. cents. 29 58 X 212 PROCESS. 19 cents. 3 4)58 3 14 cents. = 14. cents, of a pound will cost of 19 cents (or of 3 times 19 cents),* which is 142 cents. 21. What will of a bushel of corn weigh, if 1 bushel weighs 56 pounds? 22. If a box of starch weighs 283 pounds, how many pounds in 3 boxes? In of a box? In 13 boxes? 23. How many yards in g of 176 yards? In of 3435 yards? In 5 of 113 yards? 24. If 1 yard of lace is worth 88 cents, what are 16 yards worth? 25. If in 1 pane of glass there are 204 square inches, how many square inches in 43 such panes? 26. Find the weight of 3 boxes of raisins, if 1 box weighs 67 pounds? 27. If there are 904 quarts of milk in a can, how many quarts in of the can? * Either expression may be used at the option of the teacher. DIVISION OF FRACTIONS. 98. TO DIVIDE A FRACTION OR MIXED NUMBER BY AN INTEGER. PROBLEM.-(a.) I wish to divide of a cake equally among 3 boys: what should each boy receive? (b.) I wish to divide of a cake in the same way: what should each boy receive? If 3 boys are to receive of a cake, each boy should receive one-third of, or of the cake. PRINCIPLE. (a.) I. Dividing the numerator of a fraction divides the value of the fraction. The number of parts are decreased, while their size remains the same. (b.) II. Multiplying the denominator of a fraction divides the value of the fraction. The number of parts remain the same, while the size is decreased. NOTE TO THE TEACHER. The teacher, in explaining the above principles, should refer to the corresponding illustrations, remembering that children can readily learn to recite explanations, and yet be utterly ignorant of their meaning. ORAL EXERCISES. 1. What is 1 third of 3 apples? Of 6 apples? Of 12 lemons? Of 3 fifths of a pound? Of 6 eighths of an ounce? Of 12 sevenths of a gallon? 2. Divide 6 apples and 3 fifths of an apple into 3 equal parts. What is 1 third of 63 apples? What is 1 fifth of 1015 oranges? Of 15 melons? 3. Divide 1 orange into 3 equal parts. Divide of an orange into 3 equal parts. In the same way divide of an orange. Divide of an orange. Divide & of of an orange. PROBLEM. Divide 3473 by 5. PROCESS. 5)3473 69% - ANALYSIS. (1.) One-fifth of 347 is 69, with a remainder of 2 units. (2.) 2 units are equal to §, which, added to 3, equal §. One-fifth of is, which annexed to 69=6985. RULE. To divide a fraction by an integer, divide the numerator, or multiply the denominator, by the integer. NOTE. When the dividend is a mixed number, and less than the divisor, change it to an improper fraction, otherwise divide the integral number, and add the remainder, if any, to the fraction; then divide the fraction. ORAL, SLATE, AND BLACKBOARD EXERCISES. 31. If I gave of a dime for 11 boxes of matches, what did 1 box cost? W. If I gave 3 dimes? If 11 boxes cost of a dime, 1 box costs one-eleventh of of a dime, or, of a dime. 32. What will be the cost of 1 box of pencils, if 8 boxes cost $? Cost $5? W. Cost $23? $113? |