41. Frank sold 10 peaches, which was of all he had; he then divided the remainder equally among 5 companions: how many did they receive apiece? 42. Lincoln spent 60 cents for a book, which was of his money; the remainder he laid out for oranges, at 4 cents apiece: how many oranges did he buy? 43. A man paid away $35, which was of all he had; he then laid out the rest in cloth at $2 per yard: how many yards did he obtain? 44. A farmer bought a quantity of goods, and paid $20 down, which was of the bill: how many cords of wood, at $3 per cord, will it take to pay the balance? 45. A man bought a horse and paid $60 in cash, which was of the price: how many barrels of flour at $6 per barrel, will it take to pay the balance? 46.3 of 27 is a of what number? Analysis of 27 is 9. And if 9 is of a certain number, of that number is 3; and 4 is 4 times 3, which is 12, the number required. 296. It will be seen from the preceding examples, that no particular rules can be prescribed for solving questions by analysis. None in fact are requisite. The process will be easily suggested by the judgment of the pupil, and the conditions of the question. OBS. The operation of solving a question by analysis, is called an analytic solution. In reciting the following examples, the pupil QUEST.--296. Can any particular rules be prescribed for solving questions by analysis? How then will you know how to proceed? Obs. What is the operation of solving questions by analysis, called? should be required to analyze each question, and give the reason for each step, as in the preceding mental exercises. Ex. I. If 40 barrels of beef cost $320, how much will 52 barrels cost? Analytic Solution.-Since 40 bbls. cost $320, 1 bbl. will cost of $320. And of $320 is $320÷40=$8. Now if 1 bbl. cost $8, 52 bbls. will cost 52 times as much; and $8×52=$416, which is the answer required. Or thus: 52 bbls. are of 40 bbls. ; therefore 52 bbls. will cost of $320; (the cost of 40 bbls. ;) and 42 of $320 is $320×43 $416, the same result as before. (Arts. 132, 133.) OES. 1. Other solutions of this example might be given; but our present object is to show how this and similar examples may be solved by analysis. The former method is the simplest and most strictly analytic, though not so short as the latter. It contains two steps: First, we separate the given price of 40 bbls. ($320,) into 40 equal parts, to find the value of one part, or the cost of 1 bbl., which is $8. Second, we multiply the price of 1 bbl, ($8,) by 52, the number of barrels, whose cost is required, and the product is the answer sought. 2. In solving questions analytically, it may be remarked in general, that we reason from the given number to 1, then from 1 to the required number. 3. This and similar questions are usually placed under Simple Proportion, or the "Rule of Three;" but business men almost invariably solve them by analysis. 2. If 30 cows cost $360.90, how much will 47 cows cost, at the same rate? 3. If 25 barrels of apples cost $15, how much will 37 barrels cost? 4. If 15 hogsheads of molasses cost $450, how much will 21 hogsheads cost? 5. If 31 yards of cloth cost $127, how much will 89 yards cost? 6. If 55 tons of hay cost $660, what will 17 tons come to? 7. An agent paid $159 for 530 pounds of wool: how much was that per 100? 8. A man bought 30 cords of wood for $76.80: how inuch must he pay for 65 cords? 9. A gentleman bought 85 yards of carpeting for $106.25: how much would 38 yards cost? 10. A drover bought 350 sheep for $525: how much would 65 cost, at the same rate? 11. If 12 pounds of coffee cost $1.25, how much will 45 pounds cost? 12. If 16 bushels of corn are worth $8, how much are 25 bushels worth? 13. Paid $20 for 60 pounds of tea: how much would 124 pounds cost, at the same rate? 14. Bought 41 yards of flannel for $16.40: how much would 83 yards cost? 15. Bought 18 pounds of ginger for $4.50: how much will 103 pounds cost? 16. If a stage goes 84 miles in 12 hours, how far will it go in 15 hours? 17. If 8 horses eat 36 bushels of oats in a week, how many bushels will 25 horses eat in the same time? 18. If a Railroad car runs 120 miles in 5 hours, how far will it run in 123 hours? 19. If a steamboat goes 180 miles in 12 hours, how far will it go in 5 hours? 20. If 4 men can do a job of work in 12 days, how long will it take 6 men to do it? Solution Since the job requires 4 men 12 days, it will require 1 man 4 times as long; and 4 times 12 days are 48 days. Again, it requires I man 48 days, it will require 6 men as long; and 48 days÷6-8 days, which is the answer required. 21. If 6 men eat a barrel of flour in 24 days, how long will it last 10 men? 22. If a given quantity of corn lasts 9 horses 96 days, how long will the same quantity last 15 horses? 23. If 12 men can build a house in 90 days, how long will it take 20 men to build it? -24. If 100 barrels of pork last a crew of 20 men 45 months, how long will it last a crew of 28 men? 25. If 4 stacks of hay will keep 60 cattle 120 days, how long will they keep 25 cattle? 26. If of a bushel of wheat cost 30 cents, what will of a bushel cost? 27. If of a ton of hay cost $7, what will of a ton cost? 28. If of a pound of imperial tea cost 27 cents, how much will of a pound cost? 29. If of a ton of coal cost $2.61, how much will of a ton cost? 30. If of a yard of silk cost 6 shillings, how much will of a yard cost? Solution Since of a yard cost 6s., will cost 3s., and or 1 yard will cost 9s. Again, if 1 yard costs 9s., yd. will cost 14s. ; and yd. will cost 77 shillings, which is the answer required. 31. If of a cord of wood cost $1.80, how much will of a cord cost? 32. If of a yard of broadcloth cost 14 shillings, how much will of a yard cost? 33. A man bought of an acre of land for $56, and afterwards sold of an acre at cost: how much did he receive for it? 34. A grocer bought 7 barrels of vinegar for $28, and sold of a barrel at cost: how much did it come to? 35. A grocer bought a firkin of butter containing 56 lbs. for $11.20, and sold of it at cost: how much did he get a pound? 36. If 6 bushels of peas are worth $5.50, how much are 20 bushels worth? 37. If a man pays $47 for building 234 rods of ornamental fence, how much would it cost him to build 424 rods? 38. A farmer paid $45.42 for making 36% rods of stone wall: how much will it cost him to make 60 rods? 39. A man paid of a dollar for 4 pounds of veal: how much would a quarter of veal cost, which weighs 20 pounds? 40. If 5 pounds of butter cost 4 shillings, how much will 42 pounds cost? Note.-It will be seen that 44s. s. (Art. 122.) Therefore 1 pound will cost s.; and 42 lbs. will cost X42-252, or 36s. Ans. 41. If 20 pounds of cheese cost $3%, how much will 168 pounds cost? 42. If 30 yards of cotton cost $42, how much will a piece containing 19 yards cost? 43. If of a cord of wood cost of a dollar, how much will of a cord cost? Solution.-Since consequently cord cost $12, of a cord cost $5, will cost $1 Again, if 1 of a cord will cost $3; and 2-will there fore cost or $14, which is the answer required. 44. If of a yard of cloth cost £, how much will of a yard cost? 45. If of a ship cost $16000, how much is of her worth? 46. A man bought a quantity of land and sold of an acre for $63, which was only of the cost': how much did he give per acre? 47. If 7 yards of satinet cost $93, how much will 18 yards cost? 48. A ship's company of 30 men have 4500 pounds of flour; how long will it last them, allowing each man 24 lbs. per day? 49. How long will 56700 pounds of meat last a garrison of 756, allowing each man 4 lb. per day? 50. How long will the same quantity of meat last the same garrison, allowing 14 lb. apiece per day? 51. A merchant sold 12 yards of silk at 7 shillings per yard, and took his pay in wheat at 6 shillings per bushel: how many bushels did he receive? Solution. First find the cost of the silk. If 1 yd. costs 7s., 12 yds. will cost 7 times 12s., which is 84s. Now the question is, how many bushels of wheat it will take to pay this 84s. But as the wheat is 6s. a bushel, it will manifestly take as many bushels as 6s. is contained times in 84s.; and 84÷6=14. Ans. 14 bushels. |