NOTE 1.-Reduce mixed numbers to improper fractions. 8. Multiply 21 by 31. SOL.-21-2, and 31; then 2X1=63=77 Ans. = Ans. 33. 11. 21× 21. Ans. 61. 10. X17. Ans. 15. 12. 161×16. Ans. 2721. NOTE 2. After indicating the operation, employ CANCELLATION, when possible, as shown in Art. 61. 17. Multiply 3, 21, 5, and of 5 together. 18. 19. 20. 21. Ans. 417. Ans. 179. For additional problems, see Ray's Test Examples. ART. 153. MISCELLANEOUS EXAMPLES. What will be the cost 1. Of 23 lb. of meat, at 1 cts. a lb.? 2. 3. 4. Ans. 24 cts. Of 3 yd. linen, at $ a yd. ? of 5 yd.? of 7 yd.? of 61 yd.? 53 yd. ? Of 31 lb. of rice, at 44 cts. a lb. ? Ans. to last, $35. Ans. 16 cts. Of 3 tuns of iron, at $183 per T.? 5. Of 1 yd. of muslin, at $3 per yd.? Ans. $60. Ans. $. Ans. $2. Ans. $63. REVIEW.-152. How may a whole be multiplied by a mixed number? 153. How multiply one fraction by another, Rule? NOTE 1. What is required when mixed numbers occur? 2. When employ Cancellation? 8. At the rate of 5 miles an hour, how far will a man travel in 72 hours? Ans. 425 mi. 9. I own of a steamboat, and sell of my share: what part of the boat do I sell? Ans. 10. At $63 per yard, what cost of a piece of cloth containing 5 yards? 11. of of 161, × 3 of 7 of 15,-what? Ans. $81. Ans. 343. For additional problems, see Ray's Test Examples. ART. 154. DIVISION OF FRACTIONS. The object in dividing a fraction by a whole number, is to separate the fraction into a given number of equal parts, and find the value of one of the parts; or, to find what part a fraction is of a whole number. 1. If 3 yards of ribbon cost $, what cost 1 yard? Here, is divided into 3 equal parts, and the value of each part is ; thus, ++; the number of parts corresponds to the divisor, and the value of each, to the quotient. 2. At 2 dollars a yard, what part of a yard of cloth can be bought for a dollar? ANALYSIS.-Had it been required to find how many yards, at $2 a yard, could be bought for $6, the $6 should be divided by $2; and, to find the part of a yard that $will pay for, divide $1 by $2: to divide 1 OPERATION. 2 1 ÷2 2X2 4 Ans. yd. by 2, multiply the denominator (Art. 133); the quotient is one-fourth. *3. If 4 yards of muslin cost of a dollar, what will 1 yard cost? Ans. $. *4. If 1 orange cost 3 cents, what part of an orange can be purchased for a cent? Ans. Rule for Case I.-Divide the numerator by the whole number, when it can be done without a remainder; write the quotient over the denominator. Otherwise, Multiply the denominator by the whole number, and write the product under the numerator. 9 11 of a dollar, what will Ans. $3. 6. If a man travel of a mile in 3 hours, how far does he travel in 1 hour? Ans. 1 mi. 7. If 5 yards of tape cost of a dollar, what will 1 yard cost? Ans. 180 8. If 7 pounds of coffee cost of a dollar, what will 1 pound cost? Ans. $25. 9. At 4 dollars a yard for cloth, what part of a yard willig of a dollar buy? Ans. yd. 10. At 5 dollars a tun, what part of a tun of hay willa dollar purchase? Ans. ' T. 11. At 6 dollars a barrel, what part of a barrel of flour will $23 pay for? Reduce a mixed number to an improper fraction, and it may be divided by a whole number, the same as a proper fraction. 12. If 5 bushels of wheat cost 3 1 bushel? OPERATION. 23 = 12 12÷6 Ans. dollars, what cost Ans. of a dollar. 13. If 7 ounces of opium cost 8 dollars, what cost 1 ounce ? Ans. $11. 14. If 18 be divided into 9 equal parts, what will each part be? 19 Ans. 2 REVIEW.-154. What is the object in dividing a fraction by a whole number? In Ex. 1, what does the divisor show? The quotient? 154. In Ex. 2, what does the quotient show? How divide a fraction by a whole number, Rule for Case I? 15. 16. 35 44 ÷ 8= Ans. 3. 18. 337 Ans. 11. Ans. 236. 19. 4715 = = Ans. 345. 17. 12411 = 20' Ans. 14. 20. 13018 Ans. 77. CASE II. ART. 155. To divide a whole number by a fraction. Dividing a whole number by a fraction, is finding how many times the fraction is contained in the whole number. 1. At of a cent for 1 lemon, how many can be bought for 4 cents? SOLUTION.-In 4 cents there are 12 thirds of 1 cent (Art. 141). If 1 lemon costs 2 thirds of a cent, there will be as many lemons as 2 thirds are contained times in 12 thirds; that is, 6. Ans. 6 lemons. OPERATION. 4 3 Thirds 2)12 thirds. Ans. 6 Or thus: is contained in 4 as many times as there are thirds in 4, that is, 12 times; and 2 thirds in 4, one-half as many times as 1 third; 12÷2=6 times. In this operation, the whole number is reduced to the same name the same parts of a unit-as the divisor, that the divisor and dividend may be of the same denomination. Art. 41, Rem. The whole number is multiplied by the denominator of the fraction, and the product divided by the numerator. *2. At a cent each, how many apples can be bought for 3 cents? Ans. 6 apples. *3. At of a dollar per yard, how many yards of cloth can you buy for 6 dollars? Ans. 8 yards. 4 Rule for Case II.—Multiply the whole number by the denominator of the fraction, and divide the product by the numerator. REVIEW.-155. What is dividing a whole number by a fraction? How many times is one-third contained in 1? In 2? In 3? In 4? How many times two-thirds in 1? In 2? In 8? In 4? 11. At 2 dollars for 1 yard of cloth, how many yards. can be bought for $6? Ans. 21 yd. 12. At 33 cents a lb., how many pounds of rice can be bought for 30 cts. ? 13. How many times 4 in 50? Ans. 8 lb. Ans. 113. 14. Divide 56 by 51. CASE III. Ans. 103. ART. 156. To divide a fraction by a fraction. The object in dividing a fraction by a fraction, is to find how many times the divisor is contained in the dividend; or, what part the divisor is of the dividend. 0 1. At of a dollar per yard, how many yards of muslin can be bought for $? SUGGESTION.-Find how often 2 tenths are contained in 9 tenths, as you would find how often 2 cents are contained in 9 cents; that is, by dividing 9 by 2: 92=43. OPERATION. Tenths 2)9 tenths. Ans. 4. Hence, when two fractions have a common denominator, obtain their quotient by dividing the numerator of the dividend by nomination, inches; so, also, to find how often 2 thirds are |
