SUBTRACTION. 96. The process of subtracting one fiaction from another is based upon the following principles : I. One number can be subtracted from another only when the two numbers have the same unit value. Hence, II. In subtraction of fractions, the minuend and subtrahend must have a common denominator, ANALYSIS. Since the fractions have a common denominator, the difference is obtained, by subtracting the less numerator 5, from the greater 9, and writing the result over the common de nominator 12; we thus obtain, the required difference. them; how many had he left? 6. Harvey had 1g of a dollar, and gave of a dollar to a beggar; how much had he left? 7. Subtract from 3. OPERATION. ANALYSIS. As the given fracAns. tions have not a common denominator, we first reduce them to the same fractional unit, (94) and then subtract the less numerator 9, from the greater 14, and write the result over the common denominator 21. We thus obtain the required difference. Hence the following RULE. I. When the fractions have the same denominator, subtract the less numerator from the greater, and place the result over the common denominator. II. When they have not a common denominator, reduce them to a common denominator before subtracting. 23=21/2 6,7 Ans. We now add 1 to the nominator 12. Since we cannot take 2 from, we add 1-12 to, which makes 1, and from 16 leaves in the subtrahend, and say, 3 from 9 leaves 6. We thus obtain 6-7, the difference required. 2 Hence, to subtract mixed numbers, we may reduce the fractional parts to a common denominator, and then subtract the fractional and integral parts separately. 19. From 24 take 17. 20. From 147 take 491. 21. From 75 take 40%. Ans. 74. Ans. 98. Ans. 3411. Ans. 40. 22. From 63 take 228. 23. Bought flour at 65 dollars a barrel, and sold it at 7% dollars a barrel; what was the gain per barrel? Ans. of a dollar. 24. From a cask of wine containing 385 gallons, 155 gal lons were drawn; how many gallons remained? Ans. 221 gallons MULTIPLICATION. CASE I. 97. To multiply a fraction by an integer. 1. If 1 pound of sugar cost of a dollar, how much will 8 pounds cost? . ANALYSIS. If 1 pound cost of a dollar, 3 pounds, which are 3 times 1 pound, will cost 8 times or of a dollar. Therefore, 3 pounds of sugar, at of a dollar a pound, will cost 3 of a dollar. 2. If 1 horse eat of a ton of hay in 1 month, how much will 4 horses eat in the same time? 3. At of a dollar a bushel, what will be the cost of 2 bushels of pears? of 3 bushels? of 5 bushels? . 4. How many are 3 times? 5 times ? 4 times ? 6 times? 9 times? 8 times ? 5. If one yard of cloth cost of a dollar, how much will 3 yards cost? FIRST OPERATION. X3=15=21. SECOND OPERATION. X3=5=21. ANALYSIS. In the first operation we multiply the fraction by 3, by multiplying its numerator by 8, obtaining 15-21. In this case the value of the fractional unit remains the same, but we multiply the number taken, 3 times. In the second operation, we multiply the fraction by 3, by dividing its denominator by 3, obtaining §=21. In this case, the value of the fractional unit is multiplied, 8 times, but the number taken, is the same. Hence, Multiplying a fraction consists in multiplying its nu merator, or dividing its denominator. NOTE Always divide the denominator when it is exactly divisible by lae multipher. 41 OPERATION. 5 Or, 20 21 Ans. 43. Ans. 31. Ans. 5%. Ans. 4. Ans. 11. Ans. 10. ANALYSIS. In multiplying a mixed number, we first multiply the fractional part, then the integer, and then add the two products. Thus, 5X1==13; and 5X4=20, which added to 13, gives 21, the required result. Or, we may reduce the mixed number to an improper fraction, and then multiply it. 13. Multiply 63 by 8. 14. Multiply 94 by 7. Ans. 54. Ans. 683. 15. If a man earn 17 in 1 day, how much will he earn in 10 days? Ans. 18 dollars. 16. What will 14 yards of cloth cost, at of a dollar a yard? Ans. 10 dollars. 17. At 3 dollars a cord, what will be the cost of 20 cords of wood? Ans. 65 dollars. 18. If one man can mow 1 acres of grass in a day, how many acres can 5 men mow? Ans. 9 acres. 19. What will 9 dozen eggs cost, at 14 cents a dozen? Ans. 130 cents. 20. At 6 dollars a barrel, what will 30 barrels of flour oost? Ans. 204 dollars. CASE II. 98. To multiply an integer by a fraction. 1. At 9 dollars a barrel, what will of a barrel of flour cost? ANALYSIS. Since 1 barrel of flour cost 9 dollars, of a barrel will cost 2 times of 9 dollars. of 9 dollars is 3 dollars, and of 9 dollars is 2 times 3 dollars, or 6 dollars. Therefore of a barrel will cost 6 dollars. 2. If a yard of cloth be worth 8 dollars, what is of a yard worth? 3. If an acre of land produce 25 bushels of wheat, how much will of an acre produce? of an acre? of an acre? 4. If a man earn 20 dollars in a month, how much can he earn in of a month? in ? in? in ? 5. If a ton of hay cost 12 dollars, how much will of a ton cost? of a ton? of a ton? § of a ton? 6. At 60 dollars an acre, what will of an acre of land cost? FIRST OPERATION. 5)60 price of 1 acre. 12 cost of of an acre. 48 cost of of an acre. SECOND OPERATION. 60 price of 1 acre. 4 5)240 cost of 4 acres. 48 cost of of an acre, ANALYSIS. 4 fifths of an acre will cost 4 times as much as 1 fifth of an acre, or 4 times of 60 dollars. of 60 dollars is 12 dollars, and is 4 times 12, or 48 dollars, the cost of of an acre. In the second operation, we multiply the price of 1 acre by 4, and obtain 240 dollars, the cost of 4 acres; but as of 1 acre is the same as of 4 acres, we divide 240 dol lars, the cost of 4 acres, by 5, and obtain 48 dollars, the cost of of of acre. Hence, |