(a.) From the above solutions, we infer that, to multiply by a decimal fraction, we multiply as though it were a whole number, and remove the decimal point in the product as many places further to the left than it is in the multiplicand as there are places of decimal fractions in the multiplier. The product, then, will contain as many places of decimal fractions as there are in both the multiplier and multiplicand. (b.) Find each of the following products : 2. .3 x .3? 12. 8 x .3 x .07 ? 3. 4.5 x .007 ? 13. 653 X .4 x 9.5 ? 4. .25 x .25 ? 14. .25 x .25 x .25 ? 5. .48 x 6.3 ? 15. 12.5 X .5 x 1.6? 6. .006 x .03? 16. 3.07 x 2.05 x 40.3 ? 7. .005 X .005 ? 17. 14 x 1.4 x .14? 8. .878 x 1.06? 18. .06 X, .7 x .005 X 600 ? 9. 3243 x 1.04? 19. 3.4 x .007 X 8.03 x 50 ? 10. 8 x .3 x 7 ? 20. 500 x .005 x .05 X 5000? 11. .08 x .03 x .07 ? 21. 375.0623 X 6.423 X 50.0329 ? 93. Practical Problems. 1. How much will is of a ton of iron cost at $54.25 per ton? Solution. — If 1 ton of iron costs $54.25, is of a ton will cost of $54.25, which, found as before explained, is $22.60,4 = $22.604 +. NOTE. - It will often, if not usually, be found best to write the work for these problems in form for cancelling. How much will 2. of a barrel of flour cost at $11.28 per bbl.? per cord ? 7. .73 of a hogshead of wine at $137.50 per hhd.? 8. .297 of a stack of hay weigh if the whole weighs 4,36 tons? 9. 13į rods of fence cost at j of a dollar per rod ? 10. A lot 17; rods long and 16 jj ft. wide cost at $2 per rod ? 11. A block 8.3 ft. long, 2.9 ft. wide, and 12 ft. high, cost at $1.53 per ft. ? 12. 4 of a sq. rd. cost at of a dollar per sq. yd. ? 13. 53 rd. of wall cost at $1ți per yd. ? 14. George has $148.75, Joseph has as much money as George, and Edward has j as much as Joseph. How many dollars does each have ? 15. A man bought a house for $3483, and sold it so as to gain .75 of the cost? What was his gain ? 16. A man who owned an acre of land sold .3 of it at the rate of $124.74 per acre, and the remainder at the rate of $137.96 per acre ? How much did he receive for each lot and for both ? 17. I bought . 7 of a ton of coal at $8.43 per ton, 1.16 cords of wood at $7.39 per cord, and 5.5 baskets of charcoal at $.35 per basket. What was the amount of my purchase ? 18. If it takes 14 bushels of seed to sow an acre of land, how many bushels will it take to sow 3} acres ? 19. How many square feet are there in the four walls of a room 16 ft. long, 15. feet wide, and 8; ft. high? 20. If 12 lb. of chocolate cost 31, how much will 75 lb. cost? 15 SUGGESTION. 1 lb. will cost $ 15 X 23 and 7 lb. will cost $. 4 X 12' 4 X 12 X 3 21. If 18 acres of land are worth $1745, how much are 81 acres worth? 22. 3 men bought a cargo of coal for $875.625. The first took ã of it, the second took twice as much as the first, and the third took the remainder. How many dollars ought each to pay? 23. For how much must I sell a lot of goods costing $3244 in order to gain .15 of the cost ? 24. For how much must I sell a lot of goods costing $838.54 in order to lose .27 of their cost? 25. I bought 27 yards of broadcloth at $5.83 per yard, 53 yards of cassimere at $1.65 per yard, and 64 yards of doeskin at $1.90 per yard, giving in payment a hundred-dollar bill. How much change ought I to receive ? 26. I sold 85 tons of hay at $19.37 per ton, receiving in payment 8fcwt. of sugar at $8.69 per cwt., and the rest in money. How much did I receive in money? 27. In a cornfield there are 87 rows of corn, each row containing 138 bills. Now, allowing that each hill produces 5 ears of corn, and that each ear contains 1% gills, how many bushels of corn will the field contain ? 28. In a certain house there are 25 windows, each window containing 12 panes of glass, each pane 16, inches long and 114 inches wide. How many square inches of glass are there in all ? 29. I bought 4 loads of hay, the first weighing 13 tons, the second 13 tons, the third 2 tons, and the fourth lý tons. How many tons did I buy, and what did it all cost, at $18; per ton ? 94. To find a Number from its Fractional Part. 1. 36 is į of what number? 1st Solution. — If 36 is § of some number, s of that number must be 8 times 36, or 288. Hence, 36 is of 288. 2ND SOLUTION. - 36 is f of 8 times 36, or of 288. (a.) Of what number 2. Does 87 = $? 4. Does 5.06 = .001? =.1? 6. 42 equals if of what number? Solution.—If 42 = tf of some number, it of that number must equal i of 42, which is tt, and if of the number must equal 13 times this result, which, found by making 13 a factor of the numerator and reducing, is 454. Hence, 42 equals tig of 454. 7 12 454. 7. 1 = of what number? 3 9 SOLUTION. — If 14 = of some number, 36 15 X 96 of that number must equal as of 18, which is 16 X 25 196 expressed by making 25 a factor of the denomi4 5 nator; and job of the number must equal 36 times the last result, which is expressed by making 36 a factor of the numerator, as in the written work in the margin. Hence, 1 = tf of 170. (b.) Of what number 8. Does 84 分? ? 15. Does f=1? 9. Does 593 16. Does 84 = ? 10. Does 487 17. Does 36 11. Does 863 = .9? 18. Does off 12. Does 47.85 = .05 ? 19. Does 13. Does .56 =.56? 20. Does of 64 = 18? 14. Does 5.84 :.012? 21. Does ij of 22. If te of an acre of land cost $149.88, what will an acre cost? if? ? = ? 1st Solution. — If }} of an 24.98 149.88 X 25 624.50 acre cost $149.88, z's of an acre - $208.16 18 will cost is of $149.88, which 3 is $140,68, and 15 of an acre will cost 25 times the last result, which may be expressed by making 25 a factor of the numerator, as in the written work in the margin. How much will 23. A bale of cotton weigh if of a bale weighs 327 lb. ? 24. A cargo of coal cost if f of it costs $1528.35 ? 25. A load of wood cost if 1 of it costs $9.73? 26. A hogshead of molasses cost if 4 of a hhd. costs $17.25 ? 27. A barrel of flour cost if 1% barrel costs $12.50 ? 28. A load of hay cost if 2f loads costs $35}? 29. A barrel of pork cost if 4f bbl. costs $72.96 ? 30. 45 yards of cloth cost if 44 yd. costs $58 ? 31. of a yard cost if of a yard costs $177? 32. How much will a man earn in 736 days if he earns $15 in 41 days ? 33. If 3} bushels of corn are worth as much as 5% bushels of oats, how many bushels of oats are 84 bushels of corn worth? 34. If 6 lb. of coffee can be bought for $£, how many pounds can be bought for $34 ? 35. I owned f of a vessel and sold of my share for $4728. What was the whole vessel worth at that rate ? 36. A prize was to be divided between the captain and the first, second, and third mates of a vessel, in such a manner that the first mate was to have as much as the captain, the second mate as much as the first mate, and the third mate as much as the second mate. Now, allowing that the share of the third mate was $1000, what was the share of each of the others ? 37. If į of a yard of cloth is worth as much as 34 pecks of salt at of a dollar per peck, how many dollars are 151 yards of cloth worth? 95. To divide by a Fraction. (a.) From the nature of division, it is obvious that — 1st. With the same dividend, the larger the divisor the smaller will be the quotient, and the smaller the divisor the larger will be the quotient. 2d. The quotient of any number divided by 1 equals the number itself. Hence, dividing by a number greater than 1 must give for a quotient a number less than the dividend, and dividing by a number less than 1 must give for a quotient a number greater than the dividend. ILLUSTRATIONS. — The quotient of 24 + 12 is 2, but the quotient of 24 + of 12, or 6, is 2 times 2, or 4, and the quotient of 24 = f of 12, or 4, is 3 times 2, or 6. In like manner, since the quotient of 8 • 1 is 8, the quotient of 8 + tis 2 times 8, or 16, and the quotient of 8 + is 3 times 8, or 24. |