7. Multiply by 17 8. Multiply 9. What cost 90. by 38. Ans.. Ans.. Ans.. Ans. Ans.. of a bushel of corn at of a dollar per bushel? 10. If a man travels he travel in of an hour? Ans. of a dollar. of a mile in an hour, how far would 11. If a bushel of corn will buy Ans. of a mile. of a bushel of salt, how much salt might be bought for 2 of a bushel of corn? Ans. 2 of a bushel. 12. If of 3 of a dollar buy one bushel of corn, what will of of a bushel cost? Ans. of a dollar. 13. If 3 of 4 of of an acre of land cost one dollar, how much may be bought with of $18? Ans. 197 acres. ART. 157. To multiply a mixed number by a mixed number, it is only necessary to reduce them to improper fractions, and then proceed as in the foregoing rule. Ex. 1. Multiply 43 by 63. OPERATION. 43=23; 63=20. Ans. 30. 6. What cost 7 cords of wood at $5 per cord? Ans. $4131. 7. What cost 78yd. of cloth at $3 per yard? Ans. 2518. 8. What cost 6 gallons of molasses at 232 cents per gallon? Ans. $1.5218. 9. If a man travel 3 miles in one hour, how far will he travel in 97 hours? Ans. 34. QUESTION. Art. 157. How do you multiply a mixed number by a mixed number? 10. What cost 36111 acres of land at $25 per acre? Ans. $9167118. 11. How many square rods of land in a garden, which is 97 rods long, and 493 rods wide? Ans. 4810 rods. DIVISION OF COMMON FRACTIONS. ART. 158. DIVISION of Fractions is the process of dividing when the divisor or dividend, or both, are fractions. ART. 159. To divide a fraction by a whole number. FIRST OPERATION. 8 ÷4 = 2 9 9 Ans. . In this operation we divide the numerator of the fraction by 4, and write the quotient, 2, over the denominator. It is evident this process divides the fraction by 4, since the number and size of the parts into which the whole number is divided remain the same, while only of the number of parts is expressed by the fraction. Therefore, Dividing the numerator of a fraction by any number divides the fraction by that number. Ex. 2. Divide by 9. SECOND OPERATION. 517 9= 5 63 Ans. We multiply the denominator of the fraction by the divisor, 9, and write the product under the numerator. It is evident this process divides the fraction, since multiplying the denominator by 9 makes the number of parts into which the whole number is divided 9 times as many as before, and consequently each part can have but of its former value. Now, if each part has but of its former value, while only the same number of parts is expressed by the fraction, it is plain the fraction has been divided by 9. Therefore, Multiplying the denominator of a fraction by any number divides the fraction by that number. RULE. Divide the numerator of the fraction by the whole number, when it can be done without a remainder, and write the quotient over the denominator. Or, Multiply the denominator of the fraction by the whole number, and write the product under the numerator. QUESTIONS. -Art. 158. What is division of common fractions? — Art. 159. How is the fraction divided by the first operation? What inference may be drawn from this operation? How is a fraction divided by the second operation? What inference is drawn from this operation? What is the rule? $117; this he bequeaths to his five children. share will each receive? 12. Divide by 15. 23 13. Divide by 28. Ans. 23: 14. James Page's estate is valued at $10,000, and he has given of it to the Seamen's Society; of the remainder he gave to his good minister; and the remainder he divided equally among his 4 sons and 3 daughters. What sum will each of his children receive? Ans. $680,0 ART. 160. To divide a whole number by a fraction. The reason of this operation is evident, since 13 will contain as many times as there are sevenths in 13, equal 91 sevenths. Now, if 13 contain 1 seventh 91 times, it will contain as many times as 91 will contain 3, equal to 30§. RULE. · Multiply the whole number by the denominator of the fraction, and divide the product by the numerator. QUESTIONS. Art. 160. What is the rule for dividing a whole number by a fraction? Give the reason for the rule. 5. Divide 5 by . 6. Divide 12 by 2. 7. Divide 16 by 1. line it? 9. I have 50 square yards of cloth; yard wide, will be sufficient to 10. A. Poor can walk 37 can walk as fast as Poor. to walk the same distance? Ans. 25. Ans. 16. Ans. 32. Ans. 1111. how many yards, g of a Ans. 83 yards. miles in 60 minutes; Benjamin How long will it take Benjamin Ans. 73 minutes. ART. 161. To divide a mixed number by a whole number. Having divided the whole number as in simple division, we have a remainder of 5g, which we reduce to an improper fraction, and divide it by the divisor, as in Art. 159. Annexing this frac tion to the quotient 2, we obtain 243 for the answer. Hence, to divide a mixed number by a whole number, Divide the integral part of the mixed number; and the remainder, reduced if necessary to a simple fraction, divide as in Art. 159. EXAMPLES FOR PRACTICE. 2. Divide 17 by 7. Ans. 218. 3. Divide 183 by 8. 7 men. 10. Divide $1063 among 8 boys. 11. What is the value of 3 of a dollar? 12. Divide $1077 among 4 boys and 3 girls twice as much as the boys. Ans. 217. Ans. 3. 91 Ans. 29. Ans. 617. Ans. 478. Ans. 35. Ans. $24. Ans. $135. Ans. $0.3413. girls, and give the Ans. Boy's share, $1042; Girl's share, 212. 13. If $14 will purchase 17 of a ton of copperas, what quantity will $1 purchase? Ans. 1cwt. QUESTION. Art. 161. How do you divide a mixed number by a whole number? ART. 162. To divide a whole number by a mixed number. Ex. 1. Divide 25 by 4%. OPERATION. 43125 5 5 23)125(518 115 Ans. 51%. We first reduce the divisor and dividend to fifths, and then divide as in whole numbers. The reason why the answer is in whole numbers, and not in fifths, is because the divisor and dividend were both multiplied by the same number, 5, and therefore their relation to each other is the same as before, and the quotient will not be altered. Hence, Reduce the divisor and dividend to the same parts as are denoted by the denominator of the fraction in the divisor, and then divide as in whole numbers. 10 5. Divide 342 by 1447. Ans. 23. 6. There is a board 19 feet in length, which I wish to saw into pieces 23 feet long; what will be the number of pieces, and how many feet will remain ? Ans. 74 pieces. ART. 163. To divide a fraction by a fraction. Ex. 1. Divide 7 by . OPERATION. = = Ans. 131. In this operation, we invert 813. the terms of the divisor, and then proceed as in Art. 156. The reason of this process will be seen, when we consider that the divisor, §, is an expression denoting that 4 is to be divided by 9. Now, regarding 4 as a whole number, we divide the fraction multiplying the denominator; thus, 8 × 4 32 by it, by But the divisor, 4, is 9 times too great, since it was to be divided by 9, as seen in the original fraction; therefore the quotient,, is 9 times too small, and must be multiplied by 9; thus, 13. By this operation, we have multiplied the denominator of the dividend by the numerator of the divisor, and the numerator of the dividend by the denominator of the divisor. Hence the QUESTIONS. Art. 162. How do you divide a whole by a mixed number? How does it appear that this process does not alter the quotient? Art. 163. How do you divide a fraction by a fraction? why this process divides the fraction of the dividend. Give the reason |