23 by 28. 9. What cost of a bushel of corn at of a dollar per bushel? 10. If a man travels of a mile in an hour, how far would he travel in of an hour? 11. If a bushel of corn will buy how much salt might be bought for of a bushel of salt, of a bushel of corn? 12. If of of a dollar buy one bushel of corn, what will of of a bushel cost? 13. If of of of an acre of land cost one dollar, how much may be bought with of $ 18? 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. 2. Multiply 7 by 83. 3. Multiply 47 by 91. 4. Multiply 114 by 8. 5. Multiply 12 by 113. 6. What cost 72 cords of wood at $53 per cord ? 7. What cost 7 yd. of cloth at $3 per yard? 8. What cost 6 gallons of molasses at 23 cents per gallon ? 9. If a man travel 3 miles in one hour, how far will he travel in 97 hours? QUESTIONS. Art. 157. How do you multiply a mixed number by a mixed number? 10. What cost 36111 acres of land at $253 per acre? 11. How many square rods of land in a garden, which is 97 rods long, and 493 rods wide? DIVISION OF VULGAR FRACTIONS. ART. 158. DIVISION of Vulgar Fractions is the process of dividing fractions by fractions, or whole numbers and fractions by each other. ART. 159. To divide a fraction by a whole number. Ex. 1. Divide FIRST OPERATION. 9 2 9 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. 7X9 63 We multiply the denominator of the fraction by the divisor, 9, and write the product under the nu merator. 10 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 I. — 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, RULE II. Multiply the denominator of the fraction by the whole number, and write the product under the numerator. QUESTIONS. Art. 158. What is division of vulgar fractions? - Art. 159. How is the fraction divided by the first operation? How does it appear that this process divides the fraction? What inference may be drawn from this operation? How is a fraction divided by the second operation? Will you explain how this process divides the fraction? What inference is drawn from this operation? What is the first rule for dividing a fraction by a whole number? The second? 11. John Jones owns of a share in a railroad valued at $117; this he bequeathes to his five children. What part of a share will each receive? 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. $68047 ART. 160. To divide a whole number by a fraction. Ans. 301. For convenience, we invert the terms of the divisor, and then multiply the whole number by the original denominator, and divide the product by the numer ator. 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. EXAMPLES FOR PRACTICE. 2. Divide 18 by 7. 3. Divide 27 by 11. 4. Divide 23 by 4. 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. 8. Divide 100 by 17. 9. I have 50 square yards yard wide, will be sufficient to 10. A. Poor can walk 37 can walk as fast as Poor. to walk the same distance? of cloth; how many yards, g of a line it? miles in 60 minutes; Benjamin How long will it take Benjamin ART. 161. To divide a mixed number by a whole number. Ex. 1. Divide 173 by 6. 6)1793 OPERATION. 43 2-53-4; x6 RULE. 43 48 Ans. 24. Having divided the whole number as in simple division, we have a remainder of 5%, which we reduce to an improper fraction, and divide it by the divisor as in Art. 159. Annexing this fraction to the quotient 2, we obtain 243 for the answer. -Divide as in whole numbers, as far as the division can be carried, and, if the remainder is a mixed number, reduce it to an improper fraction, and then divide it by the divisor; but if the remainder is a simple fraction. only, merely divide it by the divisor. (Art. 159.) 10. Divide $1067 among 8 boys. 11. What is the value of of a dollar. 12. Divide $1077 among 4 boys and 3 girls, and give the girls twice as much as the boys. Ans. Boy's share $ Girl's share $ 13. If $14 will purchase of a ton of copperas, what quantity will $1 purchase? QUESTIONS. Art. 161. What is the rule for dividing a mixed number by a whole number? ART. 162. To divide a whole number by a mixed number. Ans. 51. Ex. 1. Divide 25 by 43. 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. RULE. - 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. EXAMPLES FOR PRACTICE. 2. Divide 36 by 97. 3. Divide 97 by 1311. 4. Divide 113 by 214. 5. Divide 342 by 147 6. There is a board 19 feet in length, which I wish to saw into pieces 2 feet long; what will be the number of pieces, and how many feet will remain? ART. 163. To divide a fraction by a fraction. 7 7 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 by it, by multiplying the denominator; thus, 8x432 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, 32, is 9 times too small, and must be multiplied by 9; thus, 32 132. 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 7 X 9 63 = QUESTIONS. Art. 162. What is the rule for dividing 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? Give the reason why process divides the fraction of the dividend. this |