Division of Fractions. 251. The reciprocal of a fraction is the fraction with its terms interchanged. Thus, the reciprocal of & is }, for } × & = 1. To write the reciprocal of a whole or a mixed number, we reduce the number to an improper fraction, and write the improper fraction with its terms interchanged. Thus, the reciprocal of 4 is ; the reciprocal of 23 is 3. Multiplying by the reciprocal of a number gives the same result as dividing by the number ($101). Hence, 252. To Divide by a Whole Number or a Fraction, Multiply the dividend by the reciprocal of the divisor. A Short Method of Dividing a Mixed Number by a In the first problem we simply divide the whole number 16 by 4, and then the fraction ₫ by 4, and obtain the result at once, 4}. In the second problem we divide the 16 by 7, and obtain the quotient 2 and a remainder 2. The remainder 2 is joined with the 3, making 2, or, and ÷ 7 = • This method is the shortest method of dividing a mixed number by a whole number. EXERCISE 54. Find the quotient of: 1. 317÷5. 5. 426. 9. 48312. 10. 2411. 11. 1937. 12. 2918. 254. Examples. 1. Find the value of (24÷§) × §. We have to divide 2 by, and to multiply the result by . But 21 is divided by by multiplying 21 by §. Hence, we invert the divisor and find the product of the three fractions. We have to divide 2 by of; hence we may find the product of and 3, and invert this product, or we may invert both factors of this product and multiply by the inverted factors and §. -16 mt cleve 255. Example. what is the cost of barrel? SOLUTION. Since cost of $3, or $1. If or $4. Hence, If of 3 of÷(×2 of 3). 7 13. of of ÷ of of 17. of a barrel of flour costs $3, of a barrel? What is the cost of a of a barrel of flour costs $3, of a barrel will of a barrel costs $1, a barrel will cost 4 × $1, 256. To Find the Whole when a Fractional Part is Given, Divide the given part by the numerator of the fraction and multiply the quotient by the denominator. EXERCISE 56. 1. If of a ton of hay costs $15, what is the cost of one ton ? 2. 15 is of what number? 3. If of a roll of carpeting is worth $75, what is the whole roll worth? 4. A man sold 63 yards of cloth, which was of the whole piece. How many yards were there in the piece? 5. A farmer sold of his hay for $195.60. What was the value of his entire crop of hay? 9. If of an acre of land is worth $32, what is the value of an acre? 10. If of a bushel of wheat is worth 48 cents, what is the value of 22 bushels of wheat? 11. If of a ton of hay is worth $15, what is the value of 7 tons of hay? 12. If g of a cord of wood is worth $4, find the value of 7 cords of wood? 13. If of a barrel of apples is worth 44 cents, what is the value of 12 barrels of apples? 14. $125 is money? 15. $132 is more than (that is, of) what sum of less than what sum of money? 20. If of a yard of silk is worth $1, find the value of 4 yards of silk. 21. If of a yard of linen is worth 60 cents, what is the value of 24 yards of linen? 22. If a man who owned of a schooner sold 3 of his share for $1200, what was the value of the schooner? 23. One fourth of one third of three sevenths of a number is 60. What is the number? 24. Three fourths of two ninths of six sevenths of a number is 125. What is the number? 25. If of the goods in a store were sold for $1000, what was the value of the whole stock of goods? 2 26. If of a farm is worth $1200, what is the value of the whole farm? Least Common Denominator. 257. Similar Fractions. Fractions that have a common denominator are called similar fractions. 258. Least Common Denominator. The L. C. M. of the denominators of a series of fractions is called their least common denominator. The letters L. C. D. stand for the words Least Common Denominator. 259. Example. Change, 2, to similar fractions. The L. C. D. of the fractions is 12. If both terms of are multiplied by 4, the value of the fraction will not be altered, but the form will be changed to 12. If both terms of are multiplied by 3, the equivalent fraction will be iz. And if both terms of are multiplied by 2, the equivalent fraction will be 12. The multipliers, 4, 3, and 2, are obtained by dividing 12, the L. C. D. of the fractions, by the respective denominators of the given fractions. Therefore, 260. To Change Fractions to Similar Fractions, Divide the least common denominator of the fractions by the denominator of the first fraction, and multiply both terms of this fraction by the quotient. Proceed in the same way with each of the other given fractions. Change 2, 3, 11 to similar fractions. 4 = 22; 8=23; 12 = 22 x 3. Hence, the L. C. D. is 28 x 3, or 24. |