ARITHMETICAL ANALYSIS. 189. Analysis is the process of solving problems by a comparison of their elements. In comparing, we reason to the unit and from the unit; the unit being the basis of the reasoning process. CASE I. 190. To pass from one integer to another integer. 1. If 5 oranges cost 12 cents, what will 8 oranges cos at the same rate? SOLUTION.-If 5 oranges cost 12 or 45 cents, 1 orange cost of 25, which is cents; and if 1 orange cost cents, 8 oranges will cost 8 times cents, or 20 cents. OPERATION. 5 cost 25 1 66 66 8 §×8=20, Ans. MENTAL EXERCISES. 1. If 4 apples cost 5 cents, what will 5 apples cost at the same rate? 2. If 3 girls earn $24 in a day, how much could 5 girls earn at the same rate? 3. How long will it take a boy to plow 8 furrows, if it takes him 13 minutes to plow 5 furrows? 4. How much will 9 barrels of flour cost, if 2 barrels cost $103? 5. If 5 hats can be bought for $114, how much will 8 hats cost? 6. What cost 5 pounds of butter at the rate of $3 for 10 pounds? WRITTEN EXERCISES. 2. If 6 hens cost $3.54, what will 7 hens cost at the same rate? Ans..$4.13. Ans. $24. 3. If 8 turkeys cost $19, what will 10 turkey cost at the same rate? 4. How much must I pay for 22 yards of muslin, at the rate of 5 yards for $18? Ans. $4.18. 5. How much will 21 books cost at the rate of 9 books for 15? Ans. $36. 6. What must I pay for 8 arithmetics, at the rate of 5 arithmetics for $4.75? Ans. $7.60. 7. If 3 tons of hay cost $473, how much will 12 tons cost at the same rate? Ans. $189. 8. Required the cost of 21 barrels of flour, at the rate of $17.50 for 3 barrels. Ans. $122.50 9. A gave $43.50 for pigs, at the rate of $7.25 for 3 pigs; how many did he buy? Ans. 18 pigs. 10. B gave $54 for flour, at the rate of $6 for 4 bushels; how many bushels did he buy? CASE II. Ans. 39 bushels. 191. To pass from a fraction to an integer. 1. If of an acre of land cost $108, what will one acre 1. If of a yard of cloth is worth 24 dimes, what are 3 yards of the cloth worth? 2. If of a barrel of sugar cost $12, what will 7 barrels cost? 3. I bought 5 yards of summer silk at the rate of $10 for 34 yards · what was the cost? 4. Mary bought 9 apples, at the rate of of an apple for of a cent; what did they cost? 5. Henry bought 25 peaches, at the rate of of a peach for ‡ of a cent; required the cost. 6. Three-fifths of what I paid for a cow was $24; what did I pay for a horse whose value was 5 times that of the cow? WRITTEN EXERCISES. Ans. $22.05. 2. If of the cost of A's farm is $2480, what did the farm cost? Ans. $3720. 3. If of a barrel of flour cost $2.50, what will 5 barrels cost at the same rate? An's. $21.87. 4. If of a yard of cloth cost $2.10, what will enough for suit containing 9 yards cost? 5. If § of a cwt. of sugar cost $12.45, what will be the cost of 20 cwt, or 1 ton of sugar? 6. What will 21 tons of hay cost, if of a ton of hay cost $12.85 ? Ans. $308.40. 7. If $467 is of the price I pay for a horse, what would be the cost of 4 pairs of horses at the same rate? Ans. $298.80. Ans. $1200. 8. If a lawyer's clerk could write 60 pages in 10 hours, how much would he write in a week, working 10 hours a day? Ans. 337 pages. 9. How much will it cost to transport 25 hundredweight of freight from Philadelphia to Pittsburgh at $4 for 2 hundredweight? Ans. $458. 10. How much will I pay for 15 acres of land, at the rate of 2 acres for $425.25? Ans. $2551.50. 11. A bought 40 bushels of grain, at the rate of 31 bushels for $3.20; what did it cost? Ans. $38.40. 12. B sold 77 gallons of milk, at the rate of $1.25 for 51 gallons; required the cost. Ans. $17.50. 13. C bought 39 cords of wood, at the rate of $12.45 for 31 cords; how much did it cost? Ans. $149.40. 14. D bought 81 tons of coal, at the rate of 5 tons for $26.25; how much did it cost? Ans. $393.75. CASE III. 192. To pass from an integer to a fraction. 1. If 1 barrel of apples cost $8, what will of a barrel cost? SOLUTION.-If 1 barrel of apples cost $8, of a barrel cost of $8, which is $2; and if of a barrel cost $2, of a barrel will cost 3 times $2, which are $6. MENTAL EXERCISES. OPERATION. 1 cost $8 " $2 1. What cost of a pound of beef, if 2 pounds cost 32 cents? 2. If a man can earn $54 in 3 days, how much will he earn in of a day? 3. If 2 harrels of flour cost $103, for what can I sell of a barrel at the rate of cost? 4. If Harry walks 63 miles in 2 hours, how far will he walk at this rate in 3 hours? 5. If 7 pounds of sugar cost 56 cents, for what can I sell 41⁄2 pounds, and neither gain nor lose? 6. A vessel sailed 60 miles in 12 hours; how far at this rate did it sail in 4 hours? 7. If a seamstress earns $1 in 2 days, how much will she earn in 10 days? WRITTEN EXERCISES. 2. Henry paid $24.60 for a cow; how much would he Aave paid had he given as much? Ans. $19.68. 3. If 3 yards of cloth cost $7.50, what will of a yard of the same cloth cost? Ans. $2.18. 4. If 7 barrels of apples cost $193, what will § of a barrel cost at the same rate? Ans. $2. 5. How much will 103 tons of coal cost at the rate of $139 for 8 tons? Ans. $184 6. A bought 98 acres of land, at the rate of $1605 for 5 acres; required the cost. Ans. $31741. 7. How much must I pay for 1815 cords of wood, at the rate of $6 for 3 cords? Ans. $421. 8. If 9 tons of hay are worth $100, what must I give for 1413 tons? Ans. $167. 9. If 12 bushels of beans cost $114, how much will 151 bushels cost? Ans. $148. 10. How much must I pay for 1818 tons of iron ore, at the rate of 17 tons for $334? Ans. $37. 11. A gave $1968 for 21 acres of land; for what would he sell 1 of an acre, at a gain of $4 an acre ? Ans. $924. 12. B gave $444 for 24 tons of hay; he sold C 1 of a ton. at $2 less a ton; what did he receive? Ans. $15.20. NOTE. The solution of several of the above problems is simplified by expressing the operations and cancelling. CASE IV. 193. To pass from a fraction to a fraction. 1. If of an acre of land cost $150, what will of an acre cost? SOLUTION.-If of an acre cost $150, of an acre cost of $150, which is $75; and of an acre will cost 3 times $75, which are $225. If 1 acre cost $225, of an acre cost of $225, which is $45, and of an acre will cost 4 times $45, which are $180. SOLUTION 2D.-If of an acre cost $150, of an acre cost of $150, and of an acre will cost $150; and of an acre will cost XX$150, which by cancellation we find is $180. OPERATION. XX$150 $180, Ans. NOTE.-The 2d method, by cancellation, is preferred, especially with the more difficult problems. MENTAL EXERCISES. 1. If of an apple cost of a cent, what will § of an apple cost? 2. If § of a barrel of flour is worth $69, what is 7 of a barrel worth? 3. What is of an acre of land worth, if of an acre is worth $126? 4. How much must I pay for f of a cord of wood, if § of a cord is worth $1.50? 5. What must I pay for of a bushel of berries, if of a bushel is worth $6? 6. What will } of a yard of cloth cost, if § of a yard cost $74 ? 7. I paid $1400 for 43 acres of land; how much will 5 acres cost at the same rate? 8. A bought 7 yards of cloth for $10%; what would he have paid for 8 yards at the same rate? 9. How many apples can you get for 53 cents, if 21 apples are worth 63 cents? WRITTEN EXERCISES. 2. When $189 are paid for 83 acres of land, how much must be paid for 10% acres? Ans. $238. SOLUTION.-1891 — 246; 83 = 43; 10565; the cost, we find by analysis, is 946 × 5 × 65 = 2381; reducing by cancellation. tons? 3. A gave 65 yards of cloth for 13 cords of wood; how many cords could be got for 9 yards? Ans. 19 cords. 4. B gave 121 tons of iron for 321 barrels of flour; how many barrels would he get for 14 Ans. 381. 5. C gave 5 tons of hay for 28 coal would he get for 7 tons of hay? 6. If 6 horses eat 7 bushels of oats in a day, how many. norses would eat 33g bushels in a day? Ans. 28. 7. If 9 pigs cost $123, how many pigs can you buy for 115 dollars? tons of coal; how much Ans. 41 tons. Ans. 84. Ans. 9. 8. If 2 barrels of flour cost $248, how many barrels can you buy for $87? 9. How much must I pay for 6 pounds of sugar, at the rate of $0.443 for 24 pounds? Ans. $1.28. 10. How much must I pay for 10 bushels of rice at the rate of $15 for 5 bushels? Ans $30. 11. If 19 tons of iron cost $429, how many tons can ou buy for $3574? Ans. 163 tons. |