RULE I. Multiply the whole number by the denominator of the fraction, and divide the product by the numerator. RULE II. Divide the whole number by the numerator of the fraction, and multiply the quotient by the denominator. 1. If a pair of shoes be 2. If a yard of linen be worth of a dollar, how many worth of a dollar, how many 2 pair can you buy with 6 dol-yards can you buy with 5 lars? dollars? 3. How many times is g contained in 6? 5. When apples are worth of a dollar a bushel, how many bushels can you purchase with 7 dollars? 7. How many times is contained in 7? 4. How many times is contained in 10? 6. If of a barrel of flour last a family 1 month, how long will 5 barrels last the same family? 8. How many times is contained in 11? 9. If of a cord of wood be worth 6 dollars, what is the value of a whole cord? 10. How many times is 11. If Ans. 8 dollars. contained in 25? Ans. 53 times. of an acre of land be worth 80 dollars, what is the value of an acre? Ans. 100 dollars. 12. If 75 be of some number, what is the whole of that number? Ans. 180. 13. If of a cotton manufactory be worth 8500 dollars, what is the whole of it worth? Ans. 22500 dollars. 14. How many times is 164, or 32, contained in 132? Ans. S times. 15. If g of of a ship is worth 2100 dollars, what is the value of the whole ship? Ans. 4000 dollars. TO DIVIDE A FRACTION BY A FRACTION. Art. 100. Dividing a fraction by a fraction is the method of finding what part of I time, or what number of times, one fraction is contained in another. As the operations in dividing one fraction by another are the reverse of those in multiplying one fraction by another, we may divide the numerator of the dividend by the numerator of the divisor for the numerator of the quotient, and the denominator of the dividend by the denominator of the divisor for the denominator of the quotient; thus, } } = 1. One fraction may be divided by another by dividing the numerator of the dividend by the fractional divisor, and writing the quotient over the denominator of the dividend; 12÷=20 4 thus, Again, one fraction may be divided by another by multiplying the denominator of the dividend by the fractional divisor, and writing the product under the numerator of the 12 4 dividend; thus, 12 25 X 15 5 As the result of each of the above methods of operation will often be a complex fraction, the following methods of operation will be more convenient. Let it be required to divide by . As these fractions have a common denominator, it is plain that is contained in as many times as 3, the numerator of the divisor, is contained in 7, the numerator of the dividend, which is 24 times. Again, let it be required to divide by . 3, and $ = • Then, dividing 8, the numerator of the dividend, by 9, the numerator of the divisor; thus, §; we find that is contained in of 1 time. Hence it is plain, that one fraction can be divided by another, when they have a common denominator, by dividing the numerator of the dividend by the numerator of the divisor. When the two fractions have not a common denominator, and as the common denominator is not used in the operation, it is only necessary to find the new numerators; these new numerators will express the quotient. From the above illustration we derive the following rule for dividing one fraction by another. RULE. Reduce compound and complex fractions to simple fractions, mixed numbers to improper fractions, fractions of different denominations to the same. If the two fractions have the same denominator, divide the numerator of the dividend by the numerator of the divisor. If the two fractions have different denominators, multiply the numerator of the dividend by the denominator of the divithe product will be the numerator of the quotient; then multiply the numerator of the divisor by the denominator of the dividend, the product will be the denominator of the quo sor, tient. 1. How many pounds of raisins, at of a dollar a pound, can you buy with of a dollar? 2. How many pounds of tea, at 2 of a dollar a pound, can you purchase with 51 dollars? 3. How many times is contained in ? 5. How many yards of cloth, at of a dollar a yard, can be purchased with 4 dollars? 7. How many times is 1g contained in 9g? 9. When figs are worth of a dollar a pound, how many pounds can you purchase with of a dollar? 11. How many times is contained in ? 4. How many times is contained in §? 6. How many boxes of oranges, at 3§ dollars a box, can be purchased with 127 dollars? 8. How many times is contained in 9§ ? 10. If a bushel of potatoes cost of a dollar, how many bushels can you buy with of a dollar? 12. How many times is contained in ? 13. At 3 dollars a yard, how many yards of broadcloth can be purchased with 25 dollars? Ans. 7 yards. 14. If of a yard of silk cost of a dollar, what will 1 yard cost? Ans. 1 dollars. 15. A man purchased of an acre of land for 244 dollars; what is the value of an acre at the same rate? Ans. 86 dollars. 16. A baker purchased 15 barrels of flour for 854 dollars; how much did the flour cost him a barrel? 17. If of of a yard of linen cost will be the value of 1 yard? 18. How many tons of coal, at 5g purchased with 125 dollars? of Ans. 5 dollars. of a dollar, what Ans. 1 dollars. dollars a ton, can be Ans. 2313 tons. you purchase with Ans. 228 barrels. dollars, what is the Ans. 2451 dollars. contained in 19. How many barrels of apples can 484 dollars, at 2 dollars a barrel ? 20. If of of a ship is worth 1225 value of the whole of the ship? 21. How many times is mile? of a rod 22. How many bottles, containing 1 filled with 27 gallons of beer? of a Ans. 336 times. pints each, can be Ans. 148 bottles. 23. If a yard of ribbon cost of a shilling, how many yards can you purchase with of a pound? Ans. 14 yards. 24. If the wheels of a railroad car are 11 feet in circumference, how many times will each of them turn round in running 424 miles? Ans. 19509 times. PRACTICAL QUESTIONS IN VULGAR FRAC TIONS. Art. 101. 1. What is the sum of 7 and? What is their difference? 2. What is the sum of of and of? What is their difference? 3. What is the sum of 17 and 12? difference? What is their 4. What is the sum of § of a pound and § of a shilling? What is their difference? 5. What is the amount of of an English ell and of a yard? What is their difference? 6. What is the amount of 15 acres and 3 roods? What is their difference? 7. What is the amount of 123 tons and 15 pounds? What is their difference? 8. What is the amount of of a day and of an hour? What is their difference? 9. What will be the product of of of t? multiplied by 10. What will be the quotient of of divided by & of ? 11. What will be the product of 253 multiplied by 12? 12. What will be the quotient of 257 divided by 12? 13. If a yard of silk is worth of a dollar, what is of a yard worth? 14. If of a yard of silk is worth of a dollar, what is the value of a yard? 15. If a yard of broadcloth is worth 37 dollars, what are 3 yards of the same cloth worth? 16. If 3 yards of broadcloth are worth 1437 dollars, what is the value of one yard of the same kind of cloth? 17. If of a cord of wood is worth 5 dollars, what is the value of a cord? What is the value of 12 cords? 18. If 31 gallons of molasses are worth 93 dollars, what is one gallon worth? What are 5 gallons worth? 19. When coal is worth 7 dollars a ton, what is of a ton worth? What is the value of 12 tons? 20. A merchant purchased of a ship; he has since sold of his share. What part of the whole ship did he sell? What part does he still own? 21. If of of an acre of land is worth 15 dollars, what is the value of an acre? What is the value of a farm containing 325 acres? DECIMAL FRACTIONS. Art. 102. A decimal fraction is any number of tenths, hundredths, thousandths, &c., of a unit or integer; hence, the denominator of a decimal fraction is a unit with so many ciphers annexed as there are figures in the numerator. A decimal fraction is expressed by writing only the numerator, or number of parts, and then placing a point at its left to distinguish it from a whole number. If the figures of the numerator do not equal in number the ciphers of the denominator, they must be made equal by prefixing ciphers. are written thus: .5 tenths. Tooo are written .05 hundredths. It is a principle of whole numbers that they increase in a tenfold proportion from the place of units towards the left; so it is a principle of decimal fractions that they decrease in a tenfold proportion from the place of units towards the right; which is shown in the following table. The figures at the left of the decimal point, in the table, express whole numbers. The 5 in the 1st place at the right of the point expresses 5 tenths; 5 in the 2nd place, 5 hundredths; 5 in the 3rd place, 5 thousandths; 5 in the 4th place, 5 ten thousandths; 5 in the 5th place, 5 hundred |